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// than the currently stored value / / Throws BadArgumentException if newVal is not small enough
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6 template <class Comparable>
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7 void PairingHeap<Comparable>: :
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8 decreaseKey( Position p, const Comparable 9 I
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newVal
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if( p->element < newVal 1 throw ~adArgument~xception(; ) p->element = newVal; if ( p ! = root )
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if( p->nextsibling ! = NULL ) p->nextsibling->prev= p->prev; if( p->prev->leftchild== p ) p->prev->leftchild p->nextsibling; = else p->prev->nextSibling = p->nextsibling; p->nextsibling = NULL; compareAndLink( root, p 1 Figure 2312 The decreaseKey method for the PairingHeap class
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children may exist, doing so would be inefficient Thus we assume that it is always an error to attempt to increase the key by using the decreaseKey (In Exercise 239 you are asked to describe an algorithm for increase~ey) After performing this test, we lower the value in the node If the node is the root, we are done Otherwise we splice the node out of the list of children that it is in, using the code in lines 16 to 23 After doing that, we merely merge the resulting tree with the root The two remaining routines are compareAndLink,which combines two trees, and combinesiblings,which combines all the siblings, when given the first sibling Figure 2313 shows how two subheaps are combined The procedure is generalized to allow the second subheap to have siblings (which is needed for the second pass in the two-pass merge) As mentioned earlier in the chapter, the subheap with the larger root is made a leftmost child of the other subheap, the code for which is shown in Figure 2314 Note that in several instances a pointer is tested against NULL before it accesses its prev data member This action suggests that having a nullNode sentinel-as was customary in the advanced search tree implementations-might be useful This possibility is left for you to explore as Exercise 2314
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Merging Priority Queues
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Figure 2313 The compareAndLink method merges two trees
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Finally, Figure 2315 implements combinesib1 ingsWe use the vector treeArray to store the subtrees We begin by separating the subtrees and storing them in treeArray,using the loop at lines 15 to 20 Assuming that we have more than one sibling to merge, we make a left-toright pass at lines 26 and 27 The special case of an odd number of trees is handled at lines 3 1-33 We finish the merging with a right-to-left pass at lines 37 and 38 Once we have finished, the result appears in array position 0 and can be returned As a practical matter, dynamically allocating and deallocating the vector can be unduly expensive and is probably not always needed We may be better off using a static vector object that can be resized as needed and build the entries without using push-back
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2323 Application: Dijkstra's Shortest Weighted Path Algorithm
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ThedecreaseKey operation is an improvement for Dijkstra,salgorithm in instances for which there are calls to it
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As an example of how the decreaseKey operation is used, we rewrite Dijkstra's algorithm (see Section 153) Recall that at any point we are maintaining a priority queue of Path objects, ordered by the d i st data member For each vertex in the graph, we needed only one Path object in the priority queue at any instant, but for convenience we had many In this section, we rework the code so that if a vertex u's distance is lowered, its position in the priority queue is found, and a decreaseKey operation is performed for its corresponding Path object
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The Paring
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Internal method that is the basic operation to maintain order Links first and second together to satisfy heap order first is root of tree 1, which may not be NULL first->nextsibling MUST be NULL on entry second is root of tree 2 , which may be NULL first becomes the result of the tree merge template <class Comparable> void PairingHeap<Comparable>: : compareAndLink( Node * & first, Node *second const i if ( second = = NULL ) return; if( second->element < first->element
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/ / Attach first as leftmost child of second second->prev = first->prev; first->prev = second; first->nextsibling = second->leftchild; if( first->nextsibling ! = NULL ) first->nextsibling->prev = first; second->leftchild = first; first = second;