# Heap (data structure)

For the memory heap (in low-level computer programming), see C dynamic memory allocation. Heap (data structure)_sentence_0

In computer science, a heap is a specialized tree-based data structure which is essentially an almost complete tree that satisfies the heap property: in a max heap, for any given node C, if P is a 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. The node at the "top" of the heap (with no parents) is called the root node. Heap (data structure)_sentence_1

The heap is one maximally efficient implementation of an abstract data type called a priority queue, and in fact, priority queues are often referred to as "heaps", regardless of how they may be implemented. Heap (data structure)_sentence_2

In a heap, the highest (or lowest) priority element is always stored at the root. Heap (data structure)_sentence_3

However, a heap is not a sorted structure; it can be regarded as being partially ordered. Heap (data structure)_sentence_4

A heap is a useful data structure when it is necessary to repeatedly remove the object with the highest (or lowest) priority. Heap (data structure)_sentence_5

A common implementation of a heap is the binary heap, in which the tree is a binary tree (see figure). Heap (data structure)_sentence_6

The heap data structure, specifically the binary heap, was introduced by J. Heap (data structure)_sentence_7 W. J. Williams in 1964, as a data structure for the heapsort sorting algorithm. Heap (data structure)_sentence_8

Heaps are also crucial in several efficient graph algorithms such as Dijkstra's algorithm. Heap (data structure)_sentence_9

When a heap is a complete binary tree, it has a smallest possible height—a heap with N nodes and for each node a branches always has loga N height. Heap (data structure)_sentence_10

Note that, as shown in the graphic, there is no implied ordering between siblings or cousins and no implied sequence for an in-order traversal (as there would be in, e.g., a binary search tree). Heap (data structure)_sentence_11

The heap relation mentioned above applies only between nodes and their parents, grandparents, etc. Heap (data structure)_sentence_12

The maximum number of children each node can have depends on the type of heap. Heap (data structure)_sentence_13

## Operations Heap (data structure)_section_0

The common operations involving heaps are: Heap (data structure)_sentence_14

Heap (data structure)_description_list_0

Heap (data structure)_unordered_list_1

• 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. peek)Heap (data structure)_item_1_0
• insert: adding a new key to the heap (a.k.a., push)Heap (data structure)_item_1_1
• 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)Heap (data structure)_item_1_2
• delete-max (or delete-min): removing the root node of a max heap (or min heap), respectivelyHeap (data structure)_item_1_3
• replace: pop root and push a new key. More efficient than pop followed by push, since only need to balance once, not twice, and appropriate for fixed-size heaps.Heap (data structure)_item_1_4

Heap (data structure)_description_list_2

Heap (data structure)_unordered_list_3

• create-heap: create an empty heapHeap (data structure)_item_3_5
• heapify: create a heap out of given array of elementsHeap (data structure)_item_3_6
• merge (union): joining two heaps to form a valid new heap containing all the elements of both, preserving the original heaps.Heap (data structure)_item_3_7
• meld: joining two heaps to form a valid new heap containing all the elements of both, destroying the original heaps.Heap (data structure)_item_3_8

Heap (data structure)_description_list_4

Heap (data structure)_unordered_list_5

• size: return the number of items in the heap.Heap (data structure)_item_5_9
• is-empty: return true if the heap is empty, false otherwise.Heap (data structure)_item_5_10

Heap (data structure)_description_list_6

Heap (data structure)_unordered_list_7

• increase-key or decrease-key: updating a key within a max- or min-heap, respectivelyHeap (data structure)_item_7_11
• delete: delete an arbitrary node (followed by moving last node and sifting to maintain heap)Heap (data structure)_item_7_12
• 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 sieve.Heap (data structure)_item_7_13
• sift-down: move a node down in the tree, similar to sift-up; used to restore heap condition after deletion or replacement.Heap (data structure)_item_7_14

## Implementation Heap (data structure)_section_1

Heaps are usually implemented with an implicit heap data structure, which is an implicit data structure consisting of an array (fixed size or dynamic array) where each element represents a tree node whose parent/children relationship is defined implicitly by their index. Heap (data structure)_sentence_15

After an element is inserted into or deleted from a heap, the heap property may be violated and the heap must be balanced by swapping elements within the array. Heap (data structure)_sentence_16

In an implicit heap data structure, the first (or last) element will contain the root. Heap (data structure)_sentence_17

The next two elements of the array contain its children. Heap (data structure)_sentence_18

The next four contain the four children of the two child nodes, etc. Heap (data structure)_sentence_19

Thus the children of the node at position n would be at positions 2n and 2n + 1 in a one-based array, or 2n + 1 and 2n + 2 in a zero-based array. Heap (data structure)_sentence_20

(If the initial element has index 0, then there are n nodes positioned before the node with index n. Each of these has two children. Heap (data structure)_sentence_21

Apart from the root node, these children are all of the nodes positioned before the children of node n. Their number is 2n.) Heap (data structure)_sentence_22

Computing the index of the parent node of n-th element is also straightforward. Heap (data structure)_sentence_23

For one-based arrays the parent of element n is located at position n/2. Heap (data structure)_sentence_24

Similarly, for zero-based arrays, the parent is located at position (n-1)/2 (floored). Heap (data structure)_sentence_25

This allows moving up or down the tree by doing simple index computations. Heap (data structure)_sentence_26

Balancing a heap is done by sift-up or sift-down operations (swapping elements which are out of order). Heap (data structure)_sentence_27

As we can build a heap from an array without requiring extra memory (for the nodes, for example), heapsort can be used to sort an array in-place. Heap (data structure)_sentence_28

Different types of heaps implement the operations in different ways, but notably, insertion is often done by adding the new element at the end of the heap in the first available free space. Heap (data structure)_sentence_29

This will generally violate the heap property, and so the elements are then shifted up until the heap property has been reestablished. Heap (data structure)_sentence_30

Similarly, deleting the root is done by removing the root and then putting the last element in the root and sifting down to rebalance. Heap (data structure)_sentence_31

Thus replacing is done by deleting the root and putting the new element in the root and sifting down, avoiding a sifting up step compared to pop (sift down of last element) followed by push (sift up of new element). Heap (data structure)_sentence_32

Construction of a binary (or d-ary) heap out of a given array of elements may be performed in linear time using the classic Floyd algorithm, 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. This is faster than a sequence of consecutive insertions into an originally empty heap, which is log-linear. Heap (data structure)_sentence_33

## Comparison of theoretic bounds for variants Heap (data structure)_section_3

Here are time complexities of various heap data structures. Heap (data structure)_sentence_34

Function names assume a max-heap. Heap (data structure)_sentence_35

For the meaning of "O(f)" and "Θ(f)" see Big O notation. Heap (data structure)_sentence_36

Heap (data structure)_table_general_0

BinaryHeap (data structure)_header_cell_0_1_0 Θ(1)Heap (data structure)_cell_0_1_1 Θ(log n)Heap (data structure)_cell_0_1_2 O(log n)Heap (data structure)_cell_0_1_3 O(log n)Heap (data structure)_cell_0_1_4 Θ(n)Heap (data structure)_cell_0_1_5
LeftistHeap (data structure)_header_cell_0_2_0 Θ(1)Heap (data structure)_cell_0_2_1 Θ(log n)Heap (data structure)_cell_0_2_2 Θ(log n)Heap (data structure)_cell_0_2_3 O(log n)Heap (data structure)_cell_0_2_4 Θ(log n)Heap (data structure)_cell_0_2_5
BinomialHeap (data structure)_header_cell_0_3_0 Θ(1)Heap (data structure)_cell_0_3_1 Θ(log n)Heap (data structure)_cell_0_3_2 Θ(1)Heap (data structure)_cell_0_3_3 Θ(log n)Heap (data structure)_cell_0_3_4 O(log n)Heap (data structure)_cell_0_3_5
FibonacciHeap (data structure)_header_cell_0_4_0 Θ(1)Heap (data structure)_cell_0_4_1 O(log n)Heap (data structure)_cell_0_4_2 Θ(1)Heap (data structure)_cell_0_4_3 Θ(1)Heap (data structure)_cell_0_4_4 Θ(1)Heap (data structure)_cell_0_4_5
PairingHeap (data structure)_header_cell_0_5_0 Θ(1)Heap (data structure)_cell_0_5_1 O(log n)Heap (data structure)_cell_0_5_2 Θ(1)Heap (data structure)_cell_0_5_3 o(log n)Heap (data structure)_cell_0_5_4 Θ(1)Heap (data structure)_cell_0_5_5
BrodalHeap (data structure)_header_cell_0_6_0 Θ(1)Heap (data structure)_cell_0_6_1 O(log n)Heap (data structure)_cell_0_6_2 Θ(1)Heap (data structure)_cell_0_6_3 Θ(1)Heap (data structure)_cell_0_6_4 Θ(1)Heap (data structure)_cell_0_6_5
Rank-pairingHeap (data structure)_header_cell_0_7_0 Θ(1)Heap (data structure)_cell_0_7_1 O(log n)Heap (data structure)_cell_0_7_2 Θ(1)Heap (data structure)_cell_0_7_3 Θ(1)Heap (data structure)_cell_0_7_4 Θ(1)Heap (data structure)_cell_0_7_5
Strict FibonacciHeap (data structure)_header_cell_0_8_0 Θ(1)Heap (data structure)_cell_0_8_1 O(log n)Heap (data structure)_cell_0_8_2 Θ(1)Heap (data structure)_cell_0_8_3 Θ(1)Heap (data structure)_cell_0_8_4 Θ(1)Heap (data structure)_cell_0_8_5
2–3 heapHeap (data structure)_header_cell_0_9_0 O(log n)Heap (data structure)_cell_0_9_1 O(log n)Heap (data structure)_cell_0_9_2 O(log n)Heap (data structure)_cell_0_9_3 Θ(1)Heap (data structure)_cell_0_9_4 ?Heap (data structure)_cell_0_9_5

## Applications Heap (data structure)_section_4

The heap data structure has many applications. Heap (data structure)_sentence_37

Heap (data structure)_unordered_list_8

• Heapsort: One of the best sorting methods being in-place and with no quadratic worst-case scenarios.Heap (data structure)_item_8_15
• Selection algorithms: 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.Heap (data structure)_item_8_16
• Graph algorithms: By using heaps as internal traversal data structures, run time will be reduced by polynomial order. Examples of such problems are Prim's minimal-spanning-tree algorithm and Dijkstra's shortest-path algorithm.Heap (data structure)_item_8_17
• Priority Queue: 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.Heap (data structure)_item_8_18
• K-way merge: 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.)Heap (data structure)_item_8_19
• Order statistics: The Heap data structure can be used to efficiently find the kth smallest (or largest) element in an array.Heap (data structure)_item_8_20

## Implementations Heap (data structure)_section_5

Heap (data structure)_unordered_list_9

• The C++ Standard Library provides the make_heap, push_heap and pop_heap algorithms for heaps (usually implemented as binary heaps), which operate on arbitrary random access iterators. 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.Heap (data structure)_item_9_21
• The Boost C++ libraries 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.Heap (data structure)_item_9_22
• There is a for C and C++ with D-ary heap and B-heap support. It provides an STL-like API.Heap (data structure)_item_9_23
• The standard library of the D programming language includes , which is implemented in terms of D's . Instances can be constructed from any . BinaryHeap exposes an that allows iteration with D's built-in foreach statements and integration with the range-based API of the .Heap (data structure)_item_9_24
• The Java platform (since version 1.5) provides a binary heap implementation with the class in the Java Collections Framework. 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.Heap (data structure)_item_9_25
• Python has a module that implements a priority queue using a binary heap. The library exposes a heapreplace function to support k-way merging.Heap (data structure)_item_9_26
• PHP has both max-heap (SplMaxHeap) and min-heap (SplMinHeap) as of version 5.3 in the Standard PHP Library.Heap (data structure)_item_9_27
• Perl has implementations of binary, binomial, and Fibonacci heaps in the distribution available on CPAN.Heap (data structure)_item_9_28
• The Go language contains 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.Heap (data structure)_item_9_29
• Apple's Core Foundation library contains a structure.Heap (data structure)_item_9_30
• Pharo 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.Heap (data structure)_item_9_31
• The Rust programming language has a binary max-heap implementation, , in the collections module of its standard library.Heap (data structure)_item_9_32