quicksort sorting a vector of pairs the quicksort algorithm C++ code for quicksort partitioning a vector code for partition
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1 quicksort 1 Set Up sorting a vector of pairs 2 Quicksort the quicksort algorithm C++ code for quicksort 3 Partition partitioning a vector code for partition 4 Iterators and Timers partition with iterator timing C++ programs MCS 360 Lecture 32 Introduction to Data Structures Jan Verschelde, 10 November 2017 Introduction to Data Structures (MCS 360) quicksort L November / 29
2 quicksort 1 Set Up sorting a vector of pairs 2 Quicksort the quicksort algorithm C++ code for quicksort 3 Partition partitioning a vector code for partition 4 Iterators and Timers partition with iterator timing C++ programs Introduction to Data Structures (MCS 360) quicksort L November / 29
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) quicksort L November / 29
4 quicksort 1 Set Up sorting a vector of pairs 2 Quicksort the quicksort algorithm C++ code for quicksort 3 Partition partitioning a vector code for partition 4 Iterators and Timers partition with iterator timing C++ programs Introduction to Data Structures (MCS 360) quicksort L November / 29
5 quicksort Let n be the length of the sequence to sort. A recursive sorting algorithm, for n > 1: 1 Partition the sequence in two halves: 1 select a pivot, 2 elements in first half pivot, 3 elements in second half > pivot. 2 Apply quicksort to the first half. 3 Apply quicksort to the second half. Introduction to Data Structures (MCS 360) quicksort L November / 29
6 running an example 5 random items : (74,v)(86,e)(65,n)(99,c)(43,j) (74,v)(86,e)(65,n)(99,c)(43,j) (65,n)(43,j) (43,j) (86,e)(99,c) (99,c) the sorted vector : (43,j)(65,n)(74,v)(86,e)(99,c) Introduction to Data Structures (MCS 360) quicksort L November / 29
7 quicksort 1 Set Up sorting a vector of pairs 2 Quicksort the quicksort algorithm C++ code for quicksort 3 Partition partitioning a vector code for partition 4 Iterators and Timers partition with iterator timing C++ programs Introduction to Data Structures (MCS 360) quicksort L November / 29
8 code for quicksort void quicksort ( vector< pair<int,char> >& v ) { if(v.size() > 1) { vector< pair<int,char> > a,b; pair<int,char> pivot = v[0]; for(int i=1; i<v.size(); i++) if(v[i].first <= pivot.first) a.push_back(v[i]); else if(v[i].first > pivot.first) b.push_back(v[i]); if(a.size() > 0) quicksort(a); if(b.size() > 0) quicksort(b); for(int i=0; i<a.size(); i++) v[i] = a[i]; v[a.size()] = pivot; for(int i=0; i<b.size(); i++) v[i+a.size()+1] = b[i]; } } Introduction to Data Structures (MCS 360) quicksort L November / 29
9 Cost Analysis In the best case, for n elements: 1 The pivot is at the middle of sequence. 2 There are log 2 (n) levels in recursion. 3 Computing the partition costs O(n). cost of quicksort is O(n log 2 (n)) In the worst case: 1 The pivot is always smallest (or largest) element. 2 The recursion is n levels deep. 3 Computing the partition costs O(n). cost of quicksort is O(n 2 ) Introduction to Data Structures (MCS 360) quicksort L November / 29
10 quicksort 1 Set Up sorting a vector of pairs 2 Quicksort the quicksort algorithm C++ code for quicksort 3 Partition partitioning a vector code for partition 4 Iterators and Timers partition with iterator timing C++ programs Introduction to Data Structures (MCS 360) quicksort L November / 29
11 partitioning a vector up down up down up down the pivot is 22 up down swap if up > pivot down < pivot move up and down till ready to swap continue until up == down Introduction to Data Structures (MCS 360) quicksort L November / 29
12 running an example 7 random items : (22,e)(37,s)(42,x)(19,u)(45,l)(20,n)(21,i) swapping 1 with 6 swapping 2 with 5 moving pivot to 3 pivot = 3 the partitioned vector : -> first half : (19,u)(21,i)(20,n) -> the pivot : (22,e) -> second half : (45,l)(42,x)(37,s) Introduction to Data Structures (MCS 360) quicksort L November / 29
13 quicksort 1 Set Up sorting a vector of pairs 2 Quicksort the quicksort algorithm C++ code for quicksort 3 Partition partitioning a vector code for partition 4 Iterators and Timers partition with iterator timing C++ programs Introduction to Data Structures (MCS 360) quicksort L November / 29
14 code for partition void partition ( vector< pair<int,char> >& v, int lower, int upper, int& pivot ) { pair<int,char> x = v[lower]; int up = lower+1; int down = upper; while(up < down) { while((up < down) && (v[up].first <= x.first)) up++; while((up < down) && (v[down].first > x.first)) down--; if(up == down) break; pair<int,char> tmp = v[up]; v[up] = v[down]; v[down] = tmp; } if(v[up].first > x.first) up--; v[lower] = v[up]; v[up] = x; pivot = up; } Introduction to Data Structures (MCS 360) quicksort L November / 29
15 using partition void quicksort ( vector< pair<int,char> >& v, int k, int lb, int ub ) { if(ub - lb > 0) { int pivot; partition(v,lb,ub,pivot); quicksort(v,k+1,lb,pivot-1); } } quicksort(v,k+1,pivot+1,ub); Introduction to Data Structures (MCS 360) quicksort L November / 29
16 quicksort 1 Set Up sorting a vector of pairs 2 Quicksort the quicksort algorithm C++ code for quicksort 3 Partition partitioning a vector code for partition 4 Iterators and Timers partition with iterator timing C++ programs Introduction to Data Structures (MCS 360) quicksort L November / 29
17 writing with iterator void write_vector ( vector< pair<int,char> >::const_iterator first, vector< pair<int,char> >::const_iterator last ); // writes from *first to *last, *last included void write_vector ( vector< pair<int,char> >::const_iterator first, vector< pair<int,char> >::const_iterator last ) { vector< pair<int,char> >::const_iterator i; } for(i=first; i!=(last+1); i++) cout << "(" << i->first << "," << i->second << ")"; Introduction to Data Structures (MCS 360) quicksort L November / 29
18 partition with iterator vector< pair<int,char> >::iterator partition ( vector< pair<int,char> >::iterator lower, vector< pair<int,char> >::iterator upper ); // precondition: lower!= upper // return iterator to pivot equal to *lower // if pivot = partition(lower,upper) then // all elements before pivot are <= pivot->first // and // all elements after pivot are > pivot->first // where lower <= pivot <= upper Introduction to Data Structures (MCS 360) quicksort L November / 29
19 definition of partition vector< pair<int,char> >::iterator partition ( vector< pair<int,char> >::iterator lower, vector< pair<int,char> >::iterator upper ) { vector< pair<int,char> >::iterator pivot; vector< pair<int,char> >::iterator up = lower+1; vector< pair<int,char> >::iterator down = upper; pair<int,char> x = *lower; while(up!= down) { while((up!= down) && (up->first <= x.first)) up++; while((up!= down) && (down->first > x.first)) down--; if(up == down) break; Introduction to Data Structures (MCS 360) quicksort L November / 29
20 code continued if(up == down) break; // last line on previous slide copied } pair<int,char> tmp = *up; *up = *down; *down = tmp; if(up->first > x.first) up--; } *lower = *up; *up = x; pivot = up; } return pivot; Introduction to Data Structures (MCS 360) quicksort L November / 29
21 quicksort with iterator void quicksort ( vector< pair<int,char> >::iterator lower, vector< pair<int,char> >::iterator upper ) { if(lower!= upper) { vector< pair<int,char> >::iterator pivot; pivot = partition(lower,upper); } } if(lower!= pivot) quicksort(lower,pivot-1); if(upper!= pivot) quicksort(pivot+1,upper); Introduction to Data Structures (MCS 360) quicksort L November / 29
22 quicksort 1 Set Up sorting a vector of pairs 2 Quicksort the quicksort algorithm C++ code for quicksort 3 Partition partitioning a vector code for partition 4 Iterators and Timers partition with iterator timing C++ programs Introduction to Data Structures (MCS 360) quicksort L November / 29
23 timing quicksort std::clock_t tstart,tstop; tstart = std::clock(); quicksort(v.begin(),v.end()-1,0); tstop = std::clock(); cout << "time elapsed : " << (tstop - tstart)/double(clocks_per_sec) << " seconds" << endl; Introduction to Data Structures (MCS 360) quicksort L November / 29
24 verify at command line To verify whether our timer is correct, we place the input in the file /tmp/input and run at the command prompt $: $ time /tmp/quicksortv3 < /tmp/input give n : time elapsed : seconds real user sys 0m1.217s 0m1.213s 0m0.004s Introduction to Data Structures (MCS 360) quicksort L November / 29
25 timing our quicksort On Mac OS X at 2.26 Ghz memory 2 GB: n : n : n : n : n : n : time : seconds time : seconds time : seconds time : seconds time : seconds time : seconds Introduction to Data Structures (MCS 360) quicksort L November / 29
26 timing STL sort (in Fall 2010) On Mac OS X at 2.26 Ghz memory 2 GB: n : n : n : n : n : n : time : seconds time : seconds time : seconds time : seconds time : seconds time : seconds Introduction to Data Structures (MCS 360) quicksort L November / 29
27 timing STL sort (in Fall 2017) On Mac OS X at 3.1 GHz memory 16 GB: n : n : n : n : n : n : time : seconds time : seconds time : seconds time : seconds time : seconds time : seconds Introduction to Data Structures (MCS 360) quicksort L November / 29
28 turn optimizer on On Mac OS X at 2.26 Ghz memory 2 GB: g++ -O3 -o /tmp/quicksortv3 quicksortv3.cpp n : n : n : n : n : n : time : seconds time : seconds time : seconds time : seconds time : seconds time : seconds Introduction to Data Structures (MCS 360) quicksort L November / 29
29 Summary + Exercises Ended chapter 10 on sorting algorithms with quicksort. Exercises: 1 Adjust the quicksort code to count the number of comparisons and swaps. For dimensions n = 10, 20, 40, 80,... run 10 random instances and make a table with the number of comparisons and swaps. Do you notice the O(n log 2 (n)) cost? 2 Change the code for partition so that the middle element of the sequence is used as pivot. Demonstrate that your changes work in quicksort. 3 Compare quicksort with selection sort on vectors of increasing size, taking timings. For which n is the effect of O(n log 2 (n)) versus O(n 2 ) noticeable? Introduction to Data Structures (MCS 360) quicksort L November / 29
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