Merging Priority Queues in Java

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Merging Priority Queues
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The order in which pairing heap subtrees are merged is importantThe simplest algorithm is two-pass merging
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Several alternatives have been proposed Most are indistinguishable, but using a single left-toright pass is a bad idea
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nodes that are children of the root Consequently, deleteMin is O(N) time The best that we can hope to do is to arrange the merges so that we do not have repeatedly expensive deleteMin operations The order in which pairing heap subtrees are merged is important The simplest and most practical of the many variants of doing so that have been proposed is two-pass merging, in which a first scan merges pairs of children from left to right* and then a second scan, right to left, is performed to complete the merging After the first scan, we have half as many trees to merge In the second scan, at each step, we merge the rightmost tree that remains from the first scan with the current merged result For example, if we have children c, through C8, the first scan performs the merges c , and c2,c3 and c,, c5 and C6, and c, and c8 The result is d,, d2, d3, and d, We perform the second pass by merging d, and d,; d, is then merged with that result, and d , is then merged with the result of that merge, completing the deleteMin operation Figure 236 shows the result of using deleteMin on the pairing heap shown in Figure 235 Other merging strategies are possible For instance, we can place each subtree (corresponding to a child) on a queue, repeatedly dequeue two trees, and then enqueue the result of merging them After c - 1 merges, only one tree remains on the queue, which is the result of the deleteMinHowever, using a stack instead of a queue is a disaster because the root of the resulting tree may possibly have c - 1 children If that occurs in a sequence, the deleteMin operation will have linear, rather than logarithmic, amortized cost per operation In Exercise 2312 you are asked to construct such a sequence
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The grev data member points to either a left sibling or a parent The insert routine returns a pointer to the new node for use by decreaseKey
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Implementation of the Pairing Heap
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The PairingHeap class interface is shown in Figure 237 The nested typedef Position,declared at line 28, is declared as equivalent to a pointer The standard priority queue operations are to a PairNode<Comparable> provided, with three main differences First, insert returns a PositionSecond, we add the decreaseKey method, which requires a P o s i t i o n as a parameter Third, the Big Three cannot safely be implemented with the normal recursive algorithm because the depth of the recursion may be too large The online code has an unsafe implementation, and in Exercise 2312 you are asked to improve it The basic node of a pairing heap, PairNode,is shown in Figure 238 and consists of an item and three pointers Two of these pointers are the left
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2 Care must be exercised if there are an odd number of children When that happens, we merge the last child with the result of the rightmost merge to complete the first scan
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The Pairing Heap
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Recombination of siblings after a deleteMinIn each merge, the larger root tree is made the left child of the smaller root tree: (a) the resulting trees; (b) after the first pass; (c) after the first merge of the second pass; (d) after the second merge of the second pass
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m p ~ e r g i n Priority Queues g
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template <class Comparable> class PairNode;
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