Example of abinary max-heap with node keys being integers between 1 and 100
Incomputer science, aheap is atree-baseddata structure that satisfies theheap property: In amax heap, for any givennode C, if P is the parent node of C, then thekey (thevalue) of P is greater than or equal to the key of C. In amin heap, the key of P is less than or equal to the key of C.[1] The node at the "top" of the heap (with no parents) is called theroot node.
The heap is one maximally efficient implementation of anabstract data type called apriority queue, 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 thebinary heap, in which the tree is acomplete[2] binary tree (see figure). The heap data structure, specifically the binary heap, was introduced byJ. W. J. Williams in 1964, as a data structure for theheapsort sorting algorithm.[3] Heaps are also crucial in several efficientgraph algorithms such asDijkstra's algorithm. When a heap is a complete binary tree, it has the smallest possible height—a heap withN nodes anda branches for each node always has logaN height.
Note that, as shown in the graphic, there is no implied ordering between siblings or cousins and no implied sequence for anin-order traversal (as there would be in, e.g., abinary search tree). 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 asradix trees in that they require no additional memory beyond that used for storing the keys.
find-max (orfind-min): find a maximum item of a max-heap, or a minimum item of a min-heap, respectively (a.k.a.peek)
insert: adding a new key to the heap (a.k.a.,push[4])
extract-max (orextract-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[5])
delete-max (ordelete-min): removing the root node of a max heap (or min heap), respectively
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.[6]
Creation
create-heap: create an empty heap
heapify: create a heap out of given array of elements
merge (union): joining two heaps to form a valid new heap containing all the elements of both, preserving the original heaps.
meld: joining two heaps to form a valid new heap containing all the elements of both, destroying the original heaps.
Inspection
size: return the number of items in the heap.
is-empty: return true if the heap is empty, false otherwise.
Internal
increase-key ordecrease-key: updating a key within a max- or min-heap, respectively
delete: delete an arbitrary node (followed by moving last node and sifting to maintain heap)
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 asieve.
sift-down: move a node down in the tree, similar to sift-up; used to restore heap condition after deletion or replacement.
Heaps are usually implemented with anarray, as follows:
Each element in the array represents a node of the heap, and
The parent / child relationship isdefined implicitly by the elements' indices in the array.
Example of a complete binary max-heap with node keys being integers from 1 to 100 and how it would be stored in an array.
For abinary heap, 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 indexi, its children are at indices and, 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),heapsort 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:
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.
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.
Replacement: Remove the root and put thenew element in the root and sift down. When compared to extraction followed by insertion, this avoids a sift up step.
Construction of a binary (ord-ary) heap out of a given array of elements may be performed in linear time using the classicFloyd algorithm, with the worst-case number of comparisons equal to 2N − 2s2(N) −e2(N) (for a binary heap), wheres2(N) is the sum of all digits of the binary representation ofN ande2(N) is the exponent of 2 in the prime factorization ofN.[7] This is faster than a sequence of consecutive insertions into an originally empty heap, which is log-linear.[a]
^Each insertion takes O(log(k)) in the existing size of the heap, thus. Since, a constant factor (half) of these insertions are within a constant factor of the maximum, so asymptotically we can assume; formally the time is. This can also be readily seen fromStirling's approximation.
^make-heap is the operation of building a heap from a sequence ofn unsorted elements. It can be done inΘ(n) time whenevermeld runs inO(log n) time (where both complexities can be amortized).[9][10] Another algorithm achievesΘ(n) for binary heaps.[11]
^abcForpersistent heaps (not supportingincrease-key), a generic transformation reduces the cost ofmeld to that ofinsert, while the new cost ofdelete-max is the sum of the old costs ofdelete-max andmeld.[14] Here, it makesmeld run inΘ(1) time (amortized, if the cost ofinsert is) whiledelete-max still runs inO(log n). Applied to skew binomial heaps, it yields Brodal-Okasaki queues, persistent heaps with optimal worst-case complexities.[13]
^abBrodal 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 apersistent data structure achieving the same optimum, except thatincrease-key is not supported.
Heapsort: One of the best sorting methods being in-place and with no quadratic worst-case scenarios.
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.[24]
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.
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.)
TheC++ Standard Library provides themake_heap,push_heap andpop_heap algorithms for heaps (usually implemented as binary heaps), which operate on arbitrary random accessiterators. It treats the iterators as a reference to an array, and uses the array-to-heap conversion. It also provides the classstd::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.
TheBoost C++ libraries include a heaps library. Unlike the STL, it supports decrease and increase operations, and supports additional types of heap: specifically, it supportsd-ary, binomial, Fibonacci, pairing and skew heaps.
TheJava platform (since version 1.5) provides a binary heap implementation with the classjava.util.PriorityQueue in theJava 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.
Python has aheapq module that implements a priority queue using a binary heap. The library exposes a heapreplace function to support k-way merging.
PHP has both max-heap (SplMaxHeap) and min-heap (SplMinHeap) as of version 5.3 in the Standard PHP Library.
Perl has implementations of binary, binomial, and Fibonacci heaps in theHeap distribution available onCPAN.
TheGo language contains aheap 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.
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.
TheRust programming language has a binary max-heap implementation,BinaryHeap, in thecollections module of its standard library.
.NET hasPriorityQueue class which uses quaternary (d-ary) min-heap implementation. It is available from .NET 6.
^CORMEN, THOMAS H. (2009).INTRODUCTION TO ALGORITHMS. United States of America: The MIT Press Cambridge, Massachusetts London, England. pp. 151–152.ISBN978-0-262-03384-8.
