BY using empty parts the array, we can perform the sort in in Java

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If we usea max heap, items in ~ncreasing order
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Figure 2128 Max heap after the bui ldHeap phase
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Figure 2129 Heap after the first deleteMax operation
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Minor changes are required heaps0rt because the root is stored in position 0
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Figure 2129 shows the heap that results after the first deleteMaxThe last element in the heap is 2 1; 97 has been placed in a part of the heap array that is technically no longer part of the heap Figure 2130 shows that after a second deleteMax,16 becomes the last element Now only eight items remain in the heap The maximum element removed, 59, is placed in the dead spot of the array After seven more deleteMax operations, the heap represents only one element, but the elements left in the array will be sorted in increasing order Implementation of the heapsort operation is simple because it basically follows the heap operation There are three minor differences between the two operations First, because we are using a max heap, we need to reverse the logic of the comparisons from > to < Second, we can no longer assume that there is a sentinel position 0 The reason is that all our other sorting
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Figure 2130 Heap after the second deleteMax operation
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1 / / Standard heapsort 2 void heapsort( vector<Comparable> & a ) 3 I 4 for( int i = asize( ) / 2; i > = 0; i - 5 percDown( a, i , alength ( ) ) ; 6 for( int j = asize( ) - 1; j > 0; j-7 t 8 swap( a[ 0 1 , a[ j 1 ) ; 9 percDown( a, 0 , j ) ; 10 1 11 I Figure 2131 The h e a p s o r t routine
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algorithms store data at position 0, and we must assume that h e a p s o r t is no different Although the sentinel is not needed anyway (there are no percolate up operations), its absence affects calculations of the child and parent That is, for a node in position i, the parent is in position ( i - 1 ) / 2 , the left child is in position 2i + 1, and the right child is next to the left child Third, percDown needs to be informed of the current heap size (which is lowered by 1 in each iteration of d e l e t e ~ a x )The implementation of percDown is left for you to do as Exercise 2123 Assuming that we have written percDown, we can easily express h e a p s o r t as shown in Figure 2131 Although heapsort is not as fast as quicksort, it can still be useful As discussed in Section 96 (and detailed in Exercise 919), in quicksort we can keep track of each recursive call's depth, and switch to an O(N log N) worst-case sort for any recursive call that is too deep (roughly 2 log N nested calls) Exercise 919 suggested mergesort, but actually heapsort is the better candidate
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Externalsorting is used when the amount of data is too large to fit in main memory
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So far, all the sorting algorithms examined require that the input fit in main memory However the input for some applications is much too large to fit in main memory In this section we discuss external sorting, which is used to handle such very large inputs Some of the external sorting algorithms involve the use of heaps
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Most of the internal sorting algorithms take advantage of the fact that memory is directly accessible Shellsort compares elements a [ i 1 and a [ i -gap] in one time unit Heapsort compares a [ i I and a [child=iX2 in one time I unit Quicksort with median-of-three pivoting, requires comparing a [ f i r s t1 , a [center], and a [last]in a constant number of time units If the input is on a tape, all these operations lose their efficiency because elements on a tape can be accessed only sequentially Even if the data are on a disk, efficiency still suffers because of the delay required to spin the disk and move the disk head To demonstrate how slow external accesses really are, we could create a random file that is large but not too big to fit in main memory When we read in the file and sort it by using an efficient algorithm, the time needed to read the input is likely to be significant compared to the time required to sort the input, even though sorting is an O ( N log N ) operation (or worse for Shellsort) and reading the input is only O(N)
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