Figure 2113 The deleteMin methods

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deleteMin is a logarithmic operation in the worst case Not surprisingly, percolation rarely terminates more than one or two levels early, so deleteMin is logarithmic on average, too Figure 2113 shows the deleteMin methods The test for emptiness in the one-parameter deleteMin is automatically done by the call to findMin at line 17The real work is done in percolateDown,shown in Figure 2114 The code shown there is similar in spirit to the percolation up code in the insert routine However, because there are two children rather than one parent, the code is a bit more complicated The percolateDown method takes a single parameter that indicates where the hole is to be placed The item in the hole is then moved out, and the percolation begins For deleteMin,hole will be position 1 The for loop at line 9 terminates when there is no left child The third expression moves the hole to the child The smaller child is found at lines 12-1 3 We have to be careful because the last node in an evensized heap is an only child; we cannot always assume that there are two children, which is why we have the first test at line 12

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A Priority Queue: The Binary Heap

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1 2 3 4

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/ / Internal method to percolate down in the heap / / hole is the index at which the percolate begins

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template <class Comparable> void BinaryHeap~Comparable>::percolateDown( int hole ) 5 i 6 int child; 7 Comparable tmp = array[ hole 1 ; 8 9 for( ; hole * 2 < = thesize; hole = child ) 10 { 11 child = hole * 2; 12 if ( child ! = thesize && array [child+ll < array [child] 13 child++; 14 if( array[ child ] < tmp ) 15 array [ hole ] = array [ child I ; 16 else 17 break; 18 1 19 array[ hole I = tmp; 20 1

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Figure 2114 The percolateDown method used for deleteMin and buildHeap

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213 The buildHeap Operation: Linear-Time Heap Construction

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ThebuildHeap

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operation can be done in linear time by applying a percolate down routine to nodes in reverse level order

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The buildHeap operation takes a complete tree that does not have heap order and reinstates i t We want it to be a linear-time operation, since N insertions could be done in O(N log N ) time We expect that O(N) is attainable because N successive insertions take a total of O(N) time on average, based on the result stated at the end of Section 2121 The N successive insertions do more work than we require because they maintain heap order after every insertion and we need heap order only at one instant The easiest abstract solution is obtained by viewing the heap as a recursively defined structure, as shown in Figure 2115: We recursively call buildHeap on the left and right subheaps At that point, we are guaranteed that heap order has been established everywhere except at the root We can establish heap order everywhere by calling percolateDown for the root The recursive routine works by guaranteeing that when we apply percolateDown ( i ) , all descendants of i have been processed recursively by their own calls to percolateDownThe recursion, however, is not necessary, for the following reason If we call percolateDown on nodes in reverse level order, then at the point percolateDown ( i ) is processed, all descendants of node i will have been processed by a prior call to

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The buildHeap Operation L~near-Time Heap ~ o n s t r u c t l o n m

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Figure 211 5 Recursive view of the heap

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1 / / Establish heap-order property from an arbitrary 2 / / arrangement of items Runs in linear time 3 template <class Comparable> 4 void BinaryHeap<Comparable>::buildHeap( )

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for( int i = thesize 1 2; i > 0 ; i-- i percolateDown( i 1 ;

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