Heap (data structure)
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 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 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 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 priority.
A common implementation of a heap is the binary heap, in which the tree is a binary tree. 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 for each node a branches 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 in-order traversal. 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 : find a maximum item of a max-heap, or a minimum item of a min-heap, respectively
- insert: adding a new key to the heap
- extract-max : returns the node of maximum value from a max heap after removing it from the heap
- delete-max : removing the root node of a max 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.
- create-heap: create an empty heap
- heapify: create a heap out of given array of elements
- merge : 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.
- size: return the number of items in the heap.
- is-empty: return true if the heap is empty, false otherwise.
- increase-key or decrease-key: updating a key within a max- or min-heap, respectively
- delete: delete an arbitrary node
- shift-up: move a node up in the tree, as long as needed; used to restore heap condition after insertion.
- shift-down: move a node down in the tree, similar to shift-up; used to restore heap condition after deletion or replacement.
Implementation
In an implicit heap data structure, the first element will contain the root. The next two elements of the array contain its children. The next four contain the four children of the two child nodes, etc. 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. Computing the index of the parent node of n-th element is also straightforward. For one-based arrays the parent of element n is located at position n/2. Similarly, for zero-based arrays, is the parent is located at position /2. This allows moving up or down the tree by doing simple index computations. Balancing a heap is done by sift-up or sift-down operations. As we can build a heap from an array without requiring extra memory, heapsort can be used to sort an array in-place.
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. This will generally violate the heap property, and so the elements are then shifted up until the heap property has been reestablished. Similarly, deleting the root is done by removing the root and then putting the last element in the root and sifting down to rebalance. 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 followed by push.
Construction of a binary 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 − e2, where s2 is the sum of all digits of the binary representation of N and e2 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 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.
- Order statistics: The Heap data structure can be used to efficiently find the kth smallest element in an array.
Implementations
- The C++ Standard Library provides the make_heap, push_heap and pop_heap algorithms for 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.
- 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 a 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 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 .
- The Java platform 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 a 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 the distribution available on CPAN.
- 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.
- Apple's Core Foundation library contains a 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 collections module of its standard library.