In computer science, a heap is a tree-based data structure that satisfies the heap property: In a max heap, for any given node C, if P is the 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 complete 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 the smallest possible height—a heap with N nodes and a branches for each node 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 (as there would be in, e.g., a binary 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 as radix trees in that they require no additional memory beyond that used for storing the keys.
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 [or minimum value from a min heap] after 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. 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. 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 using arrays 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 i, its children are at indices 2 i + 1 {\displaystyle 2i+1} and 2 i + 2 {\displaystyle 2i+2} , and its parent is at index ⌊(i−1)/2⌋ in an array starting from index 0 {\displaystyle 0} , or at 2 i {\displaystyle 2i} , 2 i + 1 {\displaystyle 2i+1} , and ⌊i/2⌋, respectively, in an array starting from 1 {\displaystyle 1} . This simple indexing scheme makes it efficient to walk "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:
… excerpt ends here. Continue reading the full article.
