Better Sorting Algorithms
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1 Better Sorting Algorithms 1 Set Up sorting a vector of pairs 2 Shell Sort the Shell sort algorithm C++ code for Shell sort 3 Merge Sort the merge sort algorithm C++ code for merge sort 4 Heap Sort the heap sort algorithm C++ code for heap sort MCS 360 Lecture 31 Introduction to Data Structures Jan Verschelde, 8 November 2017 Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
2 Better Sorting Algorithms 1 Set Up sorting a vector of pairs 2 Shell Sort the Shell sort algorithm C++ code for Shell sort 3 Merge Sort the merge sort algorithm C++ code for merge sort 4 Heap Sort the heap sort algorithm C++ code for heap sort Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
3 sorting a vector of pairs Consider vector< pair<int,char> > v as a frequency table. We sort only on the int key. 5 random items : (82,i)(42,d)(42,x)(54,r)(31,z) With char as second data field we can check if the sort is stable. In a stable sort, equal keys retain their relative order We assume we sort container with random-access iterator. For convenience: use subscripting operator []on vectors. We generate random vectors of 2-digit keys. Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
4 Better Sorting Algorithms 1 Set Up sorting a vector of pairs 2 Shell Sort the Shell sort algorithm C++ code for Shell sort 3 Merge Sort the merge sort algorithm C++ code for merge sort 4 Heap Sort the heap sort algorithm C++ code for heap sort Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
5 Shell Sort The idea of Donald Shell is to apply insertion sort to smaller subsequences separated by a gap value. For example, when gap equals 4: First, sort four sequences: [0, 4, 8], [1, 5, 9], [2, 6], and [3, 7]. Then, repeat with smaller value of gap. Start with gap value equal to n/2. Divide each time gap by 2.2, at end set gap to 1. The division factor of 2.2 appears to be empirically to give performance of O(n 5/4 ). If the value for gap is 2 k 1, then the cost is O(n 3/2 ). Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
6 running an example 5 random items : (68,q)(79,f)(65,u)(58,j)(76,b) insertion sort starts at 0 with gap 2 after 2 <-> 0 : (65,u)(79,f)(68,q)(58,j)(76,b) insertion sort starts at 1 with gap 2 after 3 <-> 1 : (65,u)(58,j)(68,q)(79,f)(76,b) insertion sort starts at 0 with gap 1 after 1 <-> 0 : (58,j)(65,u)(68,q)(79,f)(76,b) after 4 <-> 3 : (58,j)(65,u)(68,q)(76,b)(79,f) the sorted vector : (58,j)(65,u)(68,q)(76,b)(79,f) Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
7 Better Sorting Algorithms 1 Set Up sorting a vector of pairs 2 Shell Sort the Shell sort algorithm C++ code for Shell sort 3 Merge Sort the merge sort algorithm C++ code for merge sort 4 Heap Sort the heap sort algorithm C++ code for heap sort Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
8 insertion sort with gap void insertion_sort ( vector< pair<int,char> >& v, int start, int gap ) { for(int i=start+gap; i<v.size(); i=i+gap) { int j=i-gap; while((j >= start) && (v[i].first < v[j].first)) j=j-gap; j = j + gap; if(j < i) { pair<int,char> tmp = v[i]; for(int k=i-gap; k>=j; k=k-gap) v[k+gap] = v[k]; v[j] = tmp; } } } Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
9 code for Shell sort void sort ( vector< pair<int,char> >& v ) { int gap = v.size()/2; while(gap > 0) { for(int i=0; i<gap; i++) insertion_sort(v,i,gap); } } if(gap == 2) gap = 1; else gap = int(gap/2.2); Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
10 Better Sorting Algorithms 1 Set Up sorting a vector of pairs 2 Shell Sort the Shell sort algorithm C++ code for Shell sort 3 Merge Sort the merge sort algorithm C++ code for merge sort 4 Heap Sort the heap sort algorithm C++ code for heap sort Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
11 Merge Sort To sort a sequence of n elements, if n > 1: 1 split sequence in two equal halves, 2 sort first and second half, 3 merge the sorted halves. Merge sort works on tapes, external data files. Some data sets are too large to be stored entirely in internal memory. Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
12 running an example 4 random items : (15,m)(90,r)(63,p)(94,w) in : (15,m)(90,r)(63,p)(94,w) in : (15,m)(90,r) in : (15,m) out : (15,m) in : (90,r) out : (90,r) out : (15,m)(90,r) in : (63,p)(94,w) in : (63,p) out : (63,p) in : (94,w) out : (94,w) out : (63,p)(94,w) out : (15,m)(63,p)(90,r)(94,w) the sorted vector : (15,m)(63,p)(90,r)(94,w) Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
13 Better Sorting Algorithms 1 Set Up sorting a vector of pairs 2 Shell Sort the Shell sort algorithm C++ code for Shell sort 3 Merge Sort the merge sort algorithm C++ code for merge sort 4 Heap Sort the heap sort algorithm C++ code for heap sort Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
14 code for merge vector< pair<int,char> > merge ( vector< pair<int,char> > u, vector< pair<int,char> > v ) { vector< pair<int,char> > w; int i = 0; int j = 0; while((i < u.size()) && (j < v.size())) if(u[i].first <= v[j].first) w.push_back(u[i++]); else w.push_back(v[j++]); while(i < u.size()) w.push_back(u[i++]); while(j < v.size()) w.push_back(v[j++]); } return w; Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
15 function merge_sort vector< pair<int,char> > merge_sort ( vector< pair<int,char> >& v ) { vector< pair<int,char> > w; if(v.size() < 2) w = v; else { int middle = v.size()/2; vector< pair<int,char> > a,b,s_a,s_b; for(int i=0; i<middle; i++) a.push_back(v[i]); for(int i=middle; i<v.size(); i++) b.push_back(v[i]); s_a = merge_sort(a); s_b = merge_sort(b); w = merge(s_a,s_b); } return w; } Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
16 Cost Analysis The cost of a sort depends on 1 the number of comparisons, 2 the number of assignments. How many levels in the recursion? For a sequence of size n: split log 2 (n) times. Splitting and merge takes O(n) operations: Best case? Worst case? n 2 + n = n 1. 4 Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
17 Better Sorting Algorithms 1 Set Up sorting a vector of pairs 2 Shell Sort the Shell sort algorithm C++ code for Shell sort 3 Merge Sort the merge sort algorithm C++ code for merge sort 4 Heap Sort the heap sort algorithm C++ code for heap sort Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
18 a heap as a vector A heap is a binary tree: top bottom For node at p: left child is at 2p + 1, right child is at 2p + 2. Parent of node at p is at (p 1)/2. Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
19 storing integer numbers pushing Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
20 sorting with a heap popping largest element Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
21 running an example 5 random items : (81,x)(74,m)(78,i)(78,j)(50,i) insert i=0 : (81,x) insert i=1 : (81,x)(74,m) insert i=2 : (81,x)(74,m)(78,i) insert i=3 : (81,x)(78,j)(78,i)(74,m) insert i=4 : (81,x)(78,j)(78,i)(74,m)(50,i) remove i=0 : (78,i)(78,j)(50,i)(74,m) remove i=1 : (78,j)(74,m)(50,i) remove i=2 : (74,m)(50,i) remove i=3 : (50,i) remove i=4 : the sorted vector : (50,i)(74,m)(78,j)(78,i)(81,x) Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
22 Better Sorting Algorithms 1 Set Up sorting a vector of pairs 2 Shell Sort the Shell sort algorithm C++ code for Shell sort 3 Merge Sort the merge sort algorithm C++ code for merge sort 4 Heap Sort the heap sort algorithm C++ code for heap sort Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
23 code for heap sort vector< pair<int,char> > sort ( vector< pair<int,char> >& v ) { vector< pair<int,char> > h,w; int bottom = -1; for(int i=0; i<v.size(); i++) insert(h,bottom,v[i]); } for(int i=0; i<v.size(); i++) { w.insert(w.begin(),h[0]); remove(h,bottom); } return w; Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
24 Cost Analysis Inserting and removing an element: depends on the depth of the binary tree, the heap is a complete binary tree, the depth is log 2 (n). Therefore: the cost for insert/remove is O(log 2 (n)). We have n elements, so the cost is 2n log 2 (n). Best case? Worst case? Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
25 Summary + Exercises More of chapter 10 on sorting algorithms with Shell sort, heap sort, and merge sort. Exercises: 1 Modify the code for Shell sort to count the #comparisons and #swaps. Run at least 10 sorts of random sequences of sizes 100, 200, and 400. Do data fitting on the counts. Do you observe O(n 3/2 )? 2 Describe an inplace merge sort. Extra input parameters are the begin and end index in the vector. 3 Give an example (with 7 numbers) of the best and worst case for heap sort. Introduction to Data Structures (MCS 360) Better Sorting Algorithms L-31 8 November / 25
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