So we can walk up and down through the tree using simple arithmetic.Īdding a new node to the heap is done by adding the element at the end of the array to preserve the shape invariant. Conversely, the parent of a node at index i is found at index ⌊(i–1)/2⌋. The shape invariant guarantees that the children of the node at index i are found at indices 2i+1 (left) and 2i+2 (right). The nice thing about this representation is that it is possible to represent the tree structure without pointers. Note that the root of each subtree contains the highest-priority element in that subtree. Here is an example of a binary heap in which smaller values have higher priority. Priority element at the root, and the left and right subtrees are also both heaps. Equivalently still, a heap stores its highest Equivalently, the priority of any node is at least as high as (Heap Invariant) Every node n in the tree has the highest priority among all nodes in the This is not how we are using the term here.)Ī binary heap is a binary tree satisfying the heap invariant: Implementation knows where to place objects in memory. A memory heap is a low-level data structure used to keep track of the computer's memory so that the programming language (In computer science, the term heap is a bit overloaded.īinary heaps should not be confused with memory heaps. However, there is a simple concrete data structure called a binary heap that allows both operations to be done in O(log n) time. Whereas unordered lists allow constant-time add and linear time extractMin. Ordered lists allow constant-time extractMin and linear time add, It is straightforward to implement priority queues with ordered or unordered lists. This can be accomplished by using a hash table to look up the location of elements. To implement updatePriority(), an implementation must have a fast way to find the element The methods described in this interface suffice to implement Dijkstra's shortest path algorithm. Priority queues can alsoīe used for sorting, since elements to be sorted can be pushed into the priority queueĪ priority queue can be described via the following interface for a min-queue: Optimal way to compress individual symbols in a stream. Handling one event can generate new future events, which getĪnother use for priority queues is for the compression algorithm known as Used to store unprocessed events, where the priority is a timestamp indicating the time of The events need to be processed in the order in which they occur, thus a min-queue is There are also max-queues in which larger numbers correspond toĪnother application in which priority queues are very useful is event-driven simulation. Such a queue is called a min-queue because the smaller the distance, In these applications, the priority is the best-known estimate of a We have already seen that priority queues are useful for implementing Dijkstra's algorithmĪnd A * search. A priority queue is an abstraction with several important uses,Īnd it can be implemented efficiently, as we will see. That is, the next element to be removed from the queue is always the element of highest In which each element has a priority and from which elements are removed in priority order For Dijkstra's shortest-path algorithm, we needed a priority queue: a queue
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