Quadratic Sorting Algorithms
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1 Quadratic Sorting Algorithms 1 Using C++ Sorting sorting a vector of pairs setting up C++ code 2 Selection Sort the selection sort algorithm C++ code for selection sort 3 Bubble Sort the bubble sort algorithm C++ code for bubble sort 4 Insertion Sort the insertion sort algorithm C++ code for insertion sort MCS 360 Lecture 30 Introduction to Data Structures Jan Verschelde, 6 November 2017 Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
2 Quadratic Sorting Algorithms 1 Using C++ Sorting sorting a vector of pairs setting up C++ code 2 Selection Sort the selection sort algorithm C++ code for selection sort 3 Bubble Sort the bubble sort algorithm C++ code for bubble sort 4 Insertion Sort the insertion sort algorithm C++ code for insertion sort Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
3 sorting a vector of pairs Consider vector< pair<int,char> > v as a frequency table. We sort on the int key. 5 random items : (82,i)(90,d)(42,x)(54,r)(31,z) We sort in increasing order of keys: 1 customized compare function, 2 illustrate stable sort. Equal keys retain their relative order in a stable sort. For example: (23,c).. (23,a) (23,c) (23,a) We assume we sort container with random-access iterator. For convenience: use subscripting operator []on vectors. Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
4 Motivation why bother with quadratic sorts? After STL sort and stable sort, we study selection, bubble, and insertion sort. The algorithms are quadratic cost sorting algorithms, i.e.: for sequences of size n, their cost is O(n 2 ). Why bother with these less efficient algorithms? 1 these algorithms are straightforward 2 for moderate dimensions, quadratic cost is not so bad 3 understand the basis for advanced sorting algorithms Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
5 Quadratic Sorting Algorithms 1 Using C++ Sorting sorting a vector of pairs setting up C++ code 2 Selection Sort the selection sort algorithm C++ code for selection sort 3 Bubble Sort the bubble sort algorithm C++ code for bubble sort 4 Insertion Sort the insertion sort algorithm C++ code for insertion sort Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
6 the include directives #include <iostream> #include <cstdlib> #include <ctime> #include <vector> #include <algorithm> using namespace std; Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
7 a random vector vector< pair<int,char> > random_vector ( int n ) vector< pair<int,char> > v; for(int i=0; i<n; i++) pair<int,char> p; p.first = 10 + (rand() % 90); p.second = a + rand() % 26; v.push_back(p); } } return v; Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
8 write and comparison void write_vector ( vector< pair<int,char> > v ) for(int i=0; i<v.size(); i++) cout << "(" << v[i].first << "," << v[i].second << ")"; } struct less_than // defines "<" bool operator()(const pair<int,char>& a, const pair<int,char>& b) return (a.first < b.first); } }; Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
9 themainprogram int main() cout << "give n : "; int n; cin >> n; srand(time(0)); vector< pair<int,char> > v = random_vector(n); cout << n << " random items : "; write_vector(v); cout << endl; cout << "stable sort? (y/n) "; char ans; cin >> ans; if(ans == y ) stable_sort(v.begin(),v.end(),less_than()); else sort(v.begin(),v.end(),less_than()); cout << "the sorted vector : "; write_vector(v); cout << endl; Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
10 Quadratic Sorting Algorithms 1 Using C++ Sorting sorting a vector of pairs setting up C++ code 2 Selection Sort the selection sort algorithm C++ code for selection sort 3 Bubble Sort the bubble sort algorithm C++ code for bubble sort 4 Insertion Sort the insertion sort algorithm C++ code for insertion sort Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
11 Selection Sort Sort a sequence of n elements: 1 get the minimum of the sequence, 2 place the minimum in front of the sequence, 3 sort the remaining n 1 elements. With an index i we keep track of the elements in the sequence still to be sorted. For i from 0 to n 1: 1 select the minimum m in v i, v i+1,...,v n ; 2 if m v i then swap m v i. Loop invariant: first i elements are smallest. Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
12 running an example 5 random items : (24,b)(96,k)(12,t)(99,g)(26,w) after 2 <-> 0 : (12,b)(96,k)(24,t)(99,g)(26,w) after 2 <-> 1 : (12,b)(24,k)(96,t)(99,g)(26,w) after 4 <-> 2 : (12,b)(24,k)(26,t)(99,g)(96,w) after 4 <-> 3 : (12,b)(24,k)(26,t)(96,g)(99,w) the sorted vector : (12,b)(24,k)(26,t)(96,g)(99,w) Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
13 Quadratic Sorting Algorithms 1 Using C++ Sorting sorting a vector of pairs setting up C++ code 2 Selection Sort the selection sort algorithm C++ code for selection sort 3 Bubble Sort the bubble sort algorithm C++ code for bubble sort 4 Insertion Sort the insertion sort algorithm C++ code for insertion sort Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
14 code for selection sort void sort ( vector< pair<int,char> >& v ) for(int i=0; i<v.size()-1; i++) int m = v[i].first; int k = i; for(int j=i+1; j<v.size(); j++) if(v[j].first < m) m = v[j].first; k = j; } if(k!= i) v[k].first = v[i].first; v[i].first = m; } } } Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
15 Cost Analysis The cost of a sort depends on 1 the number of comparisons, 2 the number of assignments. Looking at relevant parts of code: for(int i=0; i<v.size()-1; i++) for(int j=i+1; j<v.size(); j++) if(v[j].first < m) Let n = v.size(), then n i 1 comparisons are performed for i from 0 to n 1. #comparisons: n 1 + n = 1 2n(n 1) Worst case for swaps? v is given in reverse order. Best case? v is already sorted. Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
16 Quadratic Sorting Algorithms 1 Using C++ Sorting sorting a vector of pairs setting up C++ code 2 Selection Sort the selection sort algorithm C++ code for selection sort 3 Bubble Sort the bubble sort algorithm C++ code for bubble sort 4 Insertion Sort the insertion sort algorithm C++ code for insertion sort Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
17 Bubble Sort How verify that a vector is already sorted? We run through the sequence: as long as two consecutive elements are out of order: swap. Largest element get swapped to the end. For i from 0 to n 1do 1 assume sequence is sorted; 2 for j from 0 to n i 1do if vj > v j+1 then swap v j v j+1 and sequence was not sorted; 3 if no swap occurred, then leave the loop. Loop invariant: last i elements are largest. Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
18 running an example 5 random items : (62,r)(52,y)(30,k)(70,l)(31,e) after 0 <-> 1 : (52,y)(62,r)(30,k)(70,l)(31,e) after 1 <-> 2 : (52,y)(30,k)(62,r)(70,l)(31,e) after 3 <-> 4 : (52,y)(30,k)(62,r)(31,e)(70,l) after 0 <-> 1 : (30,k)(52,y)(62,r)(31,e)(70,l) after 2 <-> 3 : (30,k)(52,y)(31,e)(62,r)(70,l) after 1 <-> 2 : (30,k)(31,e)(52,y)(62,r)(70,l) the sorted vector : (30,k)(31,e)(52,y)(62,r)(70,l) Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
19 Quadratic Sorting Algorithms 1 Using C++ Sorting sorting a vector of pairs setting up C++ code 2 Selection Sort the selection sort algorithm C++ code for selection sort 3 Bubble Sort the bubble sort algorithm C++ code for bubble sort 4 Insertion Sort the insertion sort algorithm C++ code for insertion sort Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
20 code for bubble sort void sort ( vector< pair<int,char> >& v ) for(int i=0; i<v.size()-1; i++) bool sorted = true; for(int j=0; j<v.size()-i-1; j++) if(v[j].first > v[j+1].first) pair<int,char> tmp = v[j]; v[j] = v[j+1]; v[j+1] = tmp; } sorted = false; } } if(sorted) break; Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
21 Cost Analysis Again we count comparisons and assignments. Look at relevant parts of the code: for(int i=0; i<v.size()-1; i++) for(int j=0; j<v.size()-i-1; j++) if(v[j].first > v[j+1].first) Let n = v.size(), then n i 1 comparisons are performed for i from 0 to n 1. #comparisons: n 1 + n = 1 2n(n 1) Bubble sort seems just as good (or bad) as selection sort, but it swaps adjacent elements and works on linked lists. Best case? Given vector is already sorted. Worst case for swaps? v is in reverse order. Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
22 Quadratic Sorting Algorithms 1 Using C++ Sorting sorting a vector of pairs setting up C++ code 2 Selection Sort the selection sort algorithm C++ code for selection sort 3 Bubble Sort the bubble sort algorithm C++ code for bubble sort 4 Insertion Sort the insertion sort algorithm C++ code for insertion sort Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
23 Insertion Sort Imagine sorting playing cards: 1 hold in left hand cards already picked up, 2 insert newly picked card in left hand. Denote by index i the start of the sequence of n elements still to be inserted. For i from 0 to n 1do 1 find j < i: v j v i (if j = i 1, then sorted); 2 if j i 1: shift v j+1...v i 1 one spot to the right and insert v i at the j-th spot. Loop invariant: first i elements are sorted. Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
24 running an example 5 random items : (87,i)(48,v)(36,h)(73,a)(63,h) after 1 <-> 0 : (48,v)(87,i)(36,h)(73,a)(63,h) after 2 <-> 0 : (36,h)(48,v)(87,i)(73,a)(63,h) after 3 <-> 2 : (36,h)(48,v)(73,a)(87,i)(63,h) after 4 <-> 2 : (36,h)(48,v)(63,h)(73,a)(87,i) the sorted vector : (36,h)(48,v)(63,h)(73,a)(87,i) Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
25 Quadratic Sorting Algorithms 1 Using C++ Sorting sorting a vector of pairs setting up C++ code 2 Selection Sort the selection sort algorithm C++ code for selection sort 3 Bubble Sort the bubble sort algorithm C++ code for bubble sort 4 Insertion Sort the insertion sort algorithm C++ code for insertion sort Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
26 code for insertion sort void sort ( vector< pair<int,char> >& v ) for(int i=1; i<v.size(); i++) int j=i-1; while((j >= 0) && (v[i].first < v[j].first)) j--; if(++j < i) pair<int,char> tmp = v[i]; for(int k=i-1; k>=j; k--) v[k+1] = v[k]; v[j] = tmp; } } } Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
27 Cost Analysis Once more we count comparisons and assignments. We look at relevant part of the code: for(int i=1; i<v.size(); i++) int j=i-1; while((j >= 0) && (v[i].first < v[j].first)) j--; Denote n = v.size(), we insert n 1 elements and in the worst case we make n 1 + n = 1 2n(n 1) comparisons. In the best case, when sorted: n 1 comparisons. Cost of shifting elements is smaller with linked list. Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
28 comparison of quadratic sorts #comparisons #exchanges sort cases best worst best worst selection O(n 2 ) O(n 2 ) O(1) O(n) bubble O(n) O(n 2 ) O(1) O(n 2 ) insertion O(n) O(n 2 ) O(n) O(n 2 ) Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
29 Summary + Exercises Started chapter 10 on sorting algorithms with quadratic cost: selection, bubble, and insertion sort. Exercises: 1 Write a recursive function for selection sort. 2 Give code for a bubble sort on a linked list. 3 Change the code for insertion sort to take on input random-access iterators to begin and end of a vector. 4 Modify the sort functions to count the #comparisons and #swaps. Run 10 tests on random vectors of length 100 with selection, bubble, and insertion sorts. Record the #comparisons and #swaps inatable. Introduction to Data Structures (MCS 360) Quadratic Sorting Algorithms L-30 6 November / 29
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