heaps using -heaps we are able to obtain improved running times for several network optimization algorithms -"fibonacci heaps and their uses, fredman and tarjan heap is specialized binary tree specificallyit is complete binary tree as defined on page the keys must satisfy the heap property--the key at each node is at least as great as the keys stored at its children see figure (afor an example of max-heap max-heap can be implemented as an arraythe children of the node at index are at indices and the array representation for the max-heap in figure (ais max-heap supports (log ninsertionso( time lookup for the max elementand (log ndeletion of the max element the extract-max operation is defined to delete and return the maximum element see figure (bfor an example of deletion of the max element searching for arbitrary keys has (ntime complexity heap is sometimes referred to as priority queue because it behaves like queuewith one differenceeach element has "priorityassociated with itand deletion removes the element with the highest priority the min-heap is completely symmetric version of the data structure and supports ( time lookups for the minimum element (aa max-heap note that the root hold the maximum key (bafter the deletion of max of the heap in (adeletion is performed by replacing the root' key with the key at the last leaf and then recovering the heap property by repeatedly exchanging keys with children figure max-heap and deletion on that max-heap