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@ Zhigang Zhu, 2004-2014 1 CSC212 Data Structure - Section FG Lecture 22 Recursive Sorting, Heapsort & STL Quicksort Instructor: Zhigang Zhu Department of Computer Science City College of New York
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@ Zhigang Zhu, 2004-2014 2 Topics p Recursive Sorting Algorithms p Divide and Conquer technique p An O(NlogN) Sorting Alg. using a Heap p making use of the heap properties p STL Sorting Functions p C++ sort function p Original C version of qsort
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@ Zhigang Zhu, 2004-2014 3 The Divide-and-Conquer Technique p Basic Idea: p If the problem is small, simply solve it. p Otherwise, p divide the problem into two smaller sub-problems, each of which is about half of the original problem p Solve each sub-problem, and then p Combine the solutions of the sub-problems
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@ Zhigang Zhu, 2004-2014 4 The Divide-and-Conquer Sorting Paradigm 1. Divide the elements to be sorted into two groups of (almost) equal size 2. Sort each of these smaller groups of elements (by recursive calls) 3. Combine the two sorted groups into one large sorted list
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@ Zhigang Zhu, 2004-2014 5 Mergesort p Divide the array in the middle p Sort the two half-arrays by recursion p Merge the two halves void mergesort(int data[ ], size_t n) { size_t n1; // Size of the first subarray size_t n1; // Size of the first subarray size_t n2; // Size of the second subarray size_t n2; // Size of the second subarray if (n > 1) if (n > 1) { // Compute sizes of the subarrays. // Compute sizes of the subarrays. n1 = n / 2; n1 = n / 2; n2 = n - n1; n2 = n - n1; // Sort from data[0] through data[n1-1] // Sort from data[0] through data[n1-1] mergesort(data, n1); mergesort(data, n1); // Sort from data[n1] to the end // Sort from data[n1] to the end mergesort((data + n1), n2); mergesort((data + n1), n2); // Merge the two sorted halves. // Merge the two sorted halves. merge(data, n1, n2); merge(data, n1, n2); }}
![@ Zhigang Zhu, Mergesort p Divide the array in the middle p Sort the two half-arrays by recursion p Merge the two halves void mergesort(int data[ ], size_t n) { size_t n1; // Size of the first subarray size_t n1; // Size of the first subarray size_t n2; // Size of the second subarray size_t n2; // Size of the second subarray if (n > 1) if (n > 1) { // Compute sizes of the subarrays.](http://images.slideplayer.com/15/4848454/slides/slide_5.jpg "// Compute sizes of the subarrays. n1 = n / 2; n1 = n / 2; n2 = n - n1; n2 = n - n1; // Sort from data[0] through data[n1-1] // Sort from data[0] through data[n1-1] mergesort(data, n1); mergesort(data, n1); // Sort from data[n1] to the end // Sort from data[n1] to the end mergesort((data + n1), n2); mergesort((data + n1), n2); // Merge the two sorted halves. // Merge the two sorted halves. merge(data, n1, n2); merge(data, n1, n2); }}.")
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@ Zhigang Zhu, 2004-2014 6 Mergesort – an Example 161276321810 [0] [1] [2] [3] [4] [5] [6] [7]
![@ Zhigang Zhu, Mergesort – an Example [0] [1] [2] [3] [4] [5] [6] [7]](http://images.slideplayer.com/15/4848454/slides/slide_6.jpg "@ Zhigang Zhu, Mergesort – an Example [0] [1] [2] [3] [4] [5] [6] [7]")
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@ Zhigang Zhu, 2004-2014 7 Mergesort – an Example 236710121618 161276321810?
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@ Zhigang Zhu, 2004-2014 8 Mergesort – an Example 161276321810 161276321810divide
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@ Zhigang Zhu, 2004-2014 9 Mergesort – an Example 161276321810 161276321810 161276321810divide divide
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@ Zhigang Zhu, 2004-2014 10 Mergesort – an Example 161276321810 161276321810 161276321810161276321810 divide divide divide
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@ Zhigang Zhu, 2004-2014 11 Mergesort – an Example 161276321810 161276321810 161276321810 121667231018161276321810 divide divide divide merge
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@ Zhigang Zhu, 2004-2014 12 Mergesort – an Example 161276321810 161276321810 161276321810 121667231018161276321810671216231018divide divide divide merge merge
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@ Zhigang Zhu, 2004-2014 13 Mergesort – an Example 161276321810 161276321810 161276321810 121667231018161276321810671216231018 236710121618divide divide divide merge merge merge
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@ Zhigang Zhu, 2004-2014 14 Mergesort – two issues p Specifying a subarray with pointer arithmetic p int data[10]; p (data+i)[0] is the same of data[i] p (data+i][1] is the same as data[i+1] p Merging two sorted subarrays into a sorted list p need a temporary array (by new and then delete) p step through the two sub-arrays with two cursors, and copy the elements in the right order
![@ Zhigang Zhu, Mergesort – two issues p Specifying a subarray with pointer arithmetic p int data[10]; p (data+i)[0] is the same of data[i] p (data+i][1] is the same as data[i+1] p Merging two sorted subarrays into a sorted list p need a temporary array (by new and then delete) p step through the two sub-arrays with two cursors, and copy the elements in the right order](http://images.slideplayer.com/15/4848454/slides/slide_14.jpg "@ Zhigang Zhu, Mergesort – two issues p Specifying a subarray with pointer arithmetic p int data[10]; p (data+i)[0] is the same of data[i] p (data+i][1] is the same as data[i+1] p Merging two sorted subarrays into a sorted list p need a temporary array (by new and then delete) p step through the two sub-arrays with two cursors, and copy the elements in the right order")
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@ Zhigang Zhu, 2004-2014 15 Mergesort - merge 671216231018 ????????data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d
![@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d](http://images.slideplayer.com/15/4848454/slides/slide_15.jpg "@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d")
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@ Zhigang Zhu, 2004-2014 16 Mergesort - merge 671216231018 2???????data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d
![@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d](http://images.slideplayer.com/15/4848454/slides/slide_16.jpg "@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d")
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@ Zhigang Zhu, 2004-2014 17 Mergesort - merge 671216231018 23??????data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d
![@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d](http://images.slideplayer.com/15/4848454/slides/slide_17.jpg "@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d")
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@ Zhigang Zhu, 2004-2014 18 Mergesort - merge 671216231018 236?????data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d
![@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d](http://images.slideplayer.com/15/4848454/slides/slide_18.jpg "@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d")
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@ Zhigang Zhu, 2004-2014 19 Mergesort - merge 671216231018 2367????data temp [0] [1] [2] [3] [4] [5] [6] [7] c1 c2 d
![@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1 c2 d](http://images.slideplayer.com/15/4848454/slides/slide_19.jpg "@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1 c2 d")
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@ Zhigang Zhu, 2004-2014 20 Mergesort - merge 671216231018 236710???data temp [0] [1] [2] [3] [4] [5] [6] [7] c1 c2 d
![@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1 c2 d](http://images.slideplayer.com/15/4848454/slides/slide_20.jpg "@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1 c2 d")
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@ Zhigang Zhu, 2004-2014 21 Mergesort - merge 671216231018 23671012??data temp [0] [1] [2] [3] [4] [5] [6] [7] c1 c2 d
![@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1 c2 d](http://images.slideplayer.com/15/4848454/slides/slide_21.jpg "@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1 c2 d")
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@ Zhigang Zhu, 2004-2014 22 Mergesort - merge 671216231018 2367101216?data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d
![@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d](http://images.slideplayer.com/15/4848454/slides/slide_22.jpg "@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d")
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@ Zhigang Zhu, 2004-2014 23 Mergesort - merge 671216231018 236710121618data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d
![@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d](http://images.slideplayer.com/15/4848454/slides/slide_23.jpg "@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7] c1c2 d")
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@ Zhigang Zhu, 2004-2014 24 Mergesort - merge 236710121618 data temp [0] [1] [2] [3] [4] [5] [6] [7] 236710121618
![@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7]](http://images.slideplayer.com/15/4848454/slides/slide_24.jpg "@ Zhigang Zhu, Mergesort - merge data temp [0] [1] [2] [3] [4] [5] [6] [7]")
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@ Zhigang Zhu, 2004-2014 25 Mergesort - merge data [0] [1] [2] [3] [4] [5] [6] [7] 236710121618
![@ Zhigang Zhu, Mergesort - merge data [0] [1] [2] [3] [4] [5] [6] [7]](http://images.slideplayer.com/15/4848454/slides/slide_25.jpg "@ Zhigang Zhu, Mergesort - merge data [0] [1] [2] [3] [4] [5] [6] [7]")
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@ Zhigang Zhu, 2004-2014 26 void merge(int data[ ], size_t n1, size_t n2) // Precondition: The first n1 elements of data are sorted, and the // next n2 elements of data are sorted (from smallest to largest). // Postcondition: The n1+n2 elements of data are now completely sorted. QUIZ
![@ Zhigang Zhu, void merge(int data[ ], size_t n1, size_t n2) // Precondition: The first n1 elements of data are sorted, and the // next n2 elements of data are sorted (from smallest to largest).](http://images.slideplayer.com/15/4848454/slides/slide_26.jpg "// Postcondition: The n1+n2 elements of data are now completely sorted. QUIZ.")
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@ Zhigang Zhu, 2004-2014 27 Mergesort – Time Analysis p The worst-case running time, the average- case running time and the best-case running time for mergesort are all O(n log n)
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@ Zhigang Zhu, 2004-2014 28 Mergesort – an Example 161276321810 161276321810 161276321810 121667231018161276321810671216231018 236710121618divide divide divide merge merge merge
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@ Zhigang Zhu, 2004-2014 29 Mergesort – Time Analysis p At the top (0) level, 1 call to merge creates an array with n elements p At the 1 st level, 2 calls to merge creates 2 arrays, each with n/2 elements p At the 2 nd level, 4 calls to merge creates 4 arrays, each with n/4 elements p At the 3 rd level, 8 calls to merge creates 8 arrays, each with n/8 elements p At the dth level, 2 d calls to merge creates 2 d arrays, each with n/2 d elements p Each level does total work proportional to n => c n, where c is a constant p Assume at the dth level, the size of the subarrays is n/2 d =1, which means all the work is done at this level, therefore p the number of levels d = log 2 n p The total cost of the mergesort is c nd = c n log 2 n p therefore the Big-O is O(n log 2 n)
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@ Zhigang Zhu, 2004-2014 30 Heapsort p Heapsort – Why a Heap? (two properties) p Hepasort – How to? (two steps) p Heapsort – How good? (time analysis)
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@ Zhigang Zhu, 2004-2014 31 Heap Definition p A heap is a binary tree where the entries of the nodes can be compared with the less than operator of a strict weak ordering. p In addition, two rules are followed: p The entry contained by the node is NEVER less than the entries of the node’s children p The tree is a COMPLETE tree.
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@ Zhigang Zhu, 2004-2014 32 Why a Heap for Sorting? p Two properties p The largest element is always at the root p Adding and removing an entry from a heap is O(log n)
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@ Zhigang Zhu, 2004-2014 33 Heapsort – Basic Idea p Step 1. Make a heap from elements p add an entry to the heap one at a time p reheapification upward n times – O(n log n) p Step 2. Make a sorted list from the heap p Remove the root of the heap to a sorted list and p Reheapification downward to re-organize into a updated heap p n times – O(n log n)
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@ Zhigang Zhu, 2004-2014 34 Heapsort – Step 1: Make a Heap 161276321810 [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time
![@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time](http://images.slideplayer.com/15/4848454/slides/slide_34.jpg "@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time")
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@ Zhigang Zhu, 2004-2014 35 Heapsort – Step 1: Make a Heap 161276321810 [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 16
![@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 16](http://images.slideplayer.com/15/4848454/slides/slide_35.jpg "@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 16")
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@ Zhigang Zhu, 2004-2014 36 Heapsort – Step 1: Make a Heap 161276321810 [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 16 12
![@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 16 12](http://images.slideplayer.com/15/4848454/slides/slide_36.jpg "@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 16 12")
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@ Zhigang Zhu, 2004-2014 37 Heapsort – Step 1: Make a Heap 161276321810 [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 16 12 7
![@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time](http://images.slideplayer.com/15/4848454/slides/slide_37.jpg "@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time")
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@ Zhigang Zhu, 2004-2014 38 Heapsort – Step 1: Make a Heap 161276321810 [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 16 12 7 6
![@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time](http://images.slideplayer.com/15/4848454/slides/slide_38.jpg "@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time")
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@ Zhigang Zhu, 2004-2014 39 Heapsort – Step 1: Make a Heap 161276321810 [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 16 12 7 63
![@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time](http://images.slideplayer.com/15/4848454/slides/slide_39.jpg "@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time")
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@ Zhigang Zhu, 2004-2014 40 Heapsort – Step 1: Make a Heap 161276321810 [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 16 12 7 632
![@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time](http://images.slideplayer.com/15/4848454/slides/slide_40.jpg "@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time")
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@ Zhigang Zhu, 2004-2014 41 Heapsort – Step 1: Make a Heap 161276321810 [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 16 12 7 632 18 reheapification upward: push the out-of-place node upward
![@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time reheapification upward: push the out-of-place node upward](http://images.slideplayer.com/15/4848454/slides/slide_41.jpg "@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time reheapification upward: push the out-of-place node upward")
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@ Zhigang Zhu, 2004-2014 42 Heapsort – Step 1: Make a Heap 161218632710 [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 16 12 18 632 7 reheapification upward: push the out-of-place node upward
![@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time reheapification upward: push the out-of-place node upward](http://images.slideplayer.com/15/4848454/slides/slide_42.jpg "@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time reheapification upward: push the out-of-place node upward")
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@ Zhigang Zhu, 2004-2014 43 Heapsort – Step 1: Make a Heap 181216632710 [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 18 12 16 632 7 reheapification upward: push the out-of-place node upward until it is in the right place
![@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time reheapification upward: push the out-of-place node upward until it is in the right place](http://images.slideplayer.com/15/4848454/slides/slide_43.jpg "@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time reheapification upward: push the out-of-place node upward until it is in the right place")
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@ Zhigang Zhu, 2004-2014 44 Heapsort – Step 1: Make a Heap 181216632710 [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time 18 12 16 632 7 10 reheapification upward: push the out-of-place node upward until it is in the right place
![@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time reheapification upward: push the out-of-place node upward until it is in the right place](http://images.slideplayer.com/15/4848454/slides/slide_44.jpg "@ Zhigang Zhu, Heapsort – Step 1: Make a Heap [0] [1] [2] [3] [4] [5] [6] [7] add an entry to the heap one at a time reheapification upward: push the out-of-place node upward until it is in the right place")
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@ Zhigang Zhu, 2004-2014 45 Heapsort – Step 1: Make a Heap add an entry to the heap one at a time 18 12 16 1032 7 6 reheapification upward: push the out-of-place node upward until it is in the right place 181216103276 [0] [1] [2] [3] [4] [5] [6] [7]
![@ Zhigang Zhu, Heapsort – Step 1: Make a Heap add an entry to the heap one at a time reheapification upward: push the out-of-place node upward until it is in the right place [0] [1] [2] [3] [4] [5] [6] [7]](http://images.slideplayer.com/15/4848454/slides/slide_45.jpg "@ Zhigang Zhu, Heapsort – Step 1: Make a Heap add an entry to the heap one at a time reheapification upward: push the out-of-place node upward until it is in the right place [0] [1] [2] [3] [4] [5] [6] [7]")
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@ Zhigang Zhu, 2004-2014 46 Heapsort – Step 1: Make a Heap A heap is created: it is saved in the original array- the tree on the right is only for illustration! 18 12 16 1032 7 6 181216103276 [0] [1] [2] [3] [4] [5] [6] [7] Sorted???
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@ Zhigang Zhu, 2004-2014 47 Heapsort – Step 2: Sorting from Heap 18 12 16 1032 7 6 181216103276 [0] [1] [2] [3] [4] [5] [6] [7] heap -> sorted list from smallest to largest Q: where is the largest entry?
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] heap -> sorted list from smallest to largest Q: where is the largest entry](http://images.slideplayer.com/15/4848454/slides/slide_47.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] heap -> sorted list from smallest to largest Q: where is the largest entry")
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@ Zhigang Zhu, 2004-2014 48 Heapsort – Step 2: Sorting from Heap 18 12 16 1032 7 6 181216103276 [0] [1] [2] [3] [4] [5] [6] [7] Idea: remove the root of the heap and place it in the sorted list => recall: how to remove the root?
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] Idea: remove the root of the heap and place it in the sorted list => recall: how to remove the root](http://images.slideplayer.com/15/4848454/slides/slide_48.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] Idea: remove the root of the heap and place it in the sorted list => recall: how to remove the root")
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@ Zhigang Zhu, 2004-2014 49 Heapsort – Step 2: Sorting from Heap 6 12 16 1032 7 612161032718 [0] [1] [2] [3] [4] [5] [6] [7] How to remove the root? move the last entry in the root... and for the sake of sorting, put the root entry in the “sorted side” almost a heap... sorted side
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] How to remove the root.](http://images.slideplayer.com/15/4848454/slides/slide_49.jpg "move the last entry in the root... and for the sake of sorting, put the root entry in the sorted side almost a heap... sorted side.")
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@ Zhigang Zhu, 2004-2014 50 Heapsort – Step 2: Sorting from Heap 6 12 16 1032 7 612161032718 [0] [1] [2] [3] [4] [5] [6] [7] How to remove the root? move the last entry in the root... then reposition the out-of place node to update the heap sorted side almost a heap...
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] How to remove the root.](http://images.slideplayer.com/15/4848454/slides/slide_50.jpg "move the last entry in the root... then reposition the out-of place node to update the heap sorted side almost a heap....")
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@ Zhigang Zhu, 2004-2014 51 Heapsort – Step 2: Sorting from Heap 16 12 6 1032 7 161261032718 [0] [1] [2] [3] [4] [5] [6] [7] How to remove the root? move the last entry in the root... then reposition the out-of place node to update the heap sorted side reheapification downward almost a heap...
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] How to remove the root.](http://images.slideplayer.com/15/4848454/slides/slide_51.jpg "move the last entry in the root... then reposition the out-of place node to update the heap sorted side reheapification downward almost a heap....")
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@ Zhigang Zhu, 2004-2014 52 Heapsort – Step 2: Sorting from Heap 16 12 7 1032 6 161271032618 [0] [1] [2] [3] [4] [5] [6] [7] How to remove the root? move the last entry in the root... then reposition the out-of place node to update the heap sorted side reheapification downward a heap in the unsorted side
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] How to remove the root.](http://images.slideplayer.com/15/4848454/slides/slide_52.jpg "move the last entry in the root... then reposition the out-of place node to update the heap sorted side reheapification downward a heap in the unsorted side.")
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@ Zhigang Zhu, 2004-2014 53 Heapsort – Step 2: Sorting from Heap 16 12 7 1032 6 161271032618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap in the unsorted side
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap in the unsorted side](http://images.slideplayer.com/15/4848454/slides/slide_53.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap in the unsorted side")
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@ Zhigang Zhu, 2004-2014 54 Heapsort – Step 2: Sorting from Heap 6 12 7 1032 612710321618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...](http://images.slideplayer.com/15/4848454/slides/slide_54.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...")
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@ Zhigang Zhu, 2004-2014 55 Heapsort – Step 2: Sorting from Heap 12 6 7 1032 12 6710321618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...](http://images.slideplayer.com/15/4848454/slides/slide_55.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...")
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@ Zhigang Zhu, 2004-2014 56 Heapsort – Step 2: Sorting from Heap 12 10 7 632 12 10 1076321618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...](http://images.slideplayer.com/15/4848454/slides/slide_56.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...")
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@ Zhigang Zhu, 2004-2014 57 Heapsort – Step 2: Sorting from Heap 12 10 7 632 12 10 1076321618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap again!
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap again!](http://images.slideplayer.com/15/4848454/slides/slide_57.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap again!")
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@ Zhigang Zhu, 2004-2014 58 Heapsort – Step 2: Sorting from Heap 2 10 7 63 2 10 10763121618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...](http://images.slideplayer.com/15/4848454/slides/slide_58.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...")
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@ Zhigang Zhu, 2004-2014 59 Heapsort – Step 2: Sorting from Heap 10 2 7 63 10 10 2763121618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...](http://images.slideplayer.com/15/4848454/slides/slide_59.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...")
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@ Zhigang Zhu, 2004-2014 60 Heapsort – Step 2: Sorting from Heap 10 6 7 23 10 10 6723121618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap again!
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap again!](http://images.slideplayer.com/15/4848454/slides/slide_60.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap again!")
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@ Zhigang Zhu, 2004-2014 61 Heapsort – Step 2: Sorting from Heap 3 6 7 2 3 67210121618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...](http://images.slideplayer.com/15/4848454/slides/slide_61.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side almost a heap...")
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@ Zhigang Zhu, 2004-2014 62 Heapsort – Step 2: Sorting from Heap 7 6 3 2 7 63210121618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap again !
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap again !](http://images.slideplayer.com/15/4848454/slides/slide_62.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap again !")
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@ Zhigang Zhu, 2004-2014 63 Heapsort – Step 2: Sorting from Heap 2 6 3 2 63710121618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side a heap ?? sorted side
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side a heap .](http://images.slideplayer.com/15/4848454/slides/slide_63.jpg "sorted side.")
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@ Zhigang Zhu, 2004-2014 64 Heapsort – Step 2: Sorting from Heap 6 2 3 6 23710121618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap !!
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap !!](http://images.slideplayer.com/15/4848454/slides/slide_64.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side a heap !!")
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@ Zhigang Zhu, 2004-2014 65 Heapsort – Step 2: Sorting from Heap 3 2 3 26710121618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side heap !
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side heap !](http://images.slideplayer.com/15/4848454/slides/slide_65.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side heap !")
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@ Zhigang Zhu, 2004-2014 66 Heapsort – Step 2: Sorting from Heap 2 2 36710121618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side heap !
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side heap !](http://images.slideplayer.com/15/4848454/slides/slide_66.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side heap !")
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@ Zhigang Zhu, 2004-2014 67 Heapsort – Step 2: Sorting from Heap 2 36710121618 [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side DONE!
![@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side DONE!](http://images.slideplayer.com/15/4848454/slides/slide_67.jpg "@ Zhigang Zhu, Heapsort – Step 2: Sorting from Heap [0] [1] [2] [3] [4] [5] [6] [7] do the same thing again for the heap in the unsorted side until all the entries have been moved to the sorted side sorted side DONE!")
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@ Zhigang Zhu, 2004-2014 68 Heapsort – Time Analysis p Step 1. Make a heap from elements p add an entry to the heap one at a time p reheapification upward n times – O(n log n) p Step 2. Make a sorted list from the heap p Remove the root of the heap to a sorted list and p Reheapification downward to re-organize the unsorted side into a updated heap p do this n times – O(n log n) p The running time is O(n log n)
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@ Zhigang Zhu, 2004-2014 69 C++ STL Sorting Functions p The C++ sort function p void sort(Iterator begin, Iterator end); p The original C version of qsort void qsort( void *base, size_t number_of_elements, size_t element_size, int compare(const void*, const void*) );
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@ Zhigang Zhu, 2004-2014 70 Summary & Homework p Recursive Sorting Algorithms p Divide and Conquer technique p An O(NlogN) Sorting Alg. using a Heap p making use of the heap properties p STL Sorting Functions p C++ sort function p Original C version of qsort p Homework p use your heap implementation to implement a heapsort!
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