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<title>Greedy algorithm</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Greedy algorithm</span></span>

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<p>A <b>greedy algorithm</b> is any <a href="Algorithm" title="Algorithm">algorithm</a> that follows the problem-solving <a href="Heuristic_(computer_science)" title="Heuristic (computer science)">heuristic</a> of making the locally optimal choice at each stage.<sup id="cite_ref-NISTg_1-0" class="reference"><a href="#cite_note-NISTg-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>For example, a greedy strategy for the <a href="Travelling_salesman_problem" title="Travelling salesman problem">travelling salesman problem</a> (which is of high <a href="Computational_complexity" title="Computational complexity">computational complexity</a>) is the following heuristic: "At each step of the journey, visit the nearest unvisited city." This heuristic does not intend to find the best solution, but it terminates in a reasonable number of steps; finding an optimal solution to such a complex problem typically requires unreasonably many steps.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>In <a href="Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a>, greedy algorithms optimally solve <a href="Combinatorics" title="Combinatorics">combinatorial</a> problems having the properties of <a href="Matroid" title="Matroid">matroids</a> and give constant-factor approximations to optimization problems with the submodular structure.

</p>

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

<p>Greedy algorithms produce good solutions on some <a href="Mathematical_problem" title="Mathematical problem">mathematical problems</a>, but not on others. Most problems for which they work will have two properties:

</p>

<dl><dt>Greedy choice property</dt>

<dd>Whichever choice seems best at a given moment can be made and then (recursively) solve the remaining sub-problems. The choice made by a greedy algorithm may depend on choices made so far, but not on future choices or all the solutions to the subproblem. It iteratively makes one greedy choice after another, reducing each given problem into a smaller one. In other words, a greedy algorithm never reconsiders its choices. This is the main difference from <a href="Dynamic_programming" title="Dynamic programming">dynamic programming</a>, which is exhaustive and is guaranteed to find the solution. After every stage, dynamic programming makes decisions based on all the decisions made in the previous stage and may reconsider the previous stage's algorithmic path to the solution.</dd>

<dt>Optimal substructure</dt>

<dd>"A problem exhibits <a href="Optimal_substructure" title="Optimal substructure">optimal substructure</a> if an optimal solution to the problem contains optimal solutions to the sub-problems."<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Correctness_Proofs">Correctness Proofs</h3></div>

<p>A common technique for proving the correctness of greedy algorithms uses an <a href="Inductive_reasoning" title="Inductive reasoning">inductive</a> exchange argument.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The exchange argument demonstrates that any solution different from the greedy solution can be transformed into the greedy solution without degrading its quality. This proof pattern typically follows these steps:

</p><p>This proof pattern typically follows these steps (by contradiction):

</p>

<ol><li>Assume there exists an optimal solution different from the greedy solution</li>

<li>Identify the first point where the optimal and greedy solutions differ</li>

<li>Prove that exchanging the optimal choice for the greedy choice at this point cannot worsen the solution</li>

<li>Conclude by induction that there must exist an optimal solution identical to the greedy solution</li></ol>

<p>In some cases, an additional step may be needed to prove that no optimal solution can strictly improve upon the greedy solution.

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<div class="mw-heading mw-heading3"><h3 id="Cases_of_failure">Cases of failure</h3></div>

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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:304px;max-width:304px"><div class="trow"><div class="theader">Examples on how a greedy algorithm may fail to achieve the optimal solution.</div></div><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">Starting from A, a greedy algorithm that tries to find the maximum by following the greatest slope will find the local maximum at "m", oblivious to the global maximum at "M".</div></div></div><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">To reach the largest sum, at each step, the greedy algorithm will choose what appears to be the optimal immediate choice, so it will choose 12 instead of 3 at the second step, and will not reach the best solution, which contains 99.</div></div></div></div></div>

<p>Greedy algorithms fail to produce the optimal solution for many other problems and may even produce the <i>unique worst possible</i> solution. One example is the <a href="Travelling_salesman_problem" title="Travelling salesman problem">travelling salesman problem</a> mentioned above: for each number of cities, there is an assignment of distances between the cities for which the nearest-neighbour heuristic produces the unique worst possible tour.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

For other possible examples, see <a href="Horizon_effect" title="Horizon effect">horizon effect</a>.

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

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<p>Greedy algorithms can be characterized as being 'short sighted', and also as 'non-recoverable'. They are ideal only for problems that have an 'optimal substructure'. Despite this, for many simple problems, the best-suited algorithms are greedy. It is important, however, to note that the greedy algorithm can be used as a selection algorithm to prioritize options within a search, or branch-and-bound algorithm. There are a few variations to the greedy algorithm:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li>Pure greedy algorithms</li>

<li>Orthogonal greedy algorithms</li>

<li>Relaxed greedy algorithms</li></ul>

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

<p>Greedy algorithms have a long history of study in <a href="Combinatorial_optimization" title="Combinatorial optimization">combinatorial optimization</a> and <a href="Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a>. Greedy heuristics are known to produce suboptimal results on many problems,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and so natural questions are:

</p>

<ul><li>For which problems do greedy algorithms perform optimally?</li>

<li>For which problems do greedy algorithms guarantee an approximately optimal solution?</li>

<li>For which problems are greedy algorithms guaranteed <i>not</i> to produce an optimal solution?</li></ul>

<p>A large body of literature exists answering these questions for general classes of problems, such as <a href="Matroid" title="Matroid">matroids</a>, as well as for specific problems, such as <a href="Set_cover" class="mw-redirect" title="Set cover">set cover</a>.

</p>

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

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Matroid" title="Matroid">Matroid</a></div>

<p>A <a href="Matroid" title="Matroid">matroid</a> is a mathematical structure that generalizes the notion of <a href="Linear_independence" title="Linear independence">linear independence</a> from <a href="Vector_spaces" class="mw-redirect" title="Vector spaces">vector spaces</a> to arbitrary sets. If an optimization problem has the structure of a matroid, then the appropriate greedy algorithm will solve it optimally.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Submodular_functions">Submodular functions</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Submodular_set_function#Optimization_problems" title="Submodular set function">Submodular set function § Optimization problems</a></div>

<p>A function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> defined on subsets of a set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> is called <a href="Submodular" class="mw-redirect" title="Submodular">submodular</a> if for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S,T\subseteq \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S,T\subseteq \Omega }</annotation>

</semantics>

</math></span><img src="./c0d257a9a2eb787c5690a306ab0cdccba7c2893b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.946ex; height:2.509ex;" alt="{\displaystyle S,T\subseteq \Omega }" loading="lazy"></span> we have that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)+f(T)\geq f(S\cup T)+f(S\cap T)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(S)+f(T)\geq f(S\cup T)+f(S\cap T)}</annotation>

</semantics>

</math></span><img src="./f21068db00908ee3727f8f8f64990b2a68ea7c09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.702ex; height:2.843ex;" alt="{\displaystyle f(S)+f(T)\geq f(S\cup T)+f(S\cap T)}" loading="lazy"></span>.

</p><p>Suppose one wants to find a set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span> which maximizes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span>. The greedy algorithm, which builds up a set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span> by incrementally adding the element which increases <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> the most at each step, produces as output a set that is at least <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-1/e)\max _{X\subseteq \Omega }f(X)}">

</mrow>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1-1/e)\max _{X\subseteq \Omega }f(X)}</annotation>

</semantics>

</math></span><img src="./80e68143eac3b3c1ce6906a7f2646793bc940fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.388ex; height:4.343ex;" alt="{\displaystyle (1-1/e)\max _{X\subseteq \Omega }f(X)}" loading="lazy"></span>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> That is, greedy performs within a constant factor of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-1/e)\approx 0.63}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1-1/e)\approx 0.63}</annotation>

</semantics>

</math></span><img src="./91728e8230b87f795fbfc68ec7c8775e4430b169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.453ex; height:2.843ex;" alt="{\displaystyle (1-1/e)\approx 0.63}" loading="lazy"></span> as good as the optimal solution.

</p><p>Similar guarantees are provable when additional constraints, such as cardinality constraints,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> are imposed on the output, though often slight variations on the greedy algorithm are required. See <sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> for an overview.

</p>

<div class="mw-heading mw-heading3"><h3 id="Other_problems_with_guarantees">Other problems with guarantees</h3></div>

<p>Other problems for which the greedy algorithm gives a strong guarantee, but not an optimal solution, include

</p>

<ul><li><a href="Set_cover_problem#Greedy_algorithm" title="Set cover problem">Set cover</a></li>

<li>The <a href="Steiner_tree_problem" title="Steiner tree problem">Steiner tree problem</a></li>

<li><a href="Load_balancing_(computing)" title="Load balancing (computing)">Load balancing</a><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Independent_set_(graph_theory)#Approximation_algorithms" title="Independent set (graph theory)">Independent set</a></li></ul>

<p>Many of these problems have matching lower bounds; i.e., the greedy algorithm does not perform better than the guarantee in the worst case.

</p>

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

<p>Greedy algorithms typically (but not always) fail to find the globally optimal solution because they usually do not operate exhaustively on all the data. They can make commitments to certain choices too early, preventing them from finding the best overall solution later. For example, all known <a href="Greedy_coloring" title="Greedy coloring">greedy coloring</a> algorithms for the <a href="Graph_coloring_problem" class="mw-redirect" title="Graph coloring problem">graph coloring problem</a> and all other <a href="NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a> problems do not consistently find optimum solutions. Nevertheless, they are useful because they are quick to think up and often give good approximations to the optimum.

</p><p>If a greedy algorithm can be proven to yield the global optimum for a given problem class, it typically becomes the method of choice because it is faster than other optimization methods like <a href="Dynamic_programming" title="Dynamic programming">dynamic programming</a>. Examples of such greedy algorithms are <a href="Kruskal's_algorithm" title="Kruskal's algorithm">Kruskal's algorithm</a> and <a href="Prim's_algorithm" title="Prim's algorithm">Prim's algorithm</a> for finding <a href="Minimum_spanning_tree" title="Minimum spanning tree">minimum spanning trees</a> and the algorithm for finding optimum <a href="Huffman_tree" class="mw-redirect" title="Huffman tree">Huffman trees</a>.

</p><p>Greedy algorithms appear in network <a href="Routing" title="Routing">routing</a> as well. Using greedy routing, a message is forwarded to the neighbouring node which is "closest" to the destination. The notion of a node's location (and hence "closeness") may be determined by its physical location, as in <a href="Geographic_routing" title="Geographic routing">geographic routing</a> used by <a href="Ad_hoc_network" title="Ad hoc network">ad hoc networks</a>. Location may also be an entirely artificial construct as in <a href="Small_world_routing" class="mw-redirect" title="Small world routing">small world routing</a> and <a href="Distributed_hash_table" title="Distributed hash table">distributed hash table</a>.

</p>

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

<ul><li>The <a href="Activity_selection_problem" title="Activity selection problem">activity selection problem</a> is characteristic of this class of problems, where the goal is to pick the maximum number of activities that do not clash with each other.</li>

<li>In the <a href="Macintosh_computer" class="mw-redirect" title="Macintosh computer">Macintosh computer</a> game <i><a href="Crystal_Quest" title="Crystal Quest">Crystal Quest</a></i> the objective is to collect crystals, in a fashion similar to the <a href="Travelling_salesman_problem" title="Travelling salesman problem">travelling salesman problem</a>. The game has a demo mode, where the game uses a greedy algorithm to go to every crystal. The <a href="Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a> does not account for obstacles, so the demo mode often ends quickly.</li>

<li>The <a href="Matching_pursuit" title="Matching pursuit">matching pursuit</a> is an example of a greedy algorithm applied on signal approximation.</li>

<li>A greedy algorithm finds the optimal solution to <a href="Malfatti_circles" title="Malfatti circles">Malfatti's problem</a> of finding three disjoint circles within a given triangle that maximize the total area of the circles; it is conjectured that the same greedy algorithm is optimal for any number of circles.</li>

<li>A greedy algorithm is used to construct a Huffman tree during <a href="Huffman_coding" title="Huffman coding">Huffman coding</a> where it finds an optimal solution.</li>

<li>In <a href="Decision_tree_learning" title="Decision tree learning">decision tree learning</a>, greedy algorithms are commonly used, however they are not guaranteed to find the optimal solution.

<ul><li>One popular such algorithm is the <a href="ID3_algorithm" title="ID3 algorithm">ID3 algorithm</a> for decision tree construction.</li></ul></li>

<li><a href="Dijkstra's_algorithm" title="Dijkstra's algorithm">Dijkstra's algorithm</a> and the related <a href="A*_search_algorithm" title="A* search algorithm">A* search algorithm</a> are verifiably optimal greedy algorithms for <a href="Graph_search" class="mw-redirect" title="Graph search">graph search</a> and <a href="Shortest_path_problem" title="Shortest path problem">shortest path finding</a>.

<ul><li>A* search is conditionally optimal, requiring an "<a href="Admissible_heuristic" title="Admissible heuristic">admissible heuristic</a>" that will not overestimate path costs.</li></ul></li>

<li><a href="Kruskal's_algorithm" title="Kruskal's algorithm">Kruskal's algorithm</a> and <a href="Prim's_algorithm" title="Prim's algorithm">Prim's algorithm</a> are greedy algorithms for constructing <a href="Minimum_spanning_tree" title="Minimum spanning tree">minimum spanning trees</a> of a given <a href="Connected_graph" class="mw-redirect" title="Connected graph">connected graph</a>. They always find an optimal solution, which may not be unique in general.</li>

<li>The <a href="Sequitur_algorithm" title="Sequitur algorithm">Sequitur</a> and <a href="Lempel-Ziv-Welch_algorithm" class="mw-redirect" title="Lempel-Ziv-Welch algorithm">Lempel-Ziv-Welch</a> algorithms are <a href="Grammar_induction#Grammatical_inference_by_greedy_algorithms" title="Grammar induction">greedy algorithms for grammar induction</a>.</li></ul>

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

<li><a href="Multi-armed_bandit#Semi-uniform_strategies" title="Multi-armed bandit">Epsilon-greedy strategy</a></li>

<li><a href="Greedy_algorithm_for_Egyptian_fractions" title="Greedy algorithm for Egyptian fractions">Greedy algorithm for Egyptian fractions</a></li>

<li><a href="Greedy_source" title="Greedy source">Greedy source</a></li>

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

<li><a href="Horizon_effect" title="Horizon effect">Horizon effect</a></li>

<li><a href="Matroid" title="Matroid">Matroid</a></li></ul>

</div>

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

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</style><cite id="CITEREFBlack2005" class="citation web cs1">Black, Paul E. (2 February 2005). <a rel="nofollow" class="external text" href="http://xlinux.nist.gov/dads//HTML/greedyalgo.html">"greedy algorithm"</a>. <i>Dictionary of Algorithms and Data Structures</i>. <a href="National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">U.S. National Institute of Standards and Technology</a> (NIST)<span class="reference-accessdate">. Retrieved <span class="nowrap">17 August</span> 2012</span>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFvan_Melkebeek" class="citation web cs1">van Melkebeek, Dieter. <a rel="nofollow" class="external text" href="https://pages.cs.wisc.edu/~dieter/Courses/2004F-CS787/Scribes/greedy-approx.pdf">"Greedy Approximations"</a> <span class="cs1-format">(PDF)</span>. <i>University of Wisconsin–Madison</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2025-07-25</span></span>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite web}}</code>: CS1 maint: url-status (link)</span></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFFriezePegden2023" class="citation journal cs1 cs1-prop-long-vol">Frieze, Alan; Pegden, Wesley (2023-10-05). <a rel="nofollow" class="external text" href="https://arxiv.org/pdf/2310.03222">"The bright side of simple heuristics for the TSP"</a>. <i>arXiv preprint</i>. arXiv:2310.03222<span class="reference-accessdate">. Retrieved <span class="nowrap">2025-07-25</span></span>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFCormenLeisersonRivestStein2001">Cormen et al. 2001</a>, Ch. 16</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFErickson2019" class="citation book cs1">Erickson, Jeff (2019). "Greedy Algorithms". <a rel="nofollow" class="external text" href="https://jeffe.cs.illinois.edu/teaching/algorithms/"><i>Algorithms</i></a>. University of Illinois at Urbana-Champaign.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFGutinYeoZverovich2002" class="citation journal cs1">Gutin, Gregory; Yeo, Anders; Zverovich, Alexey (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0166-218X%2801%2900195-0">"Traveling salesman should not be greedy: Domination analysis of greedy-type heuristics for the TSP"</a>. <i>Discrete Applied Mathematics</i>. <b>117</b> (<span class="nowrap">1–</span>3): <span class="nowrap">81–</span>86. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0166-218X%2801%2900195-0">10.1016/S0166-218X(01)00195-0</a></span>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFDeVoreTemlyakov1996" class="citation journal cs1">DeVore, R. A.; Temlyakov, V. N. (1996-12-01). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF02124742">"Some remarks on greedy algorithms"</a></span>. <i>Advances in Computational Mathematics</i>. <b>5</b> (1): <span class="nowrap">173–</span>187. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02124742">10.1007/BF02124742</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1572-9044">1572-9044</a>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFFeige1998">Feige 1998</a></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFPapadimitriouSteiglitz1998">Papadimitriou & Steiglitz 1998</a></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFNemhauserWolseyFisher1978">Nemhauser, Wolsey & Fisher 1978</a></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFBuchbinderFeldmanNaorSchwartz2014">Buchbinder et al. 2014</a></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFKrauseGolovin2014">Krause & Golovin 2014</a></span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.win.tue.nl/~mdberg/Onderwijs/AdvAlg_Material/Course%20Notes/lecture5.pdf">"Lecture 5: Introduction to Approximation Algorithms"</a> <span class="cs1-format">(PDF)</span>. <i>Advanced Algorithms (2IL45) — Course Notes</i>. TU Eindhoven. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.win.tue.nl/~mdberg/Onderwijs/AdvAlg_Material/Course%20Notes/lecture5.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite></span>

</li>

</ol></div></div>

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

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.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}

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<ul><li><cite id="CITEREFCormenLeisersonRivestStein2001" class="citation book cs1">Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NLngYyWFl_YC&pg=PA370">"16 Greedy Algorithms"</a>. <a href="Introduction_to_Algorithms" title="Introduction to Algorithms"><i>Introduction To Algorithms</i></a>. MIT Press. pp. 370–. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-262-03293-3</bdi>.</cite></li>

<li><cite id="CITEREFGutinYeoZverovich2002" class="citation journal cs1">Gutin, Gregory; Yeo, Anders; Zverovich, Alexey (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0166-218X%2801%2900195-0">"Traveling salesman should not be greedy: Domination analysis of greedy-type heuristics for the TSP"</a>. <i>Discrete Applied Mathematics</i>. <b>117</b> (<span class="nowrap">1–</span>3): <span class="nowrap">81–</span>86. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0166-218X%2801%2900195-0">10.1016/S0166-218X(01)00195-0</a></span>.</cite></li>

<li><cite id="CITEREFBang-JensenGutinYeo2004" class="citation journal cs1">Bang-Jensen, Jørgen; Gutin, Gregory; Yeo, Anders (2004). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.disopt.2004.03.007">"When the greedy algorithm fails"</a>. <i>Discrete Optimization</i>. <b>1</b> (2): <span class="nowrap">121–</span>127. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.disopt.2004.03.007">10.1016/j.disopt.2004.03.007</a></span>.</cite></li>

<li><cite id="CITEREFBendallMargot2006" class="citation journal cs1">Bendall, Gareth; Margot, François (2006). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.disopt.2006.03.001">"Greedy-type resistance of combinatorial problems"</a>. <i>Discrete Optimization</i>. <b>3</b> (4): <span class="nowrap">288–</span>298. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.disopt.2006.03.001">10.1016/j.disopt.2006.03.001</a></span>.</cite></li>

<li><cite id="CITEREFFeige1998" class="citation journal cs1">Feige, U. (1998). <a rel="nofollow" class="external text" href="https://www.cs.duke.edu/courses/cps296.2/spring07/papers/p634-feige.pdf">"A threshold of ln n for approximating set cover"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of the ACM</i>. <b>45</b> (4): <span class="nowrap">634–</span>652. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F285055.285059">10.1145/285055.285059</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:52827488">52827488</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/https://www.cs.duke.edu/courses/cps296.2/spring07/papers/p634-feige.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite></li>

<li><cite id="CITEREFNemhauserWolseyFisher1978" class="citation journal cs1">Nemhauser, G.; Wolsey, L.A.; Fisher, M.L. (1978). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/242914003">"An analysis of approximations for maximizing submodular set functions—I"</a>. <i>Mathematical Programming</i>. <b>14</b> (1): <span class="nowrap">265–</span>294. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01588971">10.1007/BF01588971</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:206800425">206800425</a>.</cite></li>

<li><cite id="CITEREFBuchbinderFeldmanNaorSchwartz2014" class="citation book cs1">Buchbinder, Niv; Feldman, Moran; Naor, Joseph (Seffi); Schwartz, Roy (2014). <a rel="nofollow" class="external text" href="http://theory.epfl.ch/moranfe/Publications/SODA2014.pdf">"Submodular maximization with cardinality constraints"</a> <span class="cs1-format">(PDF)</span>. <i>Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms</i>. Society for Industrial and Applied Mathematics. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9781611973402.106">10.1137/1.9781611973402.106</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-61197-340-2</bdi>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://theory.epfl.ch/moranfe/Publications/SODA2014.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite></li>

<li><cite id="CITEREFKrauseGolovin2014" class="citation book cs1">Krause, A.; Golovin, D. (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YxliAgAAQBAJ&pg=PA71">"Submodular Function Maximization"</a>. In Bordeaux, L.; Hamadi, Y.; Kohli, P. (eds.). <i>Tractability: Practical Approaches to Hard Problems</i>. Cambridge University Press. pp. <span class="nowrap">71–</span>104. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9781139177801.004">10.1017/CBO9781139177801.004</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781139177801</bdi>.</cite></li>

<li><cite id="CITEREFPapadimitriouSteiglitz1998" class="citation book cs1"><a href="Christos_Papadimitriou" title="Christos Papadimitriou">Papadimitriou, Christos H.</a>; <a href="Kenneth_Steiglitz" title="Kenneth Steiglitz">Steiglitz, Kenneth</a> (1998). <i>Combinatorial Optimization: Algorithms and Complexity</i>. Dover.</cite></li></ul>

</div>

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

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Greedy_algorithms" class="extiw external" title="commons:Category:Greedy algorithms">Greedy algorithms</a></span>.</div></div>

</div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Greedy_algorithm">"Greedy algorithm"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><cite id="CITEREFGift" class="citation web cs1">Gift, Noah. <a rel="nofollow" class="external text" href="http://www.oreillynet.com/onlamp/blog/2008/04/python_greedy_coin_changer_alg.html">"Python greedy coin example"</a>.</cite></li></ul>

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</style><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<br> 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<br><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="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<br> 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<br> 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="Dijkstra's_algorithm" title="Dijkstra's algorithm">Dijkstra</a></li>

<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 class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Data_structures_and_algorithms145" 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="2"><div id="Data_structures_and_algorithms145" style="font-size:114%;margin:0 4em"><a href="Data_structure" title="Data structure">Data structures</a> and <a href="Algorithm" title="Algorithm">algorithms</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Data structures</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="Array_(data_structure)" title="Array (data structure)">Array</a></li>

<li><a href="Associative_array" title="Associative array">Associative array</a></li>

<li><a href="Binary_search_tree" title="Binary search tree">Binary search tree</a></li>

<li><a href="Fenwick_tree" title="Fenwick tree">Fenwick tree</a></li>

<li><a href="Graph_(abstract_data_type)" title="Graph (abstract data type)">Graph</a></li>

<li><a href="Hash_table" title="Hash table">Hash table</a></li>

<li><a href="Linked_list" title="Linked list">Linked list</a></li>

<li><a href="Queue_(abstract_data_type)" title="Queue (abstract data type)">Queue</a></li>

<li><a href="Segment_tree" title="Segment tree">Segment tree</a></li>

<li><a href="Stack_(abstract_data_type)" title="Stack (abstract data type)">Stack</a></li>

<li><a href="String_(computer_science)" title="String (computer science)">String</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algorithms and <a href="Algorithmic_paradigm" title="Algorithmic paradigm">algorithmic paradigms</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="Backtracking" title="Backtracking">Backtracking</a></li>

<li><a href="Binary_search" title="Binary search">Binary search</a></li>

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

<li><a href="Brute-force_search" title="Brute-force search">Brute-force search</a></li>

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

<li><a href="Divide-and-conquer_algorithm" title="Divide-and-conquer algorithm">Divide and conquer</a></li>

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

<li><a href="Graph_traversal" title="Graph traversal">Graph traversal</a></li>

<li><a href="Hash_function" title="Hash function">Hash function</a></li>

<li><a href="Minimax" title="Minimax">Minimax</a></li>

<li><a href="Online_algorithm" title="Online algorithm">Online</a></li>

<li><a href="Randomized_algorithm" title="Randomized algorithm">Randomized</a></li>

<li><a href="Recursion_(computer_science)" title="Recursion (computer science)">Recursion</a></li>

<li><a href="Root-finding_algorithm" title="Root-finding algorithm">Root-finding</a></li>

<li><a href="Sorting_algorithm" title="Sorting algorithm">Sorting</a></li>

<li><a href="Streaming_algorithm" title="Streaming algorithm">Streaming</a></li>

<li><a href="Sweep_line_algorithm" title="Sweep line algorithm">Sweep line</a></li>

<li><a href="String-searching_algorithm" title="String-searching algorithm">String-searching</a></li>

<li><a href="Topological_sorting" title="Topological sorting">Topological sorting</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="List_of_data_structures" title="List of data structures">List of data structures</a></li>

<li><a href="List_of_algorithms" title="List of algorithms">List of algorithms</a></li></ul>

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