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The decrease-key operation requires a reference to the node we wish to decrease the key of. However, the decrease-key operation itself sometimes swaps the key of a node and the key root. Assume that the insert operation returns some opaque reference that we can call decrease-key on, as part of the public API.
The decrease key operation replaces the value of a node with a given value with a lower value, and the increase key operation does the same but with a higher value. This involves finding the node with the given value, changing the value, and then down-heapifying or up-heapifying to restore the heap property. Decrease key can be done as follows:
A node is marked if at least one of its children was cut, since this node was made a child of another node (all roots are unmarked). The amortized time for an operation is given by the sum of the actual time and c {\displaystyle c} times the difference in potential, where c is a constant (chosen to match the constant factors in the big O ...
A pairing heap is either an empty heap, or a pairing tree consisting of a root element and a possibly empty list of pairing trees. The heap ordering property requires that parent of any node is no greater than the node itself. The following description assumes a purely functional heap that does not support the decrease-key operation.
Example of a binary max-heap with node keys being integers between 1 and 100. 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.
To delete the minimum element from the heap, first find this element, remove it from the root of its binomial tree, and obtain a list of its child subtrees (which are each themselves binomial trees, of distinct orders). Transform this list of subtrees into a separate binomial heap by reordering them from smallest to largest order.
In an O(k) preprocessing step the heap is created using the standard heapify procedure. Afterwards, the algorithm iteratively transfers the element that the root pointer points to, increases this pointer and executes the standard decrease key procedure upon the root element. The running time of the increase key procedure is bounded by O(log k).
The d-ary heap consists of an array of n items, each of which has a priority associated with it. These items may be viewed as the nodes in a complete d-ary tree, listed in breadth first traversal order: the item at position 0 of the array (using zero-based numbering) forms the root of the tree, the items at positions 1 through d are its children, the next d 2 items are its grandchildren, etc.