Heap (data structure) Totally Explained

Everything about Heap Data Structure totally explained

In computer science, a heap is a specialized tree-based data structure that satisfies the heap property: if B is a child node of A, then key(A) ≥ key(B). This implies that an element with the greatest key is always in the root node, and so such a heap is sometimes called a max heap. (Alternatively, if the comparison is reversed, the smallest element is always in the root node, which results in a min heap.) This is why heaps are used to implement priority queues. The efficiency of heap operations is crucial in several graph algorithms.
The operations commonly performed with a heap are

delete-max or delete-min: removing the root node of a max- or min-heap, respectively
increase-key or decrease-key: updating a key within a max- or min-heap, respectively
insert: adding a new key to the heap
merge: joining two heaps to form a valid new heap containing all the elements of both.

Heaps are used in the sorting algorithm heapsort.

Variants

Binary heap

D-ary heap

Soft heap

2-3 heap

Ternary heap

Treap

Beap

Skew heap

Comparison of theoretic bounds for variants

The following complexities are worst-case for binary and binomial heaps and amortized complexity for Fibonacci heap. O(f) gives asymptotic upper bound and Θ(f) is asymptotically tight bound (see Big O notation). Function names assume a min-heap. Binary

Binomial	Fibonacci
createHeap
findMin
deleteMin
insert
decreaseKey
merge

For pairing heaps the insert and merge operations are conjectured to be O(1) amortized complexity but this hasn't yet been proven. decreaseKey isn't O(1) amortized complexity (External Link

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Heap applications

Heaps are favorite data structures for many applications.

Heapsort: One of the best sorting methods being in-place and with no quadratic worst case scenarios.

Selection algorithms: Finding the min, max or both of them, median or even any k-th element in sublinear time can be done dynamically with heaps.

Graph algorithms: By using heaps as internal traversal data structures, run time will be reduced by an order of polynomial. Examples of such problems are Prim's minimal spanning tree algorithm and Dijkstra's shortest path problem. Interestingly, binary heaps may be represented using an array alone. The first (or last) 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. Balancing a heap is done by 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.
One more advantage of heaps over trees in some applications is that construction of heaps can be done in linear time using Tarjan's algorithm.

Heap implementations

The C++ Standard Template Library provides the make_heap, push_heap and pop_heap algorithms for 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 detailed above.Further Information

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