2.3 Quicksort Two classic sorting algorithms Critical components in the world’s computational infrastructure. • Full scientific understanding of their properties has enabled us to develop them into practical system sorts. • Quicksort honored as one of top 10 algorithms of 20th century in science and engineering. ‣ quicksort Mergesort. last lecture ‣ selection • Java sort for objects. ‣ duplicate keys • Perl, C++ stable sort, Python stable sort, Firefox JavaScript, ... ‣ system sorts Quicksort. this lecture • Java sort for primitive types. • C qsort, Unix, Visual C++, Python, Matlab, Chrome JavaScript, ... th Algorithms, 4 Edition · Robert Sedgewick and Kevin Wayne · Copyright © 2002–2010 · January 30, 2011 12:46:16 PM 2 Quicksort Basic plan. • Shuffle the array. • Partition so that, for some j - element a[j] is in place - no larger element to the left of j j - no smaller element to the right of Sir Charles Antony Richard Hoare • Sort each piece recursively. 1980 Turing Award ‣ quicksort ‣ selection input Q U I C K S O R T E X A M P L E ‣ duplicate keys shu!e K R A T E L E P U I M Q C X O S ‣ system sorts partitioning element partition E C A I E K L P U T M Q R X O S not greater not less sort left A C E E I K L P U T M Q R X O S sort right A C E E I K L M O P Q R S T U X result A C E E I K L M O P Q R S T U X Quicksort overview 3 4 Quicksort partitioning Quicksort: Java code for partitioning Basic plan. private static int partition(Comparable[] a, int lo, int hi) • Scan i from left for an item that belongs on the right. { int i = lo, j = hi+1; • Scan j from right for item item that belongs on the left. while (true) • Exchange a[i] and a[j]. { while (less(a[++i], a[lo])) find item on left to swap • Repeat until pointers cross. if (i == hi) break; v a[i] while (less(a[lo], a[--j])) find item on right to swap i j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 if (j == lo) break; initial values 0 16 K R A T E L E P U I M Q C X O S if (i >= j) break; check if pointers cross scan left, scan right 1 12 K R A T E L E P U I M Q C X O S exch(a, i, j); swap } exchange 1 12 K C A T E L E P U I M Q R X O S scan left, scan right 3 9 K C A T E L E P U I M Q R X O S exch(a, lo, j); swap with partitioning item exchange 3 9 K C A I E L E P U T M Q R X O S return j; return index of item now known tobefore be in placev } scan left, scan right 5 6 K C A I E L E P U T M Q R X O S lo hi exchange 5 6 K C A I E E L P U T M Q R X O S before v during v Յ v Ն v scan left, scan right 6 5 K C A I E E L P U T M Q R X O S lo hi i j !nal exchange 6 5 E C A I E K L P U T M Q R X O S after before v during v Յ v Ն v Յ v v Ն v result 6 5 E C A I E K L P U T M Q R X O S lo hi i j lo j hi Partitioning trace (array contents before and after each exchange) during v Յ v Ն v after Յ v v Ն v Quicksort partitioning overview 5 6 i j lo j hi after Յ v v Ն v Quicksort partitioning overview lo j hi Quicksort partitioning overview Quicksort: Java implementation Quicksort trace public class Quick lo j hi 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 { initial values Q U I C K S O R T E X A M P L E private static int partition(Comparable[] a, int lo, int hi) random shu!e K R A T E L E P U I M Q C X O S { /* see previous slide */ } 0 5 15 E C A I E K L P U T M Q R X O S 0 3 4 E C A E I K L P U T M Q R X O S public static void sort(Comparable[] a) 0 2 2 A C E E I K L P U T M Q R X O S { 0 0 1 A C E E I K L P U T M Q R X O S StdRandom.shuffle(a); 1 1 A C E E I K L P U T M Q R X O S shuffle needed for 4 4 A C E E I K L P U T M Q R X O S sort(a, 0, a.length - 1); performance guarantee 6 6 15 A C E E I K L P U T M Q R X O S } (stay tuned) no partition 7 9 15 A C E E I K L M O P T Q R X U S for subarrays of size 1 7 7 8 A C E E I K L M O P T Q R X U S private static void sort(Comparable[] a, int lo, int hi) 8 8 A C E E I K L M O P T Q R X U S { 10 13 15 A C E E I K L M O P S Q R T U X if (hi <= lo) return; 10 12 12 A C E E I K L M O P R Q S T U X int j = partition(a, lo, hi); 10 11 11 A C E E I K L M O P Q R S T U X sort(a, lo, j-1); 10 10 A C E E I K L M O P Q R S T U X sort(a, j+1, hi); 14 14 15 A C E E I K L M O P Q R S T U X } 15 15 A C E E I K L M O P Q R S T U X } result A C E E I K L M O P Q R S T U X Quicksort trace (array contents after each partition) 7 8 Quicksort animation Quicksort: implementation details 50 random elements Partitioning in-place. Using an extra array makes partitioning easier (and stable), but is not worth the cost. Terminating the loop. Testing whether the pointers cross is a bit trickier than it might seem. Staying in bounds. The (j == lo) test is redundant (why?), but the (i == hi) test is not. Preserving randomness. Shuffling is needed for performance guarantee. Equal keys. When duplicates are present, it is (counter-intuitively) better algorithm position to stop on elements equal to the partitioning element. in order current subarray not in order http://www.sorting-algorithms.com/quick-sort 9 10 Quicksort: empirical analysis Quicksort: best-case analysis Running time estimates: Best case. Number of compares is ~ N lg N. • Home PC executes 108 compares/second. • Supercomputer executes 1012 compares/second. initial values random shuffle insertion sort (N2) mergesort (N log N) quicksort (N log N) computer thousand million billion thousand million billion thousand million billion home instant 2.8 hours 317 years instant 1 second 18 min instant 0.6 sec 12 min super instant 1 second 1 week instant instant instant instant instant instant Lesson 1. Good algorithms are better than supercomputers. Lesson 2. Great algorithms are better than good ones. 11 12 Quicksort: worst-case analysis Quicksort: average-case analysis 2 Worst case. Number of compares is ~ ! N . Proposition. The average number of compares CN to quicksort an array of N distinct keys is ~ 2N ln N (and the number of exchanges is ~ " N ln N). C C = C = 0 N # 2 initial values Pf 1. N satisfies the recurrence 0 1 and for : random shuffle C0 + C1 + ... + CN 1 CN 1 + CN 2 + ... + C0 C =(N + 1) + − + − − N N N partitioning left right partitioning probability • Multiply both sides by N and collect terms: NCN = N(N + 1) + 2(C0 + C1 + ... + CN 1) − • Subtract this from the same equation for N - 1: NCN (N 1)CN 1 =2N +2CN 1 − − − − • Rearrange terms and divide by N (N + 1): CN CN 1 2 = − + N +1 N N +1 13 14 Quicksort: average-case analysis Quicksort: average-case analysis • Repeatedly apply above equation: Proposition. The average number of compares CN to quicksort an array of CN CN 1 2 N distinct keys is ~ 2N ln N (and the number of exchanges is ~ " N ln N). = − + N +1 N N +1 CN 2 CN 1 2 = − + − + 1 N N 1 N N +1 Pf 2. Consider BST representation of keys to . previous equation − CN 3 CN 2 CN 1 2 = − + − + − + N 2 N 1 N N +1 shufe 2 − 2 2 − 2 = + + + ..