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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 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.
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.
Figure 4. Fibonacci heap from Figure 1 after decreasing key of node 9 to 0. If decreasing the key of a node causes it to become smaller than its parent, then it is cut from its parent, becoming a new unmarked root. If it is also less than the minimum key, then the minimum pointer is updated.
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.
Min-reduce the local solutions to find the vertex v having the minimum possible value of C[v] (global solution). Broadcast the selected node to every processor. Add v to F and, if E[v] is not the special flag value, also add E[v] to F. On every processor: update and as in the sequential algorithm.
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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.