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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.
The function Join on two AVL trees t 1 and t 2 and a key k will return a tree containing all elements in t 1, t 2 as well as k. It requires k to be greater than all keys in t 1 and smaller than all keys in t 2. If the two trees differ by height at most one, Join simply create a new node with left subtree t 1, root k and right subtree t 2.
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.
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This may be done by comparing the keys at the roots of the two trees (the smallest keys in both trees). The root node with the larger key is made into a child of the root node with the smaller key, increasing its order by one: [1] [3] function mergeTree(p, q) if p.root.key <= q.root.key return p.addSubTree(q) else return q.addSubTree(p)