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<title>Heap (data structure)</title>

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Heap (data structure)

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</style><div role="note" class="hatnote navigation-not-searchable">For the memory heap (in low-level computer programming), which is unrelated to this data structure, see <a href="C_dynamic_memory_allocation" title="C dynamic memory allocation">C dynamic memory allocation</a>.</div>

In <a href="Computer_science" title="Computer science">computer science</a>, a heap is a <a href="Tree_(data_structure)" class="mw-redirect" title="Tree (data structure)">tree</a>-based <a href="Data_structure" title="Data structure">data structure</a> that satisfies the heap property: In a max heap, for any given <a href="Node_(computer_science)" title="Node (computer science)">node</a> C, if P is the parent node of C, then the key (the value) of P is greater than or equal to the key of C. In a min heap, the key of P is less than or equal to the key of C.<a href="#cite_note-1">[1]</a> The node at the "top" of the heap (with no parents) is called the root node.

The heap is one maximally efficient implementation of an <a href="Abstract_data_type" title="Abstract data type">abstract data type</a> called a <a href="Priority_queue" title="Priority queue">priority queue</a>, and in fact, priority queues are often referred to as "heaps", regardless of how they may be implemented. In a heap, the highest (or lowest) priority element is always stored at the root. However, a heap is not a sorted structure; it can be regarded as being partially ordered. A heap is a useful data structure when it is necessary to repeatedly remove the object with the highest (or lowest) priority, or when insertions need to be interspersed with removals of the root node.

A common implementation of a heap is the <a href="Binary_heap" title="Binary heap">binary heap</a>, in which the tree is a <a href="Binary_tree#Types_of_binary_trees" title="Binary tree">complete</a><a href="#cite_note-2">[2]</a> binary tree (see figure). The heap data structure, specifically the binary heap, was introduced by <a href="J._W._J._Williams" title="J. W. J. Williams">J. W. J. Williams</a> in 1964, as a data structure for the <a href="Heapsort" title="Heapsort">heapsort</a> sorting algorithm.<a href="#cite_note-3">[3]</a> Heaps are also crucial in several efficient <a href="Graph_algorithms" class="mw-redirect" title="Graph algorithms">graph algorithms</a> such as <a href="Dijkstra's_algorithm" title="Dijkstra's algorithm">Dijkstra's algorithm</a>. When a heap is a complete binary tree, it has the smallest possible height—a heap with N nodes and a branches for each node always has loga N height.

Note that, as shown in the graphic, there is no implied ordering between siblings or cousins and no implied sequence for an <a href="Inorder_traversal" class="mw-redirect" title="Inorder traversal">in-order traversal</a> (as there would be in, e.g., a <a href="Binary_search_tree" title="Binary search tree">binary search tree</a>). The heap relation mentioned above applies only between nodes and their parents, grandparents. The maximum number of children each node can have depends on the type of heap.

Heaps are typically constructed in-place in the same array where the elements are stored, with their structure being implicit in the access pattern of the operations. Heaps differ in this way from other data structures with similar or in some cases better theoretic bounds such as <a href="Radix_tree" title="Radix tree">radix trees</a> in that they require no additional memory beyond that used for storing the keys.

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

The common operations involving heaps are:

<dl><dt>Basic</dt></dl>

<ul><li>find-max (or find-min): find a maximum item of a max-heap, or a minimum item of a min-heap, respectively (a.k.a. <a href="Peek_(data_type_operation)" title="Peek (data type operation)">peek</a>)</li>

<li>insert: adding a new key to the heap (a.k.a., push<a href="#cite_note-4">[4]</a>)</li>

<li>extract-max (or extract-min): returns the node of maximum value from a max heap [or minimum value from a min heap] after removing it from the heap (a.k.a., pop<a href="#cite_note-5">[5]</a>)</li>

<li>delete-max (or delete-min): removing the root node of a max heap (or min heap), respectively</li>

<li>replace: pop root and push a new key. This is more efficient than a pop followed by a push, since it only needs to balance once, not twice, and is appropriate for fixed-size heaps.<a href="#cite_note-6">[6]</a></li></ul>

<dl><dt>Creation</dt></dl>

<ul><li>create-heap: create an empty heap</li>

<li>heapify: create a heap out of given array of elements</li>

<li>merge (union): joining two heaps to form a valid new heap containing all the elements of both, preserving the original heaps.</li>

<li>meld: joining two heaps to form a valid new heap containing all the elements of both, destroying the original heaps.</li></ul>

<dl><dt>Inspection</dt></dl>

<ul><li>size: return the number of items in the heap.</li>

<li>is-empty: return true if the heap is empty, false otherwise.</li></ul>

<dl><dt>Internal</dt></dl>

<ul><li>increase-key or decrease-key: updating a key within a max- or min-heap, respectively</li>

<li>delete: delete an arbitrary node (followed by moving last node and sifting to maintain heap)</li>

<li>sift-up: move a node up in the tree, as long as needed; used to restore heap condition after insertion. Called "sift" because node moves up the tree until it reaches the correct level, as in a <a href="Sieve" title="Sieve">sieve</a>.</li>

<li>sift-down: move a node down in the tree, similar to sift-up; used to restore heap condition after deletion or replacement.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Implementation_using_arrays">Implementation using arrays</h2></div>

Heaps are usually implemented with an <a href="Array_data_structure" class="mw-redirect" title="Array data structure">array</a>, as follows:

<ul><li>Each element in the array represents a node of the heap, and</li>

<li>The parent / child relationship is <a href="Implicit_data_structure" title="Implicit data structure">defined implicitly</a> by the elements' indices in the array.</li></ul>

For a <a href="Binary_heap" title="Binary heap">binary heap</a>, in the array, the first index contains the root element. The next two indices of the array contain the root's children. The next four indices contain the four children of the root's two child nodes, and so on. Therefore, given a node at index i, its children are at indices ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2i+1}">

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<annotation encoding="application/x-tex">{\displaystyle 2i+1}</annotation>

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</math><img src="./f077f73c2ecdf3c6e29a120f948a7255c0a65da1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.968ex; height:2.343ex;" alt="{\displaystyle 2i+1}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2i+2}">

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<annotation encoding="application/x-tex">{\displaystyle 2i+2}</annotation>

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</math><img src="./a756deaef520ee23fe2b1232c90957f55ec9d92b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.968ex; height:2.343ex;" alt="{\displaystyle 2i+2}" loading="lazy">⁠, and its parent is at index ⌊(i−1)/2⌋. This simple indexing scheme makes it efficient to move "up" or "down" the tree.

Balancing a heap is done by sift-up or sift-down operations (swapping elements which are out of order). As we can build a heap from an array without requiring extra memory (for the nodes, for example), <a href="Heapsort" title="Heapsort">heapsort</a> can be used to sort an array in-place.

After an element is inserted into or deleted from a heap, the heap property may be violated, and the heap must be re-balanced by swapping elements within the array.

Although different types of heaps implement the operations differently, the most common way is as follows:

<ul><li>Insertion: Add the new element at the end of the heap, in the first available free space. If this will violate the heap property, sift up the new element (swim operation) until the heap property has been reestablished.</li>

<li>Extraction: Remove the root and insert the last element of the heap in the root. If this will violate the heap property, sift down the new root (sink operation) to reestablish the heap property.</li>

<li>Replacement: Remove the root and put the new element in the root and sift down. When compared to extraction followed by insertion, this avoids a sift up step.</li></ul>

Construction of a binary (or d-ary) heap out of a given array of elements may be performed in linear time using the classic <a href="Heapsort#Variations" title="Heapsort">Floyd algorithm</a>, with the worst-case number of comparisons equal to 2N − 2s2(N) − e2(N) (for a binary heap), where s2(N) is the sum of all digits of the binary representation of N and e2(N) is the exponent of 2 in the prime factorization of N.<a href="#cite_note-7">[7]</a> This is faster than a sequence of consecutive insertions into an originally empty heap, which is log-linear.<a href="#cite_note-8">[a]</a>

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

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<li><a href="Binary_heap" title="Binary heap">Binary heap</a></li>

<li><a href="Binomial_heap" title="Binomial heap">Binomial heap</a></li>

<li><a href="Brodal_queue" title="Brodal queue">Brodal queue</a></li>

<li><a href="Fibonacci_heap" title="Fibonacci heap">Fibonacci heap</a></li>

<li><a href="Leftist_tree" title="Leftist tree">Leftist heap</a></li>

<li><a href="Skew_binomial_heap" title="Skew binomial heap">Skew binomial heap</a></li>

<li><a href="Strict_Fibonacci_heap" title="Strict Fibonacci heap">Strict Fibonacci heap</a></li>

<li><a href="Pairing_heap" title="Pairing heap">Pairing heap</a></li>

<li><a href="Radix_heap" title="Radix heap">Radix heap</a></li>

<li><a href="Randomized_meldable_heap" title="Randomized meldable heap">Randomized meldable heap</a></li>

<li><a href="Ternary_heap" class="mw-redirect" title="Ternary heap">Ternary heap</a></li>

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

</div>

<div class="mw-heading mw-heading2"><h2 id="Comparison_of_theoretic_bounds_for_variants">Comparison of theoretic bounds for variants</h2></div>

Here are <a href="Computational_complexity_theory" title="Computational complexity theory">time complexities</a><a href="#cite_note-CLRS-9">[8]</a> of various heap data structures. The abbreviation <abbr title="amortized complexity">am.</abbr> indicates that the given complexity is amortized, otherwise it is a worst-case complexity. For the meaning of "O(f)" and "Θ(f)" see <a href="Big_O_notation" title="Big O notation">Big O notation</a>. Names of operations assume a max-heap.

<th>Operation

</th>

<th>find-max

</th>

<th>delete-max

</th>

<th>increase-key

</th>

<th>insert

</th>

<th>meld

</th>

</th></tr>

<tr>

<th><a href="Binary_heap" title="Binary heap">Binary</a><a href="#cite_note-CLRS-9">[8]</a>

</th>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

<td style="background:#ffdddd">Θ(n)

</td>

<td style="background:#ddffdd">Θ(n)

</td></tr>

<tr>

</th>

<td style="background:#ddffdd">Θ(1)

</td>

</td>

</td>

</td>

</td>

</td></tr>

<tr>

<th><a href="Leftist_tree" title="Leftist tree">Leftist</a><a href="#cite_note-tarjan-leftist-11">[10]</a>

</th>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

<td style="background:#ddffdd">Θ(n)

</td></tr>

<tr>

<th><a href="Binomial_heap" title="Binomial heap">Binomial</a><a href="#cite_note-CLRS-9">[8]</a><a href="#cite_note-14">[12]</a>

</th>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

</td>

</td>

<td style="background:#ddffdd">Θ(n)

</td></tr>

<tr>

<th><a href="Skew_binomial_heap" title="Skew binomial heap">Skew binomial</a><a href="#cite_note-brodal-okasaki-16">[13]</a>

</th>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

<td style="background:#ddffdd">Θ(1)

</td>

</td>

<td style="background:#ddffdd">Θ(n)

</td></tr>

<tr>

</th>

<td style="background:#ddffdd">Θ(1)

</td>

</td>

<td style="background:#ddffdd">Θ(1)

</td>

</td>

</td>

<td style="background:#ddffdd">Θ(n)

</td></tr>

<tr>

<th><a href="Skew_heap" title="Skew heap">Bottom-up skew</a><a href="#cite_note-sleator-tarjan-skew-10">[9]</a>

</th>

<td style="background:#ddffdd">Θ(1)

</td>

</td>

</td>

</td>

</td>

</td></tr>

<tr>

<th><a href="Pairing_heap" title="Pairing heap">Pairing</a><a href="#cite_note-Iacono-19">[16]</a>

</th>

<td style="background:#ddffdd">Θ(1)

</td>

</td>

</td>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ddffdd">Θ(n)

</td></tr>

<tr>

<th>Rank-pairing<a href="#cite_note-23">[19]</a>

</th>

<td style="background:#ddffdd">Θ(1)

</td>

</td>

</td>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ddffdd">Θ(n)

</td></tr>

<tr>

<th><a href="Fibonacci_heap" title="Fibonacci heap">Fibonacci</a><a href="#cite_note-CLRS-9">[8]</a><a href="#cite_note-Fredman_And_Tarjan-24">[20]</a>

</th>

<td style="background:#ddffdd">Θ(1)

</td>

</td>

</td>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ddffdd">Θ(n)

</td></tr>

<tr>

<th><a href="Strict_Fibonacci_heap" title="Strict Fibonacci heap">Strict Fibonacci</a><a href="#cite_note-25">[21]</a><a href="#cite_note-optimum-26">[e]</a>

</th>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ddffdd">Θ(n)

</td></tr>

<tr>

<th><a href="Brodal_queue" title="Brodal queue">Brodal</a><a href="#cite_note-27">[22]</a><a href="#cite_note-optimum-26">[e]</a>

</th>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ffffdd">Θ(log n)

</td>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ddffdd">Θ(1)

</td>

<td style="background:#ddffdd">Θ(1)

</td>

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

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> Each insertion takes O(log(k)) in the existing size of the heap, thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}O(\log k)}">

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}O(\log k)}</annotation>

</semantics>

</math><img src="./57ee533ff7ef5300fd717d9230ca17a135c24df4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.895ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}O(\log k)}" loading="lazy">. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log n/2=(\log n)-1}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log n/2=(\log n)-1}</annotation>

</semantics>

</math><img src="./4dc03c817e6d65fe0ea9dffce246278943db343e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.743ex; height:2.843ex;" alt="{\displaystyle \log n/2=(\log n)-1}" loading="lazy">, a constant factor (half) of these insertions are within a constant factor of the maximum, so asymptotically we can assume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c4b8cc6fcba0c0b24656b0fb33414d2e6cffb83c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.704ex; height:2.176ex;" alt="{\displaystyle k=n}" loading="lazy">; formally the time is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nO(\log n)-O(n)=O(n\log n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle nO(\log n)-O(n)=O(n\log n)}</annotation>

</semantics>

</math><img src="./d96da998a5bf48e66f5960204a4610d2d9e0a226.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.765ex; height:2.843ex;" alt="{\displaystyle nO(\log n)-O(n)=O(n\log n)}" loading="lazy">. This can also be readily seen from <a href="Stirling's_approximation" title="Stirling's approximation">Stirling's approximation</a>.

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> make-heap is the operation of building a heap from a sequence of n unsorted elements. It can be done in Θ(n) time whenever meld runs in O(log n) time (where both complexities can be amortized).<a href="#cite_note-sleator-tarjan-skew-10">[9]</a><a href="#cite_note-tarjan-leftist-11">[10]</a> Another algorithm achieves Θ(n) for binary heaps.<a href="#cite_note-hayward-mcdiarmid-heap-build-12">[11]</a>

</li>

<li id="cite_note-bootstrap-meld-15">^ <a href="#cite_ref-bootstrap-meld_15-0">a</a> <a href="#cite_ref-bootstrap-meld_15-1">b</a> <a href="#cite_ref-bootstrap-meld_15-2">c</a> For <a href="Persistent_data_structure" title="Persistent data structure">persistent</a> heaps (not supporting increase-key), a generic transformation reduces the cost of meld to that of insert, while the new cost of delete-max is the sum of the old costs of delete-max and meld.<a href="#cite_note-17">[14]</a> Here, it makes meld run in Θ(1) time (amortized, if the cost of insert is) while delete-max still runs in O(log n). Applied to skew binomial heaps, it yields Brodal-Okasaki queues, persistent heaps with optimal worst-case complexities.<a href="#cite_note-brodal-okasaki-16">[13]</a>

</li>

<li id="cite_note-pairingdecreasekey-22"><a href="#cite_ref-pairingdecreasekey_22-0">^</a> Lower bound of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (\log \log n),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega (\log \log n),}</annotation>

</semantics>

</math><img src="./7caf8c32fbd99eed1338c02b7f7f1255f16ade36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.247ex; height:2.843ex;" alt="{\displaystyle \Omega (\log \log n),}" loading="lazy"><a href="#cite_note-Fredman1999-20">[17]</a> upper bound of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(2^{2{\sqrt {\log \log n}}}).}">

<msup>

<msqrt>

</msqrt>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(2^{2{\sqrt {\log \log n}}}).}</annotation>

</semantics>

</math><img src="./45c772d2ec070e49a6438ea92b1c8dc764613c5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.052ex; height:3.509ex;" alt="{\displaystyle O(2^{2{\sqrt {\log \log n}}}).}" loading="lazy"><a href="#cite_note-21">[18]</a>

</li>

<li id="cite_note-optimum-26">^ <a href="#cite_ref-optimum_26-0">a</a> <a href="#cite_ref-optimum_26-1">b</a> Brodal queues and strict Fibonacci heaps achieve optimal worst-case complexities for heaps. They were first described as imperative data structures. The Brodal-Okasaki queue is a <a href="Persistent_data_structure" title="Persistent data structure">persistent</a> data structure achieving the same optimum, except that increase-key is not supported.

</li>

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

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

The heap data structure has many applications.

<ul><li><a href="Heapsort" title="Heapsort">Heapsort</a>: One of the best sorting methods being in-place and with no quadratic worst-case scenarios.</li>

<li><a href="Selection_algorithm" title="Selection algorithm">Selection algorithms</a>: A heap allows access to the min or max element in constant time, and other selections (such as median or kth-element) can be done in sub-linear time on data that is in a heap.<a href="#cite_note-29">[24]</a></li>

<li><a href="List_of_algorithms#Graph_algorithms" title="List of algorithms">Graph algorithms</a>: By using heaps as internal traversal data structures, run time will be reduced by polynomial order. Examples of such problems are <a href="Prim's_algorithm" title="Prim's algorithm">Prim's minimal-spanning-tree algorithm</a> and <a href="Dijkstra's_algorithm" title="Dijkstra's algorithm">Dijkstra's shortest-path algorithm</a>.</li>

<li><a href="Priority_queue" title="Priority queue">Priority queue</a>: A priority queue is an abstract concept like "a list" or "a map"; just as a list can be implemented with a linked list or an array, a priority queue can be implemented with a heap or a variety of other methods.</li>

<li><a href="K-way_merge_algorithm" title="K-way merge algorithm">K-way merge</a>: A heap data structure is useful to merge many already-sorted input streams into a single sorted output stream. Examples of the need for merging include external sorting and streaming results from distributed data such as a log structured merge tree. The inner loop is obtaining the min element, replacing with the next element for the corresponding input stream, then doing a sift-down heap operation. (Alternatively the replace function.) (Using extract-max and insert functions of a priority queue are much less efficient.)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Programming_language_implementations">Programming language implementations</h2></div>

<ul><li>The <a href="C%2B%2B_Standard_Library" title="C++ Standard Library">C++ Standard Library</a> provides the <style data-mw-deduplicate="TemplateStyles:r886049734">

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</style>make_heap, push_heap and pop_heap algorithms for heaps (usually implemented as binary heaps), which operate on arbitrary random access <a href="Iterator" title="Iterator">iterators</a>. It treats the iterators as a reference to an array, and uses the array-to-heap conversion. It also provides the container adaptor priority_queue, which wraps these facilities in a container-like class. However, there is no standard support for the replace, sift-up/sift-down, or decrease/increase-key operations.</li>

<li>The <a href="Boost_(C%2B%2B_libraries)" title="Boost (C++ libraries)">Boost C++ libraries</a> include a heaps library. Unlike the STL, it supports decrease and increase operations, and supports additional types of heap: specifically, it supports d-ary, binomial, Fibonacci, pairing and skew heaps.</li>

<li>There is a <a rel="nofollow" class="external text" href="https://github.com/valyala/gheap">generic heap implementation</a> for <a href="C_(programming_language)" title="C (programming language)">C</a> and <a href="C%2B%2B" title="C++">C++</a> with <a href="D-ary_heap" title="D-ary heap">D-ary heap</a> and <a href="B-heap" title="B-heap">B-heap</a> support. It provides an STL-like API.</li>

<li>The standard library of the <a href="D_(programming_language)" title="D (programming language)">D programming language</a> includes <a rel="nofollow" class="external text" href="https://dlang.org/phobos/std_container_binaryheap.html">std.container.BinaryHeap</a>, which is implemented in terms of D's <a rel="nofollow" class="external text" href="https://tour.dlang.org/tour/en/basics/ranges">ranges</a>. Instances can be constructed from any <a rel="nofollow" class="external text" href="https://dlang.org/phobos/std_range_primitives.html#isRandomAccessRange">random-access range</a>. BinaryHeap exposes an <a rel="nofollow" class="external text" href="https://dlang.org/phobos/std_range_primitives.html#isInputRange">input range interface</a> that allows iteration with D's built-in foreach statements and integration with the range-based API of the <a rel="nofollow" class="external text" href="https://dlang.org/phobos/std_algorithm.html">std.algorithm package</a>.</li>

<li>For <a href="Haskell" title="Haskell">Haskell</a> there is the <a rel="nofollow" class="external text" href="https://hackage.haskell.org/package/heaps">Data.Heap</a> module.</li>

<li>The <a href="Java_(programming_language)" title="Java (programming language)">Java</a> platform (since version 1.5) provides a binary heap implementation with the class <code><a rel="nofollow" class="external text" href="https://docs.oracle.com/en/java/javase/24/docs/api/java.base/java/util/PriorityQueue.html">java.util.PriorityQueue</a></code> in the <a href="Java_Collections_Framework" class="mw-redirect" title="Java Collections Framework">Java Collections Framework</a>. This class implements by default a min-heap; to implement a max-heap, programmer should write a custom comparator. There is no support for the replace, sift-up/sift-down, or decrease/increase-key operations.</li>

<li><a href="Python_(programming_language)" title="Python (programming language)">Python</a> has a <a rel="nofollow" class="external text" href="https://docs.python.org/library/heapq.html">heapq</a> module that implements a priority queue using a binary heap. The library exposes a heapreplace function to support k-way merging. Python only supports a min-heap implementation.</li>

<li><a href="PHP" title="PHP">PHP</a> has both max-heap (SplMaxHeap) and min-heap (SplMinHeap) as of version 5.3 in the Standard PHP Library.</li>

<li><a href="Perl" title="Perl">Perl</a> has implementations of binary, binomial, and Fibonacci heaps in the <a rel="nofollow" class="external text" href="https://metacpan.org/module/Heap">Heap</a> distribution available on <a href="CPAN" title="CPAN">CPAN</a>.</li>

<li>The <a href="Go_(programming_language)" title="Go (programming language)">Go</a> language contains a <a rel="nofollow" class="external text" href="https://golang.org/pkg/container/heap/">heap</a> package with heap algorithms that operate on an arbitrary type that satisfies a given interface. That package does not support the replace, sift-up/sift-down, or decrease/increase-key operations.</li>

<li>Apple's <a href="Core_Foundation" title="Core Foundation">Core Foundation</a> library contains a <a rel="nofollow" class="external text" href="https://developer.apple.com/documentation/corefoundation/cfbinaryheap">CFBinaryHeap</a> structure.</li>

<li><a href="Pharo" title="Pharo">Pharo</a> has an implementation of a heap in the Collections-Sequenceable package along with a set of test cases. A heap is used in the implementation of the timer event loop.</li>

<li>The <a href="Rust_(programming_language)" title="Rust (programming language)">Rust</a> programming language has a binary max-heap implementation, <a rel="nofollow" class="external text" href="https://doc.rust-lang.org/std/collections/struct.BinaryHeap.html">BinaryHeap</a>, in the collections module of its standard library.</li>

<li><a href=".NET" title=".NET">.NET</a> has <a rel="nofollow" class="external text" href="https://docs.microsoft.com/dotnet/api/system.collections.generic.priorityqueue-2">PriorityQueue</a> class which uses quaternary (d-ary) min-heap implementation. It is available from .NET 6.</li></ul>

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

<li><a href="Search_data_structure" title="Search data structure">Search data structure</a></li>

<li><a href="Treap" title="Treap">Treap</a>, a form of binary search tree based on heap-ordered trees</li></ul>

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

<li id="cite_note-1"><a href="#cite_ref-1">^</a> Black (ed.), Paul E. (2004-12-14). Entry for heap in <a href="Dictionary_of_Algorithms_and_Data_Structures" class="mw-redirect" title="Dictionary of Algorithms and Data Structures">Dictionary of Algorithms and Data Structures</a>. Online version. U.S. <a href="National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a>, 14 December 2004. Retrieved on 2017-10-08 from <a rel="nofollow" class="external free" href="https://xlinux.nist.gov/dads/HTML/heap.html">https://xlinux.nist.gov/dads/HTML/heap.html</a>.

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<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite id="CITEREFBrodalLagogiannisTarjan2012" class="citation conference cs1"><a href="Gerth_St%C3%B8lting_Brodal" class="mw-redirect" title="Gerth Stølting Brodal">Brodal, Gerth Stølting</a>; Lagogiannis, George; <a href="Robert_Tarjan" title="Robert Tarjan">Tarjan, Robert E.</a> (2012). <a rel="nofollow" class="external text" href="http://www.cs.au.dk/~gerth/papers/stoc12.pdf">Strict Fibonacci heaps</a> (PDF). Proceedings of the 44th symposium on Theory of Computing - STOC '12. pp. 1177–1184. <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=10.1.1.233.1740">10.1.1.233.1740</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F2213977.2214082">10.1145/2213977.2214082</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4503-1245-5</bdi>.</cite>

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite id="CITEREFBrodal1996" class="citation cs2"><a href="Gerth_St%C3%B8lting_Brodal" class="mw-redirect" title="Gerth Stølting Brodal">Brodal, Gerth S.</a> (1996), <a rel="nofollow" class="external text" href="http://www.cs.au.dk/~gerth/papers/soda96.pdf">"Worst-Case Efficient Priority Queues"</a> (PDF), Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58</cite>

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite id="CITEREFGoodrichTamassia2004" class="citation book cs1"><a href="Michael_T._Goodrich" title="Michael T. Goodrich">Goodrich, Michael T.</a>; <a href="Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a> (2004). "7.3.6. Bottom-Up Heap Construction". Data Structures and Algorithms in Java (3rd ed.). pp. 338–341. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-46983-1</bdi>.</cite>

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite id="CITEREFFrederickson1993" class="citation cs2">Frederickson, Greg N. (1993), "An Optimal Algorithm for Selection in a Min-Heap", <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121203045606/http://ftp.cs.purdue.edu/research/technical_reports/1991/TR%2091-027.pdf">Information and Computation</a> (PDF), vol. 104, Academic Press, pp. 197–214, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Finco.1993.1030">10.1006/inco.1993.1030</a>, archived from <a rel="nofollow" class="external text" href="http://ftp.cs.purdue.edu/research/technical_reports/1991/TR%2091-027.pdf">the original</a> (PDF) on 2012-12-03, retrieved 2010-10-31</cite>

</li>

</ol></div></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 <a href="https://commons.wikimedia.org/wiki/Category:Heap_data_structures" class="extiw external" title="commons:Category:Heap data structures">Heap data structures</a>.</div></div>

</div>

<div class="side-box-text plainlist">The Wikibook <a href="https://en.wikibooks.org/wiki/Data_Structures" class="extiw external" title="wikibooks:Data Structures">Data Structures</a> has a page on the topic of: <a href="https://en.wikibooks.org/wiki/Data_Structures/Min_and_Max_Heaps" class="extiw external" title="wikibooks:Data Structures/Min and Max Heaps">Min and Max Heaps</a></div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Heap.html">Heap</a> at Wolfram MathWorld</li>

<li><a rel="nofollow" class="external text" href="https://www.cs.auckland.ac.nz/software/AlgAnim/heaps.html">Explanation</a> of how the basic heap algorithms work</li>

<li><cite id="CITEREFBentley2000" class="citation book cs1">Bentley, Jon Louis (2000). Programming Pearls (2nd ed.). Addison Wesley. pp. 147–162. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0201657880</bdi>.</cite></li></ul>

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</style><div id="Tree_data_structures246" style="font-size:114%;margin:0 4em"><a href="Tree_(abstract_data_type)" title="Tree (abstract data type)">Tree data structures</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Search_tree" title="Search tree">Search trees</a> (<a href="Set_(abstract_data_type)" title="Set (abstract data type)">dynamic sets</a>, <a href="Associative_array" title="Associative array">associative arrays</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="K-D-B-tree" title="K-D-B-tree">K-Dimensional</a></li></ul></li>

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

<ul><li><a href="Optimal_binary_search_tree" title="Optimal binary search tree">Optimal</a></li>

<li><a href="Self-balancing_binary_search_tree" title="Self-balancing binary search tree">Self-balancing</a></li></ul></li>

<li><a href="Dancing_tree" title="Dancing tree">Dancing</a></li>

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

<li><a href="Interval_tree" title="Interval tree">Interval</a></li>

<li><a href="Order_statistic_tree" title="Order statistic tree">Order statistic</a></li>

<li><a href="Palindrome_tree" title="Palindrome tree">Palindrome</a></li>

<li>(<a href="Left-leaning_red%E2%80%93black_tree" title="Left-leaning red–black tree">Left-leaning</a>) <a href="Red%E2%80%93black_tree" title="Red–black tree">Red–black</a></li>

<li><a href="Scapegoat_tree" title="Scapegoat tree">Scapegoat</a></li>

<li><a href="Splay_tree" title="Splay tree">Splay</a></li>

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

<li><a href="Weight-balanced_tree" title="Weight-balanced tree">Weight-balanced</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"></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="Binary_heap" title="Binary heap">Binary</a></li>

<li><a href="Binomial_heap" title="Binomial heap">Binomial</a></li>

<li><a href="Brodal_queue" title="Brodal queue">Brodal</a></li>

<li><a href="Fibonacci_heap" title="Fibonacci heap">Fibonacci</a></li>

<li><a href="Leftist_tree" title="Leftist tree">Leftist</a></li>

<li><a href="Pairing_heap" title="Pairing heap">Pairing</a></li>

<li><a href="Skew_binomial_heap" title="Skew binomial heap">Skew binomial</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Trie" title="Trie">Tries</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="Ctrie" title="Ctrie">Ctrie</a></li>

<li><a href="C-trie" title="C-trie">C-trie</a> (compressed ADT)</li>

<li><a href="Radix_tree" title="Radix tree">Radix</a></li>

<li><a href="Suffix_tree" title="Suffix tree">Suffix</a></li>

<li><a href="Ternary_search_tree" title="Ternary search tree">Ternary search</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Spatial_index" class="mw-redirect" title="Spatial index">Spatial</a> data partitioning trees</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Cartesian_tree" title="Cartesian tree">Cartesian</a></li>

<li><a href="Hilbert_R-tree" title="Hilbert R-tree">Hilbert R</a></li>

<li><a href="K-d_tree" title="K-d tree">k-d</a> (<a href="Implicit_k-d_tree" title="Implicit k-d tree">implicit k-d</a>)</li>

<li><a href="Metric_tree" title="Metric tree">Metric</a></li>

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

<li><a href="Priority_R-tree" title="Priority R-tree">Priority R</a></li>

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other trees</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="Cover_tree" title="Cover tree">Cover</a></li>

<li><a href="Exponential_tree" title="Exponential tree">Exponential</a></li>

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

<li><a href="Finger_tree" title="Finger tree">Finger</a></li>

<li><a href="Fractal_tree_index" title="Fractal tree index">Fractal index</a></li>

<li><a href="Fusion_tree" title="Fusion tree">Fusion</a></li>

<li><a href="Hash_calendar" title="Hash calendar">Hash calendar</a></li>

<li><a href="IDistance" title="IDistance">iDistance</a></li>

<li><a href="Left-child_right-sibling_binary_tree" title="Left-child right-sibling binary tree">Left-child right-sibling</a></li>

<li><a href="Log-structured_merge-tree" title="Log-structured merge-tree">Log-structured merge</a></li>

<li><a href="Merkle_tree" title="Merkle tree">Merkle</a></li>

<li><a href="Range_tree" title="Range tree">Range</a></li>

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

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Data_structures216" 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_structures216" style="font-size:114%;margin:0 4em"><a href="Data_structure" title="Data structure">Data structures</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types</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="Collection_(abstract_data_type)" title="Collection (abstract data type)">Collection</a></li>

<li><a href="Container_(abstract_data_type)" title="Container (abstract data type)">Container</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Abstract_data_type" title="Abstract data type">Abstract</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="Associative_array" title="Associative array">Associative array</a>

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

<li><a href="Retrieval_Data_Structure" title="Retrieval Data Structure">Retrieval Data Structure</a></li></ul></li>

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

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

<ul><li><a href="Double-ended_queue" title="Double-ended queue">Double-ended queue</a></li></ul></li>

<li><a href="Priority_queue" title="Priority queue">Priority queue</a>

<ul><li><a href="Double-ended_priority_queue" title="Double-ended priority queue">Double-ended priority queue</a></li></ul></li>

<ul><li><a href="Set_(abstract_data_type)#Multiset" title="Set (abstract data type)">Multiset</a></li>

<li><a href="Disjoint-set_data_structure" title="Disjoint-set data structure">Disjoint-set</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Array_(data_structure)" title="Array (data structure)">Arrays</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="Bit_array" title="Bit array">Bit array</a></li>

<li><a href="Circular_buffer" title="Circular buffer">Circular buffer</a></li>

<li><a href="Dynamic_array" title="Dynamic array">Dynamic array</a></li>

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

<li><a href="Hashed_array_tree" title="Hashed array tree">Hashed array tree</a></li>

<li><a href="Sparse_matrix" title="Sparse matrix">Sparse matrix</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Linked_data_structure" title="Linked data structure">Linked</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="Association_list" title="Association list">Association list</a></li>

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

<li><a href="Unrolled_linked_list" title="Unrolled linked list">Unrolled linked list</a></li>

<li><a href="XOR_linked_list" title="XOR linked list">XOR linked list</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Tree_(data_structure)" class="mw-redirect" title="Tree (data structure)">Trees</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

<li><a href="Red%E2%80%93black_tree" title="Red–black tree">Red–black tree</a></li>

<li><a href="Self-balancing_binary_search_tree" title="Self-balancing binary search tree">Self-balancing tree</a></li>

<li><a href="Splay_tree" title="Splay tree">Splay tree</a></li></ul></li>

<li>

<ul><li><a href="Binary_heap" title="Binary heap">Binary heap</a></li>

<li><a href="Binomial_heap" title="Binomial heap">Binomial heap</a></li>

<li><a href="Fibonacci_heap" title="Fibonacci heap">Fibonacci heap</a></li></ul></li>

<li><a href="Hilbert_R-tree" title="Hilbert R-tree">Hilbert R-tree</a></li></ul></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Graph_(abstract_data_type)" title="Graph (abstract data type)">Graphs</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="Binary_decision_diagram" title="Binary decision diagram">Binary decision diagram</a></li>

<li><a href="Directed_acyclic_graph" title="Directed acyclic graph">Directed acyclic graph</a></li>

<li><a href="Deterministic_acyclic_finite_state_automaton" title="Deterministic acyclic finite state automaton">Directed acyclic word graph</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></ul>

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

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