Min-heap

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.

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. 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 binary heap, in which the tree is a binary tree (see figure). The heap data structure, specifically the binary heap, was introduced by J. W. J. Williams in 1964, as a data structure for the heapsort sorting algorithm. Heaps are also crucial in several efficient graph algorithms such as Dijkstra's algorithm. When a heap is a complete binary tree, it has a smallest possible height—a heap with ''N'' nodes and ''a'' branches for each node always has log_''a'' ''N'' height.

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). The heap relation mentioned above applies only between nodes and their parents, grandparents, etc. The maximum number of children each node can have depends on the type of heap.

Operations

The common operations involving heaps are:
;Basic
* ''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'')
* ''insert'': adding a new key to the heap (a.k.a., ''push'')
* ''extract-max'' (or ''extract-min''): returns the node of maximum value from a max heap r minimum value from a min heapafter removing it from the heap (a.k.a., ''pop'')
* ''delete-max'' (or ''delete-min''): removing the root node of a max heap (or min heap), respectively
* ''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.

;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'' or ''decrease-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 a sieve.
* ''sift-down'': move a node down in the tree, similar to sift-up; used to restore heap condition after deletion or replacement.

Implementation

Heaps are usually implemented with an array, as follows:

* Each element in the array represents a node of the heap, and
* The parent / child relationship is defined implicitly by the elements' indices in the array.

For a binary 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 index , its children are at indices and , and its parent is at index . 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 type 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 the ''new'' element in the root and sift down. When compared to extraction followed by insertion, this avoids a sift up step.

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 2''N'' − 2''s''₂(''N'') − ''e''₂(''N'') (for a binary heap), where ''s''₂(''N'') is the sum of all digits of the binary representation of ''N'' and ''e''₂(''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.

Variants

* 2–3 heap
* B-heap
* Beap
* Binary heap
* Binomial heap
* Brodal queue
* ''d''-ary heap
* Fibonacci heap
* K-D Heap
* Leaf heap
* Leftist heap
* Pairing heap
* Radix heap
* Randomized meldable heap
* Skew heap
* Soft heap
* Ternary heap
* Treap
* Weak heap

Comparison of theoretic bounds for variants

Applications

The heap data structure has many applications.

* 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.
* 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.
*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.)
*Order statistics: The Heap data structure can be used to efficiently find the kth smallest (or largest) element in an array.

Programming language implementations

* The C++ Standard Library provides the , and 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 , 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.
* 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.
* There is
generic heap implementation
for C and C++ with D-ary heap and B-heap support. It provides an STL-like API.
* The standard library of the D programming language include

which is implemented in terms of D'
ranges
Instances can be constructed from an

exposes a

that allows iteration with D's built-in statements and integration with the range-based API of th


* For Haskell there is th

module.
* 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.
* Python has

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 () and min-heap () as of version 5.3 in the Standard PHP Library.
* Perl has implementations of binary, binomial, and Fibonacci heaps in th

distribution available on CPAN.
* The Go language contains

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.
* Apple's Core Foundation library contains

structure.
* 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.
* The Rust programming language has a binary max-heap implementation

in the module of its standard library.
* .NET ha
PriorityQueue
class which uses quarternary (d-ary) min-heap implementation. It is available from .NET 6.

See also

* Sorting algorithm
* Search data structure
* Stack (abstract data type)
* Queue (abstract data type)
* Tree (data structure)
* Treap, a form of binary search tree based on heap-ordered trees

References

External links

Heap
at Wolfram MathWorld

of how the basic heap algorithms work
*

{{DEFAULTSORT:Heap (Data Structure)