SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Dijkstra's algorithm</title>

</head>

Dijkstra's algorithm

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Dykstra's_projection_algorithm" title="Dykstra's projection algorithm">Dykstra's projection algorithm</a>.</div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}

/* end https://en.wikipedia.org/ */

</style><table class="infobox"><caption class="infobox-title">Dijkstra's algorithm</caption><tbody><tr><td colspan="2" class="infobox-image"><div class="infobox-caption">Dijkstra's algorithm to find the shortest path between a and b. It picks the unvisited vertex with the lowest distance, calculates the distance through it to each unvisited neighbor, and updates the neighbor's distance if smaller. Mark visited (set to red) when done with neighbors.</div></td></tr><tr><th scope="row" class="infobox-label">Class</th><td class="infobox-data"><a href="Search_algorithm" title="Search algorithm">Search algorithm</a> <a href="Greedy_algorithm" title="Greedy algorithm">Greedy algorithm</a> <a href="Dynamic_programming" title="Dynamic programming">Dynamic programming</a><a href="#cite_note-1">[1]</a></td></tr><tr><th scope="row" class="infobox-label">Data structure</th><td class="infobox-data"><a href="Graph_(data_structure)" class="mw-redirect" title="Graph (data structure)">Graph</a> Usually used with <a href="Priority_queue" title="Priority queue">priority queue</a> or <a href="Heap_(data_structure)" title="Heap (data structure)">heap</a> for optimization<a href="#cite_note-FOOTNOTECormenLeisersonRivestStein2001-2">[2]</a><a href="#cite_note-FOOTNOTEFredmanTarjan1987-3">[3]</a></td></tr><tr><th scope="row" class="infobox-label"><a href="Best%2C_worst_and_average_case" title="Best, worst and average case">Worst-case</a> <a href="Time_complexity" title="Time complexity">performance</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|E|+|V|\log |V|)}">

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|E|+|V|\log |V|)}</annotation>

</semantics>

Dijkstra's algorithm (/ˈdaɪkstrəz/ DYKE-strəz) is an <a href="Algorithm" title="Algorithm">algorithm</a> for finding the <a href="Shortest_path_problem" title="Shortest path problem">shortest paths</a> between <a href="Vertex_(graph_theory)" title="Vertex (graph theory)">nodes</a> in a weighted <a href="Graph_(abstract_data_type)" title="Graph (abstract data type)">graph</a>, which may represent, for example, a <a href="Road_network" class="mw-redirect" title="Road network">road network</a>. It was conceived by computer scientist <a href="Edsger_W._Dijkstra" title="Edsger W. Dijkstra">Edsger W. Dijkstra</a> in 1956 and published three years later.<a href="#cite_note-4">[4]</a><a href="#cite_note-Dijkstra_Interview2-5">[5]</a><a href="#cite_note-Dijkstra19592-6">[6]</a>

Dijkstra's algorithm finds the shortest path from a given source node to every other node.<a href="#cite_note-mehlhorn-7">[7]</a>: 196–206 It can be used to find the shortest path to a specific destination node, by terminating the algorithm after determining the shortest path to the destination node. For example, if the nodes of the graph represent cities, and the costs of edges represent the distances between pairs of cities connected by a direct road, then Dijkstra's algorithm can be used to find the shortest route between one city and all other cities. A common application of shortest path algorithms is network <a href="Routing_protocol" title="Routing protocol">routing protocols</a>, most notably <a href="IS-IS" title="IS-IS">IS-IS</a> (Intermediate System to Intermediate System) and <a href="Open_Shortest_Path_First" title="Open Shortest Path First">OSPF</a> (Open Shortest Path First). It is also employed as a <a href="Subroutine" class="mw-redirect" title="Subroutine">subroutine</a> in algorithms such as <a href="Johnson's_algorithm" title="Johnson's algorithm">Johnson's algorithm</a>.

The algorithm uses a <a href="Priority_queue#Min-priority_queue" title="Priority queue">min-priority queue</a> <a href="Data_structure" title="Data structure">data structure</a> for selecting the shortest paths known so far. Before more advanced priority queue structures were discovered, Dijkstra's original algorithm ran in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|V|^{2})}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|V|^{2})}</annotation>

</semantics>

</math><img src="./5dbb38dcf2365c1dbf786871f4014dd1a2762df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.753ex; height:3.343ex;" alt="{\displaystyle \Theta (|V|^{2})}" loading="lazy"> <a href="Time_complexity" title="Time complexity">time</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |V|}</annotation>

</semantics>

</math><img src="./9ddcffc28643ac01a14dd0fb32c3157859e365a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.081ex; height:2.843ex;" alt="{\displaystyle |V|}" loading="lazy"> is the number of nodes.<a href="#cite_note-8">[8]</a><a href="#cite_note-FOOTNOTELeyzorekGrayJohnsonLadew1957-9">[9]</a> <a href="#CITEREFFredmanTarjan1984">Fredman & Tarjan 1984</a> proposed a <a href="Fibonacci_heap" title="Fibonacci heap">Fibonacci heap</a> priority queue to optimize the running time complexity to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|E|+|V|\log |V|)}">

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|E|+|V|\log |V|)}</annotation>

</semantics>

</math><img src="./e22162be85d06b346f3b7f7aad9746da0c1019c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.435ex; height:2.843ex;" alt="{\displaystyle \Theta (|E|+|V|\log |V|)}" loading="lazy">. This is <a href="Asymptotic_computational_complexity" title="Asymptotic computational complexity">asymptotically</a> the fastest known single-source <a href="Shortest_path_problem" title="Shortest path problem">shortest-path algorithm</a> for arbitrary <a href="Directed_graph" title="Directed graph">directed graphs</a> with unbounded non-negative weights. However, specialized cases (such as bounded/integer weights, directed acyclic graphs etc.) can be <a class="mw-selflink-fragment" href="#Specialized_variants">improved further</a>. If preprocessing is allowed, algorithms such as <a href="Contraction_hierarchy" class="mw-redirect" title="Contraction hierarchy">contraction hierarchies</a> can be up to seven <a href="Orders_of_magnitude" class="mw-redirect" title="Orders of magnitude">orders of magnitude</a> faster.

Dijkstra's algorithm is commonly used on graphs where the edge weights are positive integers or real numbers. It can be generalized to any graph where the edge weights are <a href="Partially_ordered_set" title="Partially ordered set">partially ordered</a>, provided the subsequent labels (a subsequent label is produced when traversing an edge) are <a href="Monotonic_function" title="Monotonic function">monotonically</a> non-decreasing.<a href="#cite_note-Generic_Dijkstra2-10">[10]</a><a href="#cite_note-Generic_Dijkstra_correctness2-11">[11]</a>

In many fields, particularly <a href="Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>, Dijkstra's algorithm or a variant offers a <a class="mw-selflink-fragment" href="#Practical_optimizations_and_infinite_graphs">uniform cost search</a> and is formulated as an instance of the more general idea of <a href="Best-first_search" title="Best-first search">best-first search</a>.<a href="#cite_note-felner-12">[12]</a>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}

/* end https://en.wikipedia.org/ */

</style><blockquote class="templatequote">What is the shortest way to travel from <a href="Rotterdam" title="Rotterdam">Rotterdam</a> to <a href="Groningen" title="Groningen">Groningen</a>, in general: from given city to given city. <a href="Shortest_path_problem" title="Shortest path problem">It is the algorithm for the shortest path</a>, which I designed in about twenty minutes. One morning I was shopping in <a href="Amsterdam" title="Amsterdam">Amsterdam</a> with my young fiancée, and tired, we sat down on the café terrace to drink a cup of coffee and I was just thinking about whether I could do this, and I then designed the algorithm for the shortest path. As I said, it was a twenty-minute invention. In fact, it was published in '59, three years later. The publication is still readable, it is, in fact, quite nice. One of the reasons that it is so nice was that I designed it without pencil and paper. I learned later that one of the advantages of designing without pencil and paper is that you are almost forced to avoid all avoidable complexities. Eventually, that algorithm became to my great amazement, one of the cornerstones of my fame.</blockquote><div class="templatequotecite">— Edsger Dijkstra, in an interview with Philip L. Frana, Communications of the ACM, 2001<a href="#cite_note-Dijkstra_Interview2-5">[5]</a></div>

Dijkstra thought about the shortest path problem while working as a programmer at the <a href="Centrum_Wiskunde_%26_Informatica" title="Centrum Wiskunde & Informatica">Mathematical Center in Amsterdam</a> in 1956. He wanted to demonstrate the capabilities of the new ARMAC computer.<a href="#cite_note-13">[13]</a> His objective was to choose a problem and a computer solution that non-computing people could understand. He designed the shortest path algorithm and later implemented it for ARMAC for a slightly simplified transportation map of 64 cities in the Netherlands (he limited it to 64, so that 6 bits would be sufficient to encode the city number).<a href="#cite_note-Dijkstra_Interview2-5">[5]</a> A year later, he came across another problem advanced by hardware engineers working on the institute's next computer: minimize the amount of wire needed to connect the pins on the machine's back panel. As a solution, he re-discovered <a href="Prim's_algorithm" title="Prim's algorithm">Prim's minimal spanning tree algorithm</a> (known earlier to <a href="Vojt%C4%9Bch_Jarn%C3%ADk" title="Vojtěch Jarník">Jarník</a>, and also rediscovered by <a href="Robert_C._Prim" title="Robert C. Prim">Prim</a>).<a href="#cite_note-EWD841a2-14">[14]</a><a href="#cite_note-15">[15]</a> Dijkstra published the algorithm in 1959, two years after Prim and 29 years after Jarník.<a href="#cite_note-16">[16]</a><a href="#cite_note-17">[17]</a>

<div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2></div>

The algorithm requires a starting node, and computes the shortest distance from that starting node to each other node. Dijkstra's algorithm starts with infinite distances and tries to improve them step by step:

<ol><li>Create a <a href="Set_(abstract_data_type)" title="Set (abstract data type)">set</a> of all unvisited nodes: the unvisited set.</li>

<li>Assign to every node a distance from start value: for the starting node, it is zero, and for all other nodes, it is infinity, since initially no path is known to these nodes. During execution, the distance of a node N is the length of the shortest path discovered so far between the starting node and N.<a href="#cite_note-18">[18]</a></li>

<li>From the unvisited set, select the current node to be the one with the smallest (finite) distance; initially, this is the starting node (distance zero). If the unvisited set is empty, or contains only nodes with infinite distance (which are unreachable), then the algorithm terminates by skipping to step 6. If the only concern is the path to a target node, the algorithm terminates once the current node is the target node. Otherwise, the algorithm continues.</li>

<li>For the current node, consider all of its unvisited neighbors and update their distances through the current node; compare the newly calculated distance to the one currently assigned to the neighbor and assign the smaller one to it. For example, if the current node A is marked with a distance of 6, and the edge connecting it with its neighbor B has length 2, then the distance to B through A is 6 + 2 = 8. If B was previously marked with a distance greater than 8, then update it to 8 (the path to B through A is shorter). Otherwise, keep its current distance (the path to B through A is not the shortest).</li>

<li>After considering all of the current node's unvisited neighbors, the current node is removed from the unvisited set. Thus a visited node is never rechecked, which is correct because the distance recorded on the current node is minimal (as ensured in step 3), and thus final. Repeat from step 3.</li>

<li>Once the loop exits (steps 3–5), every visited node contains its shortest distance from the starting node.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Description">Description</h2></div>

The shortest path between two <a href="Intersection_(road)" title="Intersection (road)">intersections</a> on a city map can be found by this algorithm using pencil and paper. Every intersection is listed on a separate line: one is the starting point and is labeled (given a distance of) 0. Every other intersection is initially labeled with a distance of infinity. This is done to note that no path to these intersections has yet been established. At each iteration one intersection becomes the current intersection. For the first iteration, this is the starting point.

From the current intersection, the distance to every <a href="Neighbourhood_(graph_theory)" title="Neighbourhood (graph theory)">neighbor</a> (directly-connected) intersection is assessed by summing the label (value) of the current intersection and the distance to the neighbor and then <a href="Graph_labeling" title="Graph labeling">relabeling</a> the neighbor with the lesser of that sum and the neighbor's existing label. I.e., the neighbor is relabeled if the path to it through the current intersection is shorter than previously assessed paths. If so, mark the road to the neighbor with an arrow pointing to it, and erase any other arrow that points to it. After the distances to each of the current intersection's neighbors have been assessed, the current intersection is marked as visited. The unvisited intersection with the smallest label becomes the current intersection and the process repeats until all nodes with labels less than the destination's label have been visited.

Once no unvisited nodes remain with a label smaller than the destination's label, the remaining arrows show the shortest path.

<div class="mw-heading mw-heading2"><h2 id="Pseudocode">Pseudocode</h2></div>

In the following <a href="Pseudocode" title="Pseudocode">pseudocode</a>, <style data-mw-deduplicate="TemplateStyles:r886049734">

/* start https://en.wikipedia.org/ */

.mw-parser-output .monospaced{font-family:monospace,monospace}

/* end https://en.wikipedia.org/ */

</style>dist is an array that contains the current distances from the <var>source</var> to other vertices, i.e. dist[<var>u</var>] is the current distance from the source to the vertex <var>u</var>. The prev array contains pointers to previous-hop nodes on the shortest path from source to the given vertex (equivalently, it is the next-hop on the path from the given vertex to the source). The code u ← vertex in Q with min dist[u], searches for the vertex <var>u</var> in the vertex set <var>Q</var> that has the least dist[<var>u</var>] value. Graph.Edges(<var>u</var>, <var>v</var>) returns the length of the edge joining (i.e. the distance between) the two neighbor-nodes <var>u</var> and <var>v</var>. The variable <var>alt</var> on line 14 is the length of the path from the <var>source</var> node to the neighbor node <var>v</var> if it were to go through <var>u</var>. If this path is shorter than the current shortest path recorded for <var>v</var>, then the distance of <var>v</var> is updated to <var>alt</var>.<a href="#cite_note-mehlhorn-7">[7]</a>

<pre> 1 function Dijkstra(Graph, source):

3 for each vertex v in Graph.Vertices:

4 dist[v] ← INFINITY

5 prev[v] ← UNDEFINED

6 add v to Q

7 dist[source] ← 0

9 while Q is not empty:

10 u ← vertex in Q with minimum dist[v]

11 Q.remove(u)

13 for each arc (u, v) in Q:

14 alt ← dist[u] + Graph.Edges(u, v)

15 if alt < dist[v]:

16 dist[v] ← alt

17 prev[v] ← u

19 return dist[], prev[]

</pre>

To find the shortest path between vertices <var>source</var> and <var>target</var>, the search terminates after line 10 if <var>u</var> = <var>target</var>. The shortest path from <var>source</var> to <var>target</var> can be obtained by reverse iteration:

<pre>1 S ← empty sequence

2 u ← target

3 if prev[u] is defined or u = source: // Proceed if the vertex is reachable

4 while u is defined: // Construct the shortest path with a stack S

5 S.push(u) // Push the vertex onto the stack

6 u ← prev[u] // Traverse from target to source

</pre>

Now sequence <var>S</var> is the list of vertices constituting one of the shortest paths from <var>source</var> to <var>target</var>, or the empty sequence if no path exists.

A more general problem is to find all the shortest paths between <var>source</var> and <var>target</var> (there might be several of the same length). Then instead of storing only a single node in each entry of prev[] all nodes satisfying the relaxation condition can be stored. For example, if both <var>r</var> and <var>source</var> connect to <var>target</var> and they lie on different shortest paths through <var>target</var> (because the edge cost is the same in both cases), then both <var>r</var> and <var>source</var> are added to prev[<var>target</var>]. When the algorithm completes, prev[] data structure describes a graph that is a subset of the original graph with some edges removed. Its key property is that if the algorithm was run with some starting node, then every path from that node to any other node in the new graph is the shortest path between those nodes graph, and all paths of that length from the original graph are present in the new graph. Then to actually find all these shortest paths between two given nodes, a path finding algorithm on the new graph, such as <a href="Depth-first_search" title="Depth-first search">depth-first search</a> would work.

<div class="mw-heading mw-heading3"><h3 id="Using_a_priority_queue">Using a priority queue</h3></div>

A min-priority queue is an abstract data type that provides 3 basic operations: add_with_priority(), decrease_priority() and extract_min(). As mentioned earlier, using such a data structure can lead to faster computing times than using a basic queue. Notably, <a href="Fibonacci_heap" title="Fibonacci heap">Fibonacci heap</a><a href="#cite_note-FOOTNOTEFredmanTarjan1984-19">[19]</a> or <a href="Brodal_queue" title="Brodal queue">Brodal queue</a> offer optimal implementations for those 3 operations. As the algorithm is slightly different in appearance, it is mentioned here, in pseudocode as well:

<pre>1 function Dijkstra(Graph, source):

2 Q ← Queue storing vertex priority

4 dist[source] ← 0 // Initialization

5 Q.add_with_priority(source, 0) // associated priority equals dist[·]

7 for each vertex v in Graph.Vertices:

8 if v ≠ source

9 prev[v] ← UNDEFINED // Predecessor of v

10 dist[v] ← INFINITY // Unknown distance from source to v

11 Q.add_with_priority(v, INFINITY)

14 while Q is not empty: // The main loop

15 u ← Q.extract_min() // Remove and return best vertex

16 for each arc (u, v) : // Go through all v neighbors of u

17 alt ← dist[u] + Graph.Edges(u, v)

18 if alt < dist[v]:

19 prev[v] ← u

20 dist[v] ← alt

21 Q.decrease_priority(v, alt)

23 return (dist, prev)

</pre>

Instead of filling the priority queue with all nodes in the initialization phase, it is possible to initialize it to contain only source; then, inside the <code>if alt < dist[v]</code> block, the decrease_priority() becomes an add_with_priority() operation.<a href="#cite_note-mehlhorn-7">[7]</a>: 198

Yet another alternative is to add nodes unconditionally to the priority queue and to instead check after extraction (<code>u ← Q.extract_min()</code>) that it isn't revisiting, or that no shorter connection was found yet in the <code>if alt < dist[v]</code> block. This can be done by additionally extracting the associated priority <code>p</code> from the queue and only processing further <code>if p == dist[u]</code> inside the <code>while Q is not empty</code> loop.<a href="#cite_note-Note2-20">[20]</a>

These alternatives can use entirely array-based priority queues without decrease-key functionality, which have been found to achieve even faster computing times in practice. However, the difference in performance was found to be narrower for denser graphs.<a href="#cite_note-chen_072-21">[21]</a>

<div class="mw-heading mw-heading2"><h2 id="Proof">Proof</h2></div>

To prove the <a href="Correctness_(computer_science)" title="Correctness (computer science)">correctness</a> of Dijkstra's algorithm, <a href="Mathematical_induction" title="Mathematical induction">mathematical induction</a> can be used on the number of visited nodes.<a href="#cite_note-22">[22]</a>

Invariant hypothesis: For each visited node v, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[v]</code> is the shortest distance from source to v, and for each unvisited node u, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[u]</code> is the shortest distance from source to u when traveling via visited nodes only, or infinity if no such path exists. (Note: we do not assume <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[u]</code> is the actual shortest distance for unvisited nodes, while <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[v]</code> is the actual shortest distance)

The base case is when there is just one visited node, source. Its distance is defined to be zero, which is the shortest distance, since negative weights are not allowed. Hence, the hypothesis holds.

<div class="mw-heading mw-heading3"><h3 id="Induction">Induction</h3></div>

Assuming that the hypothesis holds for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> visited nodes, to show it holds for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k+1}</annotation>

</semantics>

</math><img src="./552a558062ed4c0486297b5b5531c5ee044dbd9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k+1}" loading="lazy"> nodes, let u be the next visited node, i.e. the node with minimum <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[u]</code>. The claim is that <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[u]</code> is the shortest distance from source to u.

The proof is by contradiction. If a shorter path were available, then this shorter path either contains another unvisited node or not.

<ul><li>In the former case, let w be the first unvisited node on this shorter path. By induction, the shortest paths from source to u and w through visited nodes only have costs <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[u]</code> and <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[w]</code> respectively. This means the cost of going from source to u via w has the cost of at least <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[w]</code> + the minimal cost of going from w to u. As the edge costs are positive, the minimal cost of going from w to u is a positive number. However, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[u]</code> is at most <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[w]</code> because otherwise w would have been picked by the priority queue instead of u. This is a contradiction, since it has already been established that <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[w]</code> + a positive number < <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[u]</code>.</li>

<li>In the latter case, let w be the last but one node on the shortest path. That means <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[w] + Graph.Edges[w,u] < dist[u]</code>. That is a contradiction because by the time w is visited, it should have set <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[u]</code> to at most <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[w] + Graph.Edges[w,u]</code>.</li></ul>

For all other visited nodes v, the <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[v]</code> is already known to be the shortest distance from source already, because of the inductive hypothesis, and these values are unchanged.

After processing u, it is still true that for each unvisited node w, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[w]</code> is the shortest distance from source to w using visited nodes only. Any shorter path that did not use u, would already have been found, and if a shorter path used u it would have been updated when processing u.

After all nodes are visited, the shortest path from source to any node v consists only of visited nodes. Therefore, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">dist[v]</code> is the shortest distance.

<div class="mw-heading mw-heading2"><h2 id="Running_time">Running time</h2></div>

Bounds of the running time of Dijkstra's algorithm on a graph with edges E and vertices V can be expressed as a function of the number of edges, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |E|}</annotation>

</semantics>

</math><img src="./d8c2b9637808cf805d411190b4ae017dbd4ef8d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.069ex; height:2.843ex;" alt="{\displaystyle |E|}" loading="lazy">, and the number of vertices, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |V|}</annotation>

</semantics>

</math><img src="./9ddcffc28643ac01a14dd0fb32c3157859e365a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.081ex; height:2.843ex;" alt="{\displaystyle |V|}" loading="lazy">, using <a href="Big-O_notation" class="mw-redirect" title="Big-O notation">big-O notation</a>. The complexity bound depends mainly on the data structure used to represent the set Q. In the following, upper bounds can be simplified because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |E|}</annotation>

</semantics>

</math><img src="./d8c2b9637808cf805d411190b4ae017dbd4ef8d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.069ex; height:2.843ex;" alt="{\displaystyle |E|}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(|V|^{2})}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(|V|^{2})}</annotation>

</semantics>

</math><img src="./e1e99764e23be92b694aef042c6460ff921357e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.718ex; height:3.343ex;" alt="{\displaystyle O(|V|^{2})}" loading="lazy"> for any simple graph, but that simplification disregards the fact that in some problems, other upper bounds on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |E|}</annotation>

</semantics>

For any data structure for the vertex set Q, the running time is:<a href="#cite_note-FOOTNOTECormenLeisersonRivestStein2001-2">[2]</a>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|E|\cdot T_{\mathrm {dk} }+|V|\cdot T_{\mathrm {em} }),}</annotation>

</semantics>

</math><img src="./b49340c1fc766f10e8b48815b1088cc65898efce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.674ex; height:2.843ex;" alt="{\displaystyle \Theta (|E|\cdot T_{\mathrm {dk} }+|V|\cdot T_{\mathrm {em} }),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {dk} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {dk} }}</annotation>

</semantics>

</math><img src="./9d472cd5eeba8435ca78f0662c29f549a6e68758.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.372ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {dk} }}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {em} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {em} }}</annotation>

</semantics>

</math><img src="./10b20e61c32746a67f380a74d01eb9d672371600.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.689ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {em} }}" loading="lazy"> are the complexities of the decrease-key and extract-minimum operations in Q, respectively.

The simplest version of Dijkstra's algorithm stores the vertex set Q as a linked list or array, and edges as an <a href="Adjacency_list" title="Adjacency list">adjacency list</a> or <a href="Adjacency_matrix" title="Adjacency matrix">matrix</a>. In this case, extract-minimum is simply a linear search through all vertices in Q, so the running time is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|E|+|V|^{2})=\Theta (|V|^{2})}">

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|E|+|V|^{2})=\Theta (|V|^{2})}</annotation>

</semantics>

</math><img src="./9cff9c1ad91cc6d7b764f808f068d2233520a233.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.513ex; height:3.343ex;" alt="{\displaystyle \Theta (|E|+|V|^{2})=\Theta (|V|^{2})}" loading="lazy">.

For <a href="Sparse_graph" class="mw-redirect" title="Sparse graph">sparse graphs</a>, that is, graphs with far fewer than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V|^{2}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |V|^{2}}</annotation>

</semantics>

</math><img src="./3e2a7aad37527c097101c39b0b7724fe4e1cafb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.135ex; height:3.343ex;" alt="{\displaystyle |V|^{2}}" loading="lazy"> edges, Dijkstra's algorithm can be implemented more efficiently by storing the graph in the form of adjacency lists and using a <a href="Self-balancing_binary_search_tree" title="Self-balancing binary search tree">self-balancing binary search tree</a>, <a href="Binary_heap" title="Binary heap">binary heap</a>, <a href="Pairing_heap" title="Pairing heap">pairing heap</a>, <a href="Fibonacci_heap" title="Fibonacci heap">Fibonacci heap</a> or a priority heap as a <a href="Priority_queue" title="Priority queue">priority queue</a> to implement extracting minimum efficiently. To perform decrease-key steps in a binary heap efficiently, it is necessary to use an auxiliary data structure that maps each vertex to its position in the heap, and to update this structure as the priority queue Q changes. With a self-balancing binary search tree or binary heap, the algorithm requires

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta ((|E|+|V|)\log |V|)}</annotation>

</semantics>

</math><img src="./bb876ca772aebc634b2c4087fa81b2d7c683e71c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.244ex; height:2.843ex;" alt="{\displaystyle \Theta ((|E|+|V|)\log |V|)}" loading="lazy"></dd></dl>

time in the worst case; for connected graphs this time bound can be simplified to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|E|\log |V|)}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|E|\log |V|)}</annotation>

</semantics>

</math><img src="./df01952aa6a18b33a028713dedfc2f89d84a3c81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.514ex; height:2.843ex;" alt="{\displaystyle \Theta (|E|\log |V|)}" loading="lazy">. The <a href="Fibonacci_heap" title="Fibonacci heap">Fibonacci heap</a> improves this to

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|E|+|V|\log |V|).}</annotation>

</semantics>

</math><img src="./a5f0d0d6da9c8dbd146dae568b1ce2dbf0d65d9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.082ex; height:2.843ex;" alt="{\displaystyle \Theta (|E|+|V|\log |V|).}" loading="lazy"></dd></dl>

When using binary heaps, the <a href="Best%2C_worst_and_average_case" title="Best, worst and average case">average case</a> time complexity is lower than the worst-case: assuming edge costs are drawn independently from a common <a href="Probability_distribution" title="Probability distribution">probability distribution</a>, the expected number of decrease-key operations is bounded by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|V|\log(|E|/|V|))}">

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|V|\log(|E|/|V|))}</annotation>

</semantics>

</math><img src="./23857a2943abcd1478c6435680423251fbc1625f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.179ex; height:2.843ex;" alt="{\displaystyle \Theta (|V|\log(|E|/|V|))}" loading="lazy">, giving a total running time of<a href="#cite_note-mehlhorn-7">[7]</a>: 199–200

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O\left(|E|+|V|\log {\frac {|E|}{|V|}}\log |V|\right).}</annotation>

</semantics>

</math><img src="./518426fe2ac5cff791eee3d85aa084963096e404.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.096ex; height:6.509ex;" alt="{\displaystyle O\left(|E|+|V|\log {\frac {|E|}{|V|}}\log |V|\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Practical_optimizations_and_infinite_graphs">Practical optimizations and infinite graphs</h3></div>

In common presentations of Dijkstra's algorithm, initially all nodes are entered into the priority queue. This is, however, not necessary: the algorithm can start with a priority queue that contains only one item, and insert new items as they are discovered (instead of doing a decrease-key, check whether the key is in the queue; if it is, decrease its key, otherwise insert it).<a href="#cite_note-mehlhorn-7">[7]</a>: 198 This variant has the same worst-case bounds as the common variant, but maintains a smaller priority queue in practice, speeding up queue operations.<a href="#cite_note-felner-12">[12]</a>

Moreover, not inserting all nodes in a graph makes it possible to extend the algorithm to find the shortest path from a single source to the closest of a set of target nodes on infinite graphs or those too large to represent in memory. The resulting algorithm is called uniform-cost search (UCS) in the artificial intelligence literature<a href="#cite_note-felner-12">[12]</a><a href="#cite_note-aima-23">[23]</a><a href="#cite_note-24">[24]</a> and can be expressed in pseudocode as

<pre>procedure uniform_cost_search(start) is

node ← start

frontier ← priority queue containing node only

expanded ← empty set

do

if frontier is empty then

return failure

node ← frontier.pop()

if node is a goal state then

return solution(node)

expanded.add(node)

for each of node's neighbors n do

if n is not in expanded and not in frontier then

frontier.add(n)

else if n is in frontier with higher cost

replace existing node with n

</pre>

Its complexity can be expressed in an alternative way for very large graphs: when C* is the length of the shortest path from the start node to any node satisfying the "goal" predicate, each edge has cost at least ε, and the number of neighbors per node is bounded by b, then the algorithm's worst-case time and space complexity are both in O(b1+⌊C* <style data-mw-deduplicate="TemplateStyles:r1154941027">

/* start https://en.wikipedia.org/ */

.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}

/* end https://en.wikipedia.org/ */

</style>⁄ ε⌋).<a href="#cite_note-aima-23">[23]</a>

Further optimizations for the single-target case include <a href="Bidirectional_search" title="Bidirectional search">bidirectional</a> variants, goal-directed variants such as the <a href="A*_algorithm" class="mw-redirect" title="A* algorithm">A* algorithm</a> (see <a href="#Related_problems_and_algorithms">§ Related problems and algorithms</a>), graph pruning to determine which nodes are likely to form the middle segment of shortest paths (reach-based routing), and hierarchical decompositions of the input graph that reduce s–t routing to connecting s and t to their respective "<a href="Transit_Node_Routing" class="mw-redirect" title="Transit Node Routing">transit nodes</a>" followed by shortest-path computation between these transit nodes using a "highway".<a href="#cite_note-speedup2-25">[25]</a> Combinations of such techniques may be needed for optimal practical performance on specific problems.<a href="#cite_note-26">[26]</a>

<div class="mw-heading mw-heading3"><h3 id="Optimality_for_comparison-sorting_by_distance">Optimality for comparison-sorting by distance</h3></div>

As well as simply computing distances and paths, Dijkstra's algorithm can be used to sort vertices by their distances from a given starting vertex.

In 2023, Haeupler, Rozhoň, Tětek, Hladík, and <a href="Robert_Tarjan" title="Robert Tarjan">Tarjan</a> (one of the inventors of the 1984 heap), proved that, for this sorting problem on a positively-weighted directed graph, a version of Dijkstra's algorithm with a special heap data structure has a runtime and number of comparisons that is within a constant factor of optimal among <a href="Comparison_sort" title="Comparison sort">comparison-based</a> algorithms for the same sorting problem on the same graph and starting vertex but with variable edge weights. To achieve this, they use a comparison-based heap whose cost of returning/removing the minimum element from the heap is logarithmic in the number of elements inserted after it rather than in the number of elements in the heap.<a href="#cite_note-27">[27]</a><a href="#cite_note-28">[28]</a>

<div class="mw-heading mw-heading3"><h3 id="Specialized_variants">Specialized variants</h3></div>

When arc weights are small integers (bounded by a parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">), specialized queues can be used for increased speed. The first algorithm of this type was Dial's algorithm<a href="#cite_note-FOOTNOTEDial1969-29">[29]</a> for graphs with positive integer edge weights, which uses a <a href="Bucket_queue" title="Bucket queue">bucket queue</a> to obtain a running time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(|E|+|V|C)}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(|E|+|V|C)}</annotation>

</semantics>

</math><img src="./8c106890bef1d3c3a6918c4cceb79f2b14aec738.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.34ex; height:2.843ex;" alt="{\displaystyle O(|E|+|V|C)}" loading="lazy">. The use of a <a href="Van_Emde_Boas_tree" title="Van Emde Boas tree">Van Emde Boas tree</a> as the priority queue brings the complexity to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(|E|+|V|\log C/\log \log |V|C)}">

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(|E|+|V|\log C/\log \log |V|C)}</annotation>

</semantics>

</math><img src="./7e90be6df92daa0aff31129584071b6fc43e22c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.2ex; height:2.843ex;" alt="{\displaystyle O(|E|+|V|\log C/\log \log |V|C)}" loading="lazy"> .<a href="#cite_note-FOOTNOTEAhujaMehlhornOrlinTarjan1990-30">[30]</a> Another interesting variant based on a combination of a new <a href="Radix_heap" title="Radix heap">radix heap</a> and the well-known Fibonacci heap runs in time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(|E|+|V|{\sqrt {\log C}})}">

</mrow>

</mrow>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(|E|+|V|{\sqrt {\log C}})}</annotation>

</semantics>

</math><img src="./ee2a9dfd3f6ca5471b9ecf0f2255bc49f9da6a94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.022ex; height:3.509ex;" alt="{\displaystyle O(|E|+|V|{\sqrt {\log C}})}" loading="lazy"> .<a href="#cite_note-FOOTNOTEAhujaMehlhornOrlinTarjan1990-30">[30]</a> Finally, the best algorithms in this special case run in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(|E|\log \log |V|)}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(|E|\log \log |V|)}</annotation>

</semantics>

</math><img src="./6ad6d486e8b9d44ea253d43ce5d219652073eb8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.838ex; height:2.843ex;" alt="{\displaystyle O(|E|\log \log |V|)}" loading="lazy"><a href="#cite_note-FOOTNOTEThorup2000-31">[31]</a> time and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(|E|+|V|\min\{(\log |V|)^{1/3+\varepsilon },(\log C)^{1/4+\varepsilon }\})}">

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(|E|+|V|\min\{(\log |V|)^{1/3+\varepsilon },(\log C)^{1/4+\varepsilon }\})}</annotation>

</semantics>

</math><img src="./7ee874b438d755a44a6327b14b6f8ab107ad7426.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.864ex; height:3.343ex;" alt="{\displaystyle O(|E|+|V|\min\{(\log |V|)^{1/3+\varepsilon },(\log C)^{1/4+\varepsilon }\})}" loading="lazy"> time.<a href="#cite_note-FOOTNOTERaman1997-32">[32]</a>

<div class="mw-heading mw-heading2"><h2 id="Related_problems_and_algorithms">Related problems and algorithms</h2></div>

Dijkstra's original algorithm can be extended with modifications. For example, sometimes it is desirable to present solutions which are less than mathematically optimal. To obtain a ranked list of less-than-optimal solutions, the optimal solution is first calculated. A single edge appearing in the optimal solution is removed from the graph, and the optimum solution to this new graph is calculated. Each edge of the original solution is suppressed in turn and a new shortest-path calculated. The secondary solutions are then ranked and presented after the first optimal solution.

Dijkstra's algorithm is usually the working principle behind <a href="Link-state_routing_protocol" title="Link-state routing protocol">link-state routing protocols</a>. <a href="OSPF" class="mw-redirect" title="OSPF">OSPF</a> and <a href="IS-IS" title="IS-IS">IS-IS</a> are the most common.

Unlike Dijkstra's algorithm, the <a href="Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford algorithm</a> can be used on graphs with negative edge weights, as long as the graph contains no <a href="Negative_cycle" class="mw-redirect" title="Negative cycle">negative cycle</a> reachable from the source vertex s. The presence of such cycles means that no shortest path can be found, since the label becomes lower each time the cycle is traversed. (This statement assumes that a "path" is allowed to repeat vertices. In <a href="Graph_theory" title="Graph theory">graph theory</a> that is normally not allowed. In <a href="Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a> it often is allowed.) It is possible to adapt Dijkstra's algorithm to handle negative weights by combining it with the Bellman-Ford algorithm (to remove negative edges and detect negative cycles): <a href="Johnson's_algorithm" title="Johnson's algorithm">Johnson's algorithm</a>.

The <a href="A-star_algorithm" class="mw-redirect" title="A-star algorithm">A* algorithm</a> is a generalization of Dijkstra's algorithm that reduces the size of the subgraph that must be explored, if additional information is available that provides a lower bound on the distance to the target.

The process that underlies Dijkstra's algorithm is similar to the <a href="Greedy_algorithm" title="Greedy algorithm">greedy</a> process used in <a href="Prim's_algorithm" title="Prim's algorithm">Prim's algorithm</a>. Prim's purpose is to find a <a href="Minimum_spanning_tree" title="Minimum spanning tree">minimum spanning tree</a> that connects all nodes in the graph; Dijkstra is concerned with only two nodes. Prim's does not evaluate the total weight of the path from the starting node, only the individual edges.

<a href="Breadth-first_search" title="Breadth-first search">Breadth-first search</a> can be viewed as a special-case of Dijkstra's algorithm on unweighted graphs, where the priority queue degenerates into a <a href="FIFO_(computing_and_electronics)" title="FIFO (computing and electronics)">FIFO</a> queue.

The <a href="Fast_marching_method" title="Fast marching method">fast marching method</a> can be viewed as a continuous version of Dijkstra's algorithm which computes the geodesic distance on a triangle mesh.

<div class="mw-heading mw-heading3"><h3 id="Dynamic_programming_perspective">Dynamic programming perspective</h3></div>

From a <a href="Dynamic_programming" title="Dynamic programming">dynamic programming</a> point of view, Dijkstra's algorithm is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method.<a href="#cite_note-sniedovich_062-33">[33]</a><a href="#cite_note-denardo_032-34">[34]</a><a href="#cite_note-sniedovich_102-35">[35]</a>

In fact, Dijkstra's explanation of the logic behind the algorithm:<a href="#cite_note-FOOTNOTEDijkstra1959270-36">[36]</a>

<blockquote class="templatequote">Problem 2. Find the path of minimum total length between two given nodes P and Q.

We use the fact that, if R is a node on the minimal path from P to Q, knowledge of the latter implies the knowledge of the minimal path from P to R.</blockquote>

is a paraphrasing of <a href="Richard_Bellman" class="mw-redirect" title="Richard Bellman">Bellman's</a> <a href="Bellman_equation#Bellman's_principle_of_optimality" title="Bellman equation">Principle of Optimality</a> in the context of the shortest path problem.

<ul><li><a href="A*_search_algorithm" title="A* search algorithm">A* search algorithm</a></li>

<li><a href="Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford algorithm</a></li>

<li><a href="Euclidean_shortest_path" title="Euclidean shortest path">Euclidean shortest path</a></li>

<li><a href="Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall algorithm</a></li>

<li><a href="Johnson's_algorithm" title="Johnson's algorithm">Johnson's algorithm</a></li>

<li><a href="Longest_path_problem" title="Longest path problem">Longest path problem</a></li>

<li><a href="Parallel_all-pairs_shortest_path_algorithm" title="Parallel all-pairs shortest path algorithm">Parallel all-pairs shortest path algorithm</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">^</a> Controversial, see <style data-mw-deduplicate="TemplateStyles:r1238218222">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFMoshe_Sniedovich2006" class="citation journal cs1">Moshe Sniedovich (2006). <a rel="nofollow" class="external text" href="https://www.infona.pl/resource/bwmeta1.element.baztech-article-BAT5-0013-0005/tab/summary">"Dijkstra's algorithm revisited: the dynamic programming connexion"</a>. Control and Cybernetics. 35: 599–620.</cite> and <a href="#Dynamic_programming_perspective">below part</a>.

</li>

<li id="cite_note-FOOTNOTECormenLeisersonRivestStein2001-2">^ <a href="#cite_ref-FOOTNOTECormenLeisersonRivestStein2001_2-0">a</a> <a href="#cite_ref-FOOTNOTECormenLeisersonRivestStein2001_2-1">b</a> <a href="#CITEREFCormenLeisersonRivestStein2001">Cormen et al. 2001</a>.

</li>

<li id="cite_note-FOOTNOTEFredmanTarjan1987-3">^ <a href="#cite_ref-FOOTNOTEFredmanTarjan1987_3-0">a</a> <a href="#cite_ref-FOOTNOTEFredmanTarjan1987_3-1">b</a> <a href="#CITEREFFredmanTarjan1987">Fredman & Tarjan 1987</a>.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFRichards" class="citation web cs1">Richards, Hamilton. <a rel="nofollow" class="external text" href="http://amturing.acm.org/award_winners/dijkstra_1053701.cfm">"Edsger Wybe Dijkstra"</a>. A.M. Turing Award. Association for Computing Machinery. Retrieved 16 October 2017. <q>At the Mathematical Centre a major project was building the ARMAC computer. For its official inauguration in 1956, Dijkstra devised a program to solve a problem interesting to a nontechnical audience: Given a network of roads connecting cities, what is the shortest route between two designated cities?</q></cite>

</li>

<li id="cite_note-Dijkstra_Interview2-5">^ <a href="#cite_ref-Dijkstra_Interview2_5-0">a</a> <a href="#cite_ref-Dijkstra_Interview2_5-1">b</a> <a href="#cite_ref-Dijkstra_Interview2_5-2">c</a> <cite id="CITEREFFrana2010" class="citation journal cs1">Frana, Phil (August 2010). "An Interview with Edsger W. Dijkstra". Communications of the ACM. 53 (8): 41–47. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1787234.1787249">10.1145/1787234.1787249</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:27009702">27009702</a>.</cite>

</li>

<li id="cite_note-Dijkstra19592-6"><a href="#cite_ref-Dijkstra19592_6-0">^</a> <cite id="CITEREFDijkstra1959" class="citation journal cs1"><a href="Edsger_W._Dijkstra" title="Edsger W. Dijkstra">Dijkstra, E. W.</a> (1959). <a rel="nofollow" class="external text" href="https://ir.cwi.nl/pub/9256/9256D.pdf">"A note on two problems in connexion with graphs"</a> (PDF). Numerische Mathematik. 1: 269–271. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.7577">10.1.1.165.7577</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01386390">10.1007/BF01386390</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123284777">123284777</a>.</cite>

</li>

<li id="cite_note-mehlhorn-7">^ <a href="#cite_ref-mehlhorn_7-0">a</a> <a href="#cite_ref-mehlhorn_7-1">b</a> <a href="#cite_ref-mehlhorn_7-2">c</a> <a href="#cite_ref-mehlhorn_7-3">d</a> <a href="#cite_ref-mehlhorn_7-4">e</a> <cite id="CITEREFMehlhornSanders2008" class="citation book cs1"><a href="Kurt_Mehlhorn" title="Kurt Mehlhorn">Mehlhorn, Kurt</a>; <a href="Peter_Sanders_(computer_scientist)" title="Peter Sanders (computer scientist)">Sanders, Peter</a> (2008). <a rel="nofollow" class="external text" href="http://people.mpi-inf.mpg.de/~mehlhorn/ftp/Toolbox/ShortestPaths.pdf">"Chapter 10. Shortest Paths"</a> (PDF). Algorithms and Data Structures: The Basic Toolbox. Springer. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-77978-0">10.1007/978-3-540-77978-0</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-77977-3</bdi>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFSchrijver2012" class="citation book cs1">Schrijver, Alexander (2012). <a rel="nofollow" class="external text" href="http://ftp.gwdg.de/pub/misc/EMIS/journals/DMJDMV/vol-ismp/32_schrijver-alexander-sp.pdf">"On the history of the shortest path problem"</a> (PDF). Optimization Stories. Documenta Mathematica Series. Vol. 6. pp. 155–167. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4171%2Fdms%2F6%2F19">10.4171/dms/6/19</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-936609-58-5</bdi>.</cite>

</li>

<li id="cite_note-FOOTNOTELeyzorekGrayJohnsonLadew1957-9"><a href="#cite_ref-FOOTNOTELeyzorekGrayJohnsonLadew1957_9-0">^</a> <a href="#CITEREFLeyzorekGrayJohnsonLadew1957">Leyzorek et al. 1957</a>.

</li>

<li id="cite_note-Generic_Dijkstra2-10"><a href="#cite_ref-Generic_Dijkstra2_10-0">^</a> <cite id="CITEREFSzcześniakJajszczykWoźna-Szcześniak2019" class="citation journal cs1">Szcześniak, Ireneusz; Jajszczyk, Andrzej; Woźna-Szcześniak, Bożena (2019). "Generic Dijkstra for optical networks". Journal of Optical Communications and Networking. 11 (11): 568–577. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1810.04481">1810.04481</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FJOCN.11.000568">10.1364/JOCN.11.000568</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:52958911">52958911</a>.</cite>

</li>

<li id="cite_note-Generic_Dijkstra_correctness2-11"><a href="#cite_ref-Generic_Dijkstra_correctness2_11-0">^</a> <cite id="CITEREFSzcześniakWoźna-Szcześniak2023" class="citation cs2">Szcześniak, Ireneusz; Woźna-Szcześniak, Bożena (2023), "Generic Dijkstra: Correctness and tractability", NOMS 2023-2023 IEEE/IFIP Network Operations and Management Symposium, pp. 1–7, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2204.13547">2204.13547</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FNOMS56928.2023.10154322">10.1109/NOMS56928.2023.10154322</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-6654-7716-1</bdi>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:248427020">248427020</a></cite>

</li>

<li id="cite_note-felner-12">^ <a href="#cite_ref-felner_12-0">a</a> <a href="#cite_ref-felner_12-1">b</a> <a href="#cite_ref-felner_12-2">c</a> <cite id="CITEREFFelner2011" class="citation conference cs1">Felner, Ariel (2011). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200218150924/https://www.aaai.org/ocs/index.php/SOCS/SOCS11/paper/view/4017/4357">Position Paper: Dijkstra's Algorithm versus Uniform Cost Search or a Case Against Dijkstra's Algorithm</a>. Proc. 4th Int'l Symp. on Combinatorial Search. Archived from <a rel="nofollow" class="external text" href="http://www.aaai.org/ocs/index.php/SOCS/SOCS11/paper/view/4017/4357">the original</a> on 18 February 2020. Retrieved 12 February 2015.</cite> In a route-finding problem, Felner finds that the queue can be a factor 500–600 smaller, taking some 40% of the running time.

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131113021126/http://www-set.win.tue.nl/UnsungHeroes/machines/armac.html">"ARMAC"</a>. Unsung Heroes in Dutch Computing History. 2007. Archived from <a rel="nofollow" class="external text" href="http://www-set.win.tue.nl/UnsungHeroes/machines/armac.html">the original</a> on 13 November 2013.</cite>

</li>

<li id="cite_note-EWD841a2-14"><a href="#cite_ref-EWD841a2_14-0">^</a> <cite id="CITEREFDijkstra" class="citation cs2">Dijkstra, Edsger W., <a rel="nofollow" class="external text" href="https://www.cs.utexas.edu/users/EWD/ewd08xx/EWD841a.PDF">Reflections on "A note on two problems in connexion with graphs</a> (PDF)</cite>

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFTarjan1983" class="citation cs2"><a href="Robert_Endre_Tarjan" class="mw-redirect" title="Robert Endre Tarjan">Tarjan, Robert Endre</a> (1983), Data Structures and Network Algorithms, CBMS_NSF Regional Conference Series in Applied Mathematics, vol. 44, Society for Industrial and Applied Mathematics, p. 75, <q>The third classical minimum spanning tree algorithm was discovered by Jarník and rediscovered by Prim and Dikstra; it is commonly known as Prim's algorithm.</q></cite>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFPrim1957" class="citation journal cs1">Prim, R.C. (1957). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170718230207/http://bioinfo.ict.ac.cn/~dbu/AlgorithmCourses/Lectures/Prim1957.pdf">"Shortest connection networks and some generalizations"</a> (PDF). Bell System Technical Journal. 36 (6): 1389–1401. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1957BSTJ...36.1389P">1957BSTJ...36.1389P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fj.1538-7305.1957.tb01515.x">10.1002/j.1538-7305.1957.tb01515.x</a>. Archived from <a rel="nofollow" class="external text" href="http://bioinfo.ict.ac.cn/~dbu/AlgorithmCourses/Lectures/Prim1957.pdf">the original</a> (PDF) on 18 July 2017. Retrieved 18 July 2017.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> V. Jarník: O jistém problému minimálním [About a certain minimal problem], Práce Moravské Přírodovědecké Společnosti, 6, 1930, pp. 57–63. (in Czech)

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFGassFu2013" class="citation encyclopaedia cs1">Gass, Saul; Fu, Michael (2013). "Dijkstra's Algorithm". In Gass, Saul I; Fu, Michael C (eds.). Encyclopedia of Operations Research and Management Science. Vol. 1. Springer. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4419-1153-7">10.1007/978-1-4419-1153-7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4419-1137-7</bdi> – via Springer Link.</cite>

</li>

<li id="cite_note-FOOTNOTEFredmanTarjan1984-19"><a href="#cite_ref-FOOTNOTEFredmanTarjan1984_19-0">^</a> <a href="#CITEREFFredmanTarjan1984">Fredman & Tarjan 1984</a>.

</li>

<li id="cite_note-Note2-20"><a href="#cite_ref-Note2_20-0">^</a> Observe that p < dist[u] cannot ever hold because of the update dist[v] ← alt when updating the queue. See <a rel="nofollow" class="external free" href="https://cs.stackexchange.com/questions/118388/dijkstra-without-decrease-key">https://cs.stackexchange.com/questions/118388/dijkstra-without-decrease-key</a> for discussion.

</li>

<li id="cite_note-chen_072-21"><a href="#cite_ref-chen_072_21-0">^</a> <cite id="chen" class="citation book cs1">Chen, M.; Chowdhury, R. A.; Ramachandran, V.; Roche, D. L.; Tong, L. (2007). <a rel="nofollow" class="external text" href="http://www.cs.sunysb.edu/~rezaul/papers/TR-07-54.pdf">Priority Queues and Dijkstra's Algorithm – UTCS Technical Report TR-07-54 – 12 October 2007</a> (PDF). Austin, Texas: The University of Texas at Austin, Department of Computer Sciences.</cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFCormenLeisersonRivestStein2022" class="citation book cs1"><a href="Thomas_H._Cormen" title="Thomas H. Cormen">Cormen, Thomas H.</a>; <a href="Charles_E._Leiserson" title="Charles E. Leiserson">Leiserson, Charles E.</a>; <a href="Ron_Rivest" title="Ron Rivest">Rivest, Ronald L.</a>; <a href="Clifford_Stein" title="Clifford Stein">Stein, Clifford</a> (2022) [1990]. "22". <a href="Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a> (4th ed.). MIT Press and McGraw-Hill. pp. 622–623. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-04630-X</bdi>.</cite>

</li>

<li id="cite_note-aima-23">^ <a href="#cite_ref-aima_23-0">a</a> <a href="#cite_ref-aima_23-1">b</a> <cite id="CITEREFRussellNorvig2009" class="citation book cs1"><a href="Stuart_J._Russell" title="Stuart J. Russell">Russell, Stuart</a>; <a href="Peter_Norvig" title="Peter Norvig">Norvig, Peter</a> (2009) [1995]. <a href="Artificial_Intelligence%3A_A_Modern_Approach" title="Artificial Intelligence: A Modern Approach">Artificial Intelligence: A Modern Approach</a> (3rd ed.). Prentice Hall. pp. 75, 81. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-604259-4</bdi>.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> Sometimes also least-cost-first search: <cite id="CITEREFNau1983" class="citation journal cs1">Nau, Dana S. (1983). <a rel="nofollow" class="external text" href="https://www.cs.umd.edu/~nau/papers/nau1983expert.pdf">"Expert computer systems"</a> (PDF). Computer. 16 (2). IEEE: 63–85. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Fmc.1983.1654302">10.1109/mc.1983.1654302</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7301753">7301753</a>.</cite>

</li>

<li id="cite_note-speedup2-25"><a href="#cite_ref-speedup2_25-0">^</a> <cite id="CITEREFWagnerWillhalm2007" class="citation conference cs1">Wagner, Dorothea; Willhalm, Thomas (2007). Speed-up techniques for shortest-path computations. STACS. pp. 23–36.</cite>

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFBauerDellingSandersSchieferdecker2010" class="citation journal cs1">Bauer, Reinhard; Delling, Daniel; Sanders, Peter; Schieferdecker, Dennis; Schultes, Dominik; Wagner, Dorothea (2010). <a rel="nofollow" class="external text" href="https://publikationen.bibliothek.kit.edu/1000014952">"Combining hierarchical and goal-directed speed-up techniques for Dijkstra's algorithm"</a>. ACM Journal of Experimental Algorithmics. 15: 2.1. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1671970.1671976">10.1145/1671970.1671976</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1661292">1661292</a>.</cite>

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite id="CITEREFHaeuplerHladíkRozhoňTarjan2024" class="citation arxiv cs1">Haeupler, Bernhard; Hladík, Richard; Rozhoň, Václav; Tarjan, Robert; Tětek, Jakub (28 October 2024). "Universal Optimality of Dijkstra via Beyond-Worst-Case Heaps". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2311.11793">2311.11793</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.DS">cs.DS</a>].</cite>

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite id="CITEREFBrubaker2024" class="citation web cs1">Brubaker, Ben (25 October 2024). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/computer-scientists-establish-the-best-way-to-traverse-a-graph-20241025/">"Computer Scientists Establish the Best Way to Traverse a Graph"</a>. Quanta Magazine. Retrieved 9 December 2024.</cite>

</li>

<li id="cite_note-FOOTNOTEDial1969-29"><a href="#cite_ref-FOOTNOTEDial1969_29-0">^</a> <a href="#CITEREFDial1969">Dial 1969</a>.

</li>

<li id="cite_note-FOOTNOTEAhujaMehlhornOrlinTarjan1990-30">^ <a href="#cite_ref-FOOTNOTEAhujaMehlhornOrlinTarjan1990_30-0">a</a> <a href="#cite_ref-FOOTNOTEAhujaMehlhornOrlinTarjan1990_30-1">b</a> <a href="#CITEREFAhujaMehlhornOrlinTarjan1990">Ahuja et al. 1990</a>.

</li>

<li id="cite_note-FOOTNOTEThorup2000-31"><a href="#cite_ref-FOOTNOTEThorup2000_31-0">^</a> <a href="#CITEREFThorup2000">Thorup 2000</a>.

</li>

<li id="cite_note-FOOTNOTERaman1997-32"><a href="#cite_ref-FOOTNOTERaman1997_32-0">^</a> <a href="#CITEREFRaman1997">Raman 1997</a>.

</li>

<li id="cite_note-sniedovich_062-33"><a href="#cite_ref-sniedovich_062_33-0">^</a> <cite id="CITEREFSniedovich2006" class="citation journal cs1">Sniedovich, M. (2006). <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/cc/cc35/cc3536.pdf">"Dijkstra's algorithm revisited: the dynamic programming connexion"</a> (PDF). Journal of Control and Cybernetics. 35 (3): 599–620.</cite> <a rel="nofollow" class="external text" href="http://www.ifors.ms.unimelb.edu.au/tutorial/dijkstra_new/index.html">Online version of the paper with interactive computational modules.</a>

</li>

<li id="cite_note-denardo_032-34"><a href="#cite_ref-denardo_032_34-0">^</a> <cite id="CITEREFDenardo2003" class="citation book cs1">Denardo, E.V. (2003). Dynamic Programming: Models and Applications. Mineola, NY: <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-42810-9</bdi>.</cite>

</li>

<li id="cite_note-sniedovich_102-35"><a href="#cite_ref-sniedovich_102_35-0">^</a> <cite id="CITEREFSniedovich2010" class="citation book cs1">Sniedovich, M. (2010). Dynamic Programming: Foundations and Principles. Francis & Taylor. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8247-4099-3</bdi>.</cite>

</li>

<li id="cite_note-FOOTNOTEDijkstra1959270-36"><a href="#cite_ref-FOOTNOTEDijkstra1959270_36-0">^</a> <a href="#CITEREFDijkstra1959">Dijkstra 1959</a>, p. 270.

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFCormenLeisersonRivestStein2001" class="citation book cs1"><a href="Thomas_H._Cormen" title="Thomas H. Cormen">Cormen, Thomas H.</a>; <a href="Charles_E._Leiserson" title="Charles E. Leiserson">Leiserson, Charles E.</a>; <a href="Ronald_L._Rivest" class="mw-redirect" title="Ronald L. Rivest">Rivest, Ronald L.</a>; <a href="Clifford_Stein" title="Clifford Stein">Stein, Clifford</a> (2001). "Section 24.3: Dijkstra's algorithm". <a href="Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a> (Second ed.). <a href="MIT_Press" title="MIT Press">MIT Press</a> and <a href="McGraw%E2%80%93Hill" class="mw-redirect" title="McGraw–Hill">McGraw–Hill</a>. pp. 595–601. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-03293-7</bdi>.</cite></li>

<li><cite id="CITEREFDial1969" class="citation journal cs1">Dial, Robert B. (1969). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F363269.363610">"Algorithm 360: Shortest-path forest with topological ordering [H]"</a>. <a href="Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a>. 12 (11): 632–633. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F363269.363610">10.1145/363269.363610</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6754003">6754003</a>.</cite></li>

<li><cite id="CITEREFFredmanTarjan1984" class="citation conference cs1"><a href="Michael_Fredman" title="Michael Fredman">Fredman, Michael Lawrence</a>; <a href="Robert_Tarjan" title="Robert Tarjan">Tarjan, Robert E.</a> (1984). Fibonacci heaps and their uses in improved network optimization algorithms. 25th Annual Symposium on Foundations of Computer Science. <a href="IEEE" class="mw-redirect" title="IEEE">IEEE</a>. pp. 338–346. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FSFCS.1984.715934">10.1109/SFCS.1984.715934</a>.</cite></li>

<li><cite id="CITEREFFredmanTarjan1987" class="citation journal cs1"><a href="Michael_Fredman" title="Michael Fredman">Fredman, Michael Lawrence</a>; <a href="Robert_Tarjan" title="Robert Tarjan">Tarjan, Robert E.</a> (1987). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F28869.28874">"Fibonacci heaps and their uses in improved network optimization algorithms"</a>. Journal of the Association for Computing Machinery. 34 (3): 596–615. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F28869.28874">10.1145/28869.28874</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7904683">7904683</a>.</cite></li>

<li><cite id="CITEREFZhanNoon1998" class="citation journal cs1">Zhan, F. Benjamin; Noon, Charles E. (February 1998). "Shortest Path Algorithms: An Evaluation Using Real Road Networks". <a href="Transportation_Science" title="Transportation Science">Transportation Science</a>. 32 (1): 65–73. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Ftrsc.32.1.65">10.1287/trsc.32.1.65</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14986297">14986297</a>.</cite></li>

<li><cite id="CITEREFLeyzorekGrayJohnsonLadew1957" class="citation book cs1">Leyzorek, M.; Gray, R. S.; Johnson, A. A.; Ladew, W. C.; Meaker, Jr., S. R.; Petry, R. M.; Seitz, R. N. (1957). Investigation of Model Techniques – First Annual Report – 6 June 1956 – 1 July 1957 – A Study of Model Techniques for Communication Systems. Cleveland, Ohio: Case Institute of Technology.</cite></li>

<li><cite id="CITEREFKnuth1977" class="citation journal cs1"><a href="Donald_Knuth" title="Donald Knuth">Knuth, D.E.</a> (1977). "A Generalization of Dijkstra's Algorithm". <a href="Information_Processing_Letters" title="Information Processing Letters">Information Processing Letters</a>. 6 (1): 1–5. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0020-0190%2877%2990002-3">10.1016/0020-0190(77)90002-3</a>.</cite></li>

<li><cite id="CITEREFAhujaMehlhornOrlinTarjan1990" class="citation journal cs1">Ahuja, Ravindra K.; Mehlhorn, Kurt; Orlin, James B.; Tarjan, Robert E. (April 1990). <a rel="nofollow" class="external text" href="https://dspace.mit.edu/bitstream/1721.1/47994/1/fasteralgorithms00sloa.pdf">"Faster Algorithms for the Shortest Path Problem"</a> (PDF). Journal of the ACM. 37 (2): 213–223. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F77600.77615">10.1145/77600.77615</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1721.1%2F47994">1721.1/47994</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5499589">5499589</a>.</cite></li>

<li><cite id="CITEREFRaman1997" class="citation journal cs1">Raman, Rajeev (1997). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F261342.261352">"Recent results on the single-source shortest paths problem"</a>. SIGACT News. 28 (2): 81–87. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F261342.261352">10.1145/261342.261352</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18031586">18031586</a>.</cite></li>

<li><cite id="CITEREFThorup2000" class="citation journal cs1">Thorup, Mikkel (2000). "On RAM priority Queues". SIAM Journal on Computing. 30 (1): 86–109. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0097539795288246">10.1137/S0097539795288246</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5221089">5221089</a>.</cite></li>

<li><cite id="CITEREFThorup1999" class="citation journal cs1">Thorup, Mikkel (1999). <a rel="nofollow" class="external text" href="http://www.diku.dk/~mthorup/PAPERS/sssp.ps.gz">"Undirected single-source shortest paths with positive integer weights in linear time"</a>. Journal of the ACM. 46 (3): 362–394. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F316542.316548">10.1145/316542.316548</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:207654795">207654795</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .infobox .side-box{font-size:100%}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1237033735">

/* start https://en.wikipedia.org/ */

@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}

/* end https://en.wikipedia.org/ */

</style><div class="side-box side-box-right sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">

/* start https://en.wikipedia.org/ */

.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}

/* end https://en.wikipedia.org/ */

</style>

<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Dijkstra%27s_algorithm" class="extiw external" title="commons:Category:Dijkstra's algorithm">Dijkstra's algorithm</a>.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="http://purl.umn.edu/107247">Oral history interview with Edsger W. Dijkstra</a>, <a href="Charles_Babbage_Institute" class="mw-redirect" title="Charles Babbage Institute">Charles Babbage Institute</a>, University of Minnesota, Minneapolis</li>

<li><a rel="nofollow" class="external text" href="http://blog.cleancoder.com/uncle-bob/2016/10/26/DijkstrasAlg.html">Implementation of Dijkstra's algorithm using TDD</a>, <a href="Robert_Cecil_Martin" class="mw-redirect" title="Robert Cecil Martin">Robert Cecil Martin</a>, The Clean Code Blog</li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Edsger_Dijkstra318" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div id="Edsger_Dijkstra318" style="font-size:114%;margin:0 4em"><a href="Edsger_W._Dijkstra" title="Edsger W. Dijkstra">Edsger Dijkstra</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Works</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>A Primer of ALGOL 60 Programming (book)</li>

<li>Structured Programming (book)</li>

<li>A Discipline of Programming (book)</li>

<li>A Method of Programming (book)</li>

<li>Predicate Calculus and Program Semantics (book)</li>

<li>Selected Writings on Computing: A Personal Perspective (book)</li>

<li>A Note on Two Problems in Connexion with Graphs</li>

<li>Cooperating Sequential Processes</li>

<li>Solution of a Problem in Concurrent Programming Control</li>

<li>The Structure of the 'THE'-Multiprogramming System</li>

<li><a href="Go_To_Statement_Considered_Harmful" class="mw-redirect" title="Go To Statement Considered Harmful">Go To Statement Considered Harmful</a></li>

<li>Notes on Structured Programming</li>

<li>The Humble Programmer</li>

<li>Programming Considered as a Human Activity</li>

<li>How Do We Tell Truths That Might Hurt?</li>

<li>On the Role of Scientific Thought</li>

<li>Self-stabilizing Systems in Spite of Distributed Control</li>

<li><a href="On_the_Cruelty_of_Really_Teaching_Computer_Science" title="On the Cruelty of Really Teaching Computer Science">On the Cruelty of Really Teaching Computer Science</a></li>

<li><a href="Edsger_W._Dijkstra#Selected_publications" title="Edsger W. Dijkstra">Selected papers</a></li>

<li><a href="EWDs" class="mw-redirect" title="EWDs">EWD manuscripts</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computer_science#Areas_of_computer_science" title="Computer science">Main research areas</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Theoretical_computer_science" title="Theoretical computer science">Theoretical computing science</a></li>

<li><a href="Software_engineering" title="Software engineering">Software engineering</a></li>

<li><a href="Systems_science" title="Systems science">Systems science</a></li>

<li><a href="Algorithm_design" class="mw-redirect" title="Algorithm design">Algorithm design</a> ()</li>

<li><a href="Concurrent_computing" title="Concurrent computing">Concurrent computing</a></li>

<li><a href="Distributed_computing" title="Distributed computing">Distributed computing</a></li>

<li><a href="Formal_methods" title="Formal methods">Formal methods</a></li>

<li><a href="Programming_methodology" class="mw-redirect" title="Programming methodology">Programming methodology</a></li>

<li><a href="Programming_language_research" class="mw-redirect" title="Programming language research">Programming language research</a></li>

<li><a href="Software_design" title="Software design">Program design</a> and <a href="Software_development" title="Software development">development</a></li>

<li><a href="Software_architecture" title="Software architecture">Software architecture</a></li>

<li><a href="Philosophy_of_computer_science" title="Philosophy of computer science">Philosophy of computer programming and computing science</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related people</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Shlomi_Dolev" title="Shlomi Dolev">Shlomi Dolev</a></li>

<li><a href="Per_Brinch_Hansen" title="Per Brinch Hansen">Per Brinch Hansen</a></li>

<li><a href="Tony_Hoare" title="Tony Hoare">Tony Hoare</a></li>

<li><a href="Ole-Johan_Dahl" title="Ole-Johan Dahl">Ole-Johan Dahl</a></li>

<li><a href="Leslie_Lamport" title="Leslie Lamport">Leslie Lamport</a></li>

<li><a href="David_Parnas" title="David Parnas">David Parnas</a></li>

<li><a href="Jaap_A._Zonneveld" title="Jaap A. Zonneveld">Jaap A. Zonneveld</a></li>

<li><a href="Carel_S._Scholten" title="Carel S. Scholten">Carel S. Scholten</a></li>

<li><a href="Adriaan_van_Wijngaarden" title="Adriaan van Wijngaarden">Adriaan van Wijngaarden</a></li>

<li><a href="Niklaus_Wirth" title="Niklaus Wirth">Niklaus Wirth</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow plainlist" colspan="3"><div>

<ul><li> <a href="https://en.wikiquote.org/wiki/Special:Search/Edsger_W._Dijkstra" class="extiw external" title="wikiquote:Special:Search/Edsger W. Dijkstra">Wikiquote</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Graph_and_tree_traversal_algorithms231" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graph_and_tree_traversal_algorithms231" style="font-size:114%;margin:0 4em"><a href="Graph_traversal" title="Graph traversal">Graph</a> and <a href="Tree_traversal" title="Tree traversal">tree</a> traversal algorithms</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Graph_traversal" title="Graph traversal">Search</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alpha%E2%80%93beta_pruning" title="Alpha–beta pruning">α–β pruning</a></li>

<li><a href="Best-first_search" title="Best-first search">Best-first search</a></li>

<li><a href="Beam_search" title="Beam search">Beam search</a></li>

<li><a href="Bidirectional_search" title="Bidirectional search">Bidirectional search</a></li>

<li><a href="Breadth-first_search" title="Breadth-first search">Breadth-first search</a>

<ul><li><a href="Lexicographic_breadth-first_search" title="Lexicographic breadth-first search">Lexicographic</a></li>

<li><a href="Parallel_breadth-first_search" title="Parallel breadth-first search">Parallel</a></li></ul></li>

<li><a href="Depth-first_search" title="Depth-first search">Depth-first search</a>

<ul><li><a href="Iterative_deepening_depth-first_search" title="Iterative deepening depth-first search">Iterative deepening</a></li></ul></li>

<li><a href="Fringe_search" title="Fringe search">Fringe search</a></li>

<li><a href="Jump_point_search" title="Jump point search">Jump point search</a></li>

<li><a href="Monte_Carlo_tree_search" title="Monte Carlo tree search">Monte Carlo tree search</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Shortest_path_problem" title="Shortest path problem">Shortest path</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford</a></li>

<li><a href="Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall</a></li>

<li><a href="Johnson's_algorithm" title="Johnson's algorithm">Johnson's</a></li>

<li><a href="Shortest_path_faster_algorithm" class="mw-redirect" title="Shortest path faster algorithm">Shortest path faster</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Minimum_spanning_tree" title="Minimum spanning tree">Minimum spanning tree</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bor%C5%AFvka's_algorithm" title="Borůvka's algorithm">Borůvka's</a></li>

<li><a href="Kruskal's_algorithm" title="Kruskal's algorithm">Kruskal's</a></li>

<li><a href="Reverse-delete_algorithm" title="Reverse-delete algorithm">Reverse-delete</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="List_of_algorithms#Graph_search" title="List of algorithms">List of graph search algorithms</a></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Optimization:_Algorithms,_methods,_and_heuristics381" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Optimization:_Algorithms,_methods,_and_heuristics381" style="font-size:114%;margin:0 4em"><a href="Mathematical_optimization" title="Mathematical optimization">Optimization</a>: <a href="Optimization_algorithm" class="mw-redirect" title="Optimization algorithm">Algorithms</a>, <a href="Iterative_method" title="Iterative method">methods</a>, and <a href="Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Unconstrained_nonlinear381" style="font-size:114%;margin:0 4em"><a href="Nonlinear_programming" title="Nonlinear programming">Unconstrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Function_(mathematics)" title="Function (mathematics)">Functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Golden-section_search" title="Golden-section search">Golden-section search</a></li>

<li><a href="Powell's_method" title="Powell's method">Powell's method</a></li>

<li><a href="Line_search" title="Line search">Line search</a></li>

<li><a href="Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead method</a></li>

<li><a href="Successive_parabolic_interpolation" title="Successive parabolic interpolation">Successive parabolic interpolation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Gradient" title="Gradient">Gradients</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Local_convergence" title="Local convergence">Convergence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Trust_region" title="Trust region">Trust region</a></li>

<li><a href="Wolfe_conditions" title="Wolfe conditions">Wolfe conditions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quasi-Newton_method" title="Quasi-Newton method">Quasi–Newton</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Berndt%E2%80%93Hall%E2%80%93Hall%E2%80%93Hausman_algorithm" title="Berndt–Hall–Hall–Hausman algorithm">Berndt–Hall–Hall–Hausman</a></li>

<li><a href="Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm" title="Broyden–Fletcher–Goldfarb–Shanno algorithm">Broyden–Fletcher–Goldfarb–Shanno</a> and <a href="Limited-memory_BFGS" title="Limited-memory BFGS">L-BFGS</a></li>

<li><a href="Davidon%E2%80%93Fletcher%E2%80%93Powell_formula" title="Davidon–Fletcher–Powell formula">Davidon–Fletcher–Powell</a></li>

<li><a href="Symmetric_rank-one" title="Symmetric rank-one">Symmetric rank-one (SR1)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Iterative_method" title="Iterative method">Other methods</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonlinear_conjugate_gradient_method" title="Nonlinear conjugate gradient method">Conjugate gradient</a></li>

<li><a href="Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton</a></li>

<li><a href="Gradient_descent" title="Gradient descent">Gradient</a></li>

<li><a href="Mirror_descent" title="Mirror descent">Mirror</a></li>

<li><a href="Levenberg%E2%80%93Marquardt_algorithm" title="Levenberg–Marquardt algorithm">Levenberg–Marquardt</a></li>

<li><a href="Powell's_dog_leg_method" title="Powell's dog leg method">Powell's dog leg method</a></li>

<li><a href="Truncated_Newton_method" title="Truncated Newton method">Truncated Newton</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Hessian_matrix" title="Hessian matrix">Hessians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Newton's_method_in_optimization" title="Newton's method in optimization">Newton's method</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Constrained_nonlinear381" style="font-size:114%;margin:0 4em"><a href="Nonlinear_programming" title="Nonlinear programming">Constrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Barrier_function" title="Barrier function">Barrier methods</a></li>

<li><a href="Penalty_method" title="Penalty method">Penalty methods</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Differentiable</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Augmented_Lagrangian_method" title="Augmented Lagrangian method">Augmented Lagrangian methods</a></li>

<li><a href="Sequential_quadratic_programming" title="Sequential quadratic programming">Sequential quadratic programming</a></li>

<li><a href="Successive_linear_programming" title="Successive linear programming">Successive linear programming</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Convex_optimization381" style="font-size:114%;margin:0 4em"><a href="Convex_optimization" title="Convex optimization">Convex optimization</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Convex_minimization" class="mw-redirect" title="Convex minimization">Convex minimization</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cutting-plane_method" title="Cutting-plane method">Cutting-plane method</a></li>

<li><a href="Frank%E2%80%93Wolfe_algorithm" title="Frank–Wolfe algorithm">Reduced gradient (Frank–Wolfe)</a></li>

<li><a href="Subgradient_method" title="Subgradient method">Subgradient method</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Linear_programming" title="Linear programming">Linear</a> and <a href="Quadratic_programming" title="Quadratic programming">quadratic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Linear_programming#Interior_point" title="Linear programming">Interior point</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_scaling" title="Affine scaling">Affine scaling</a></li>

<li><a href="Ellipsoid_method" title="Ellipsoid method">Ellipsoid algorithm of Khachiyan</a></li>

<li><a href="Karmarkar's_algorithm" title="Karmarkar's algorithm">Projective algorithm of Karmarkar</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Matroid" title="Matroid">Basis-</a><a href="Exchange_algorithm" class="mw-redirect" title="Exchange algorithm">exchange</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Simplex_algorithm" title="Simplex algorithm">Simplex algorithm of Dantzig</a></li>

<li><a href="Revised_simplex_method" title="Revised simplex method">Revised simplex algorithm</a></li>

<li><a href="Criss-cross_algorithm" title="Criss-cross algorithm">Criss-cross algorithm</a></li>

<li><a href="Lemke's_algorithm" title="Lemke's algorithm">Principal pivoting algorithm of Lemke</a></li>

<li><a href="Active-set_method" title="Active-set method">Active-set method</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial381" style="font-size:114%;margin:0 4em"><a href="Combinatorial_optimization" title="Combinatorial optimization">Combinatorial</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Paradigms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approximation_algorithm" title="Approximation algorithm">Approximation algorithm</a></li>

<li><a href="Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li>

<li><a href="Greedy_algorithm" title="Greedy algorithm">Greedy algorithm</a></li>

<li><a href="Integer_programming" title="Integer programming">Integer programming</a>

<ul><li><a href="Branch_and_bound" title="Branch and bound">Branch and bound</a>/<a href="Branch_and_cut" title="Branch and cut">cut</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Graph_algorithm" class="mw-redirect" title="Graph algorithm">Graph algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Minimum_spanning_tree52" scope="row" class="navbox-group" style="width:1%"><a href="Minimum_spanning_tree" title="Minimum spanning tree">Minimum spanning tree</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bor%C5%AFvka's_algorithm" title="Borůvka's algorithm">Borůvka</a></li>

<li><a href="Kruskal's_algorithm" title="Kruskal's algorithm">Kruskal</a></li></ul>

</div></td></tr></tbody></table><div>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Shortest_path39" scope="row" class="navbox-group" style="width:1%"><a href="Shortest_path_problem" title="Shortest path problem">Shortest path</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford</a>

<li><a href="Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Flow_network" title="Flow network">Network flows</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dinic's_algorithm" title="Dinic's algorithm">Dinic</a></li>

<li><a href="Edmonds%E2%80%93Karp_algorithm" title="Edmonds–Karp algorithm">Edmonds–Karp</a></li>

<li><a href="Ford%E2%80%93Fulkerson_algorithm" title="Ford–Fulkerson algorithm">Ford–Fulkerson</a></li>

<li><a href="Push%E2%80%93relabel_maximum_flow_algorithm" title="Push–relabel maximum flow algorithm">Push–relabel maximum flow</a></li></ul>

<ul><li><a href="Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithm</a></li>

<li><a href="Hill_climbing" title="Hill climbing">Hill climbing</a></li>

<li><a href="Local_search_(optimization)" title="Local search (optimization)">Local search</a></li>

<li><a href="Parallel_metaheuristic" title="Parallel metaheuristic">Parallel metaheuristics</a></li>

<li><a href="Simulated_annealing" title="Simulated annealing">Simulated annealing</a></li>

<li><a href="Spiral_optimization_algorithm" title="Spiral optimization algorithm">Spiral optimization algorithm</a></li>

<li><a href="Tabu_search" title="Tabu search">Tabu search</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li><a href="Comparison_of_optimization_software" title="Comparison of optimization software">Software</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-21" href="https://en.wikipedia.org/wiki/?title=Dijkstra's_algorithm&oldid=1301664509">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>