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, Python stable sort. ‣ system sorts Quicksort. this lecture • Java sort for primitive types. • C qsort, Unix, g++, Visual C++, Python.
th Algorithms in Java, 4 Edition · Robert Sedgewick and Kevin Wayne · Copyright © 2009 · January 22, 2010 4:05:04 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 no smaller element to the right of j - 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 partitioning element
‣ system sorts 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 • Continue 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 -1 15 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 vplace 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 0 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 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 sort(a, 0, a.length - 1); 4 4 A C E E I K L P U T M Q R X O S performance guarantee 6 6 15 A C E E I K L P U T M Q R X O S } no partition 7 9 15 A C E E I K L M O P T Q R X U S for subarrays 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) of size 1 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 a spare 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) best 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.
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 0.3 sec 6 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 / 2. Proposition I. The average number of compares CN to quicksort an array of N
elements is ~ 2N ln N (and the number of exchanges is ~ ⅓ N ln N).
Pf. CN satisfies the recurrence C0 = C1 = 0 and for N ! 2:
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: summary of performance characteristics
• Repeatedly apply above equation: Worst case. Number of compares is quadratic. N + (N-1) + (N-2) + … + 1 ~ N2 / 2. CN CN 1 2 • = − + N +1 N N +1 • More likely that your computer is struck by lightning. CN 2 2 2 = − + + previous equation N 1 N N +1 − Average case. Number of compares is ~ 1.39 N lg N. CN 3 2 2 2 = − + + + N 2 N 1 N N +1 • 39% more compares than mergesort. 2 − 2 2 − 2 = + + + ... + • But faster than mergesort in practice because of less data movement. 1 2 3 N +1
• Approximate sum by an integral: Random shuffle. Probabilistic guarantee against worst case. 1 1 1 • C 2(N + 1) 1+ + + ... N ∼ 2 3 N • Basis for math model that can be validated with experiments. N 1 2(N + 1) dx ∼ 1 x Caveat emptor. Many textbook implementations go quadratic if input: • Is sorted or reverse sorted. • Finally, the desired result: • Has many duplicates (even if randomized!) [stay tuned] C 2(N + 1) ln N 1.39N lg N N ∼ ≈ 15 16 Quicksort: practical improvements Quicksort with cutoff to insertion sort: visualization
input Median of sample. partitioning element result of • Best choice of pivot element = median. "rst partition • Estimate true median by taking median of sample.
Insertion sort small subarrays. • Even quicksort has too much overhead for tiny subarrays. • Can delay insertion sort until end.
~ 12/7 N ln N compares left subarray partially sorted Optimize parameters. ~ 12/35 N ln N exchanges • Median-of-3 random elements. • Cutoff to insertion sort for " 10 elements.
Non-recursive version. guarantees O(log N) stack size Use explicit stack. • both subarrays • Always sort smaller half first. partially sorted result
17 Quicksort with median-of-3 partitioning and cuto! for small subarrays 18
Selection
Goal. Find the kth largest element. Ex. Min (k = 0), max (k = N-1), median (k = N/2).
Applications. • Order statistics. • Find the “top k.”
‣ quicksort Use theory as a guide. ‣ selection • Easy O(N log N) upper bound. ‣ duplicate keys • Easy O(N) upper bound for k = 1, 2, 3. ‣ system sorts • Easy #(N) lower bound.
Which is true? • #(N log N) lower bound? is selection as hard as sorting? • O(N) upper bound? is there a linear-time algorithm for all k?
19 20 Quick-select Quick-select: mathematical analysis
Partition array so that: Proposition. Quick-select takes linear time on average. • Element a[j] is in place. Pf sketch. • No larger element to the left of j. • Intuitively, each partitioning step roughly splits array in half: • No smaller element to the right of j. N + N/2 + N/4 + … + 1 ~ 2N compares. • Formal analysis similar to quicksort analysis yields: Repeat in one subarray, depending on j; finished when j equals k. CN = 2 N + k ln ( N / k) + (N - k) ln (N / (N - k))
before v public static Comparable select(Comparable[] a, int k) { lo hi if a[k] is here if a[k] is here StdRandom.shuffle(a); set hi to j-1 set lo to j+1 Ex. (2 + 2 ln 2) N compares to find the median. int lo = 0, hi = a.length - 1; during v Յ v Ն v while (hi > lo) i j { int j = partition(a, lo, hi); after Յ v v Ն v if (j < k) lo = j + 1; else if (j > k) hi = j - 1; lo j hi else return a[k]; 2 Quicksort partitioning overview Remark. Quick-select uses ~ N /2 compares in worst case, } return a[k]; but as with quicksort, the random shuffle provides a probabilistic guarantee. }
21 22
Theoretical context for selection Generic methods
Challenge. Design algorithm whose worst-case running time is linear. In our select() implementation, client needs a cast.
Proposition. [Blum, Floyd, Pratt, Rivest, Tarjan, 1973] There exists a Double[] a = new Double[N]; compare-based selection algorithm whose worst-case running time is linear. for (int i = 0; i < N; i++) a[i] = StdRandom.uniform(); unsafe cast Double median = (Double) Quick.select(a, N/2); Remark. But, algorithm is too complicated to be useful in practice. required
The compiler also complains.
Use theory as a guide. % javac Quick.java • Still worthwhile to seek practical linear-time (worst-case) algorithm. Note: Quick.java uses unchecked or unsafe operations. Note: Recompile with -Xlint:unchecked for details. • Until one is discovered, use quick-select if you don’t need a full sort.
Q. How to fix?
23 24 Generic methods
Pedantic (safe) version. Compiles cleanly, no cast needed in client.
public class QuickPedantic generic type variable { (value inferred from argument a[]) public static
return type matches array type public static
private static
private static
} can declare variables of generic type
http://www.cs.princeton.edu/algs4/35applications/QuickPedantic.java.html
Remark. Obnoxious code needed in system sort; not in this course (for brevity).
25 26
Duplicate keys Duplicate keys
Often, purpose of sort is to bring records with duplicate keys together. Mergesort with duplicate keys. Always ~ N lg N compares. • Sort population by age. • Find collinear points. see Assignment 3 Quicksort with duplicate keys. • Remove duplicates from mailing list. • Algorithm goes quadratic unless partitioning stops on equal keys! • Sort job applicants by college attended. • 1990s C user found this defect in qsort(). sorted by time sorted by city (unstable) sorted by city (stable) Chicago 09:00:00 Chicago 09:25:52 Chicago 09:00:00 Phoenix 09:00:03 Chicago 09:03:13 Chicago 09:00:59 several textbook and system implementations Typical characteristics of such applications.Houston 09:00:13 Chicago 09:21:05 Chicago 09:03:13 also have this defect Chicago 09:00:59 Chicago 09:19:46 Chicago 09:19:32 • Huge array. Houston 09:01:10 Chicago 09:19:32 Chicago 09:19:46 Chicago 09:03:13 Chicago 09:00:00 Chicago 09:21:05 • Small number of key values. Seattle 09:10:11 Chicago 09:35:21 Chicago 09:25:52 Seattle 09:10:25 Chicago 09:00:59 Chicago 09:35:21 Phoenix 09:14:25 Houston 09:01:10 Houston 09:00:13 NOT S T O P O N E Q U A L K E Y S Chicago 09:19:32 Houston 09:00:13 Houston 09:01:10 sorted Chicago 09:19:46 Phoenix 09:37:44 sorted Phoenix 09:00:03 Chicago 09:21:05 Phoenix 09:00:03 Phoenix 09:14:25 Seattle 09:22:43 Phoenix 09:14:25 Phoenix 09:37:44 swap if we don't stop if we stop on Seattle 09:22:54 Seattle 09:10:25 Seattle 09:10:11 on equal keys equal keys Chicago 09:25:52 Seattle 09:36:14 Seattle 09:10:25 Chicago 09:35:21 Seattle 09:22:43 Seattle 09:22:43 Seattle 09:36:14 Seattle 09:10:11 Seattle 09:22:54 Phoenix 09:37:44 Seattle 09:22:54 Seattle 09:36:14
Stability when sorting on a second key
key
27 28 Duplicate keys: the problem 3-way partitioning
Mistake. Put all keys equal to the partitioning element on one side. Goal. Partition array into 3 parts so that: Consequence. ~ N2 / 2 compares when all keys equal. • Elements between lt and gt equal to partition element v. No larger elements to left of lt. B A A B A B B B C C C A A A A A A A A A A A • • No smaller elements to right of gt.
before v Recommended. Stop scans on keys equal to the partitioning element. lo hi Consequence. ~ N lg N compares when all keys equal. during
B A A B A B C C B C B A A A A A A A A A A A lt i gt after
lo lt gt hi
3-way partitioning
Desirable. Put all keys equal to the partitioning element in place. Dutch national flag problem. [Edsger Dijkstra] • Conventional wisdom until mid 1990s: not worth doing. A A A B B B B B C C C A A A A A A A A A A A • New approach discovered when fixing mistake in C library qsort(). • Now incorporated into qsort() and Java system sort.
29 30
3-way partitioning: Dijkstra's solution 3-way partitioning: trace
3-way partitioning. • Let v be partitioning element a[lo]. a[] • Scan i from left to right. v lt i gt 0 1 2 3 4 5 6 7 8 9 10 11 a[i] less than v: exchange a[lt] with a[i] and increment both lt and i - 0 0 11 R B W W R W B R R W B R - a[i] greater than v: exchange a[gt] with a[i] and decrement gt 0 1 11 R B W W R W B R R W B R - a[i] equal to v: increment i 1 2 11 B R W W R W B R R W B R 1 2 10 B R R W R W B R R W B W before v 1 3 10 B R R W R W B R R W B W 1 3 9 B R R B R W B R R W W W lo hi All the right properties. 2 4 9 B B R R R W B R R W W W during
3-way partitioning trace (array contents after each loop iteration)
31 32 3-way quicksort: Java implementation 3-way quicksort: visual trace
private static void sort(Comparable[] a, int lo, int hi) { if (hi <= lo) return; int lt = lo, gt = hi; Comparable v = a[lo]; int i = lo; while (i <= gt) { int cmp = a[i].compareTo(v); equal to partitioning element if (cmp < 0) exch(a, lt++, i++); else if (cmp > 0) exch(a, i, gt--); else i++; }
before v sort(a, lo, lt - 1); lo hi sort(a, gt + 1, hi); } during
lt i gt Visual trace of quicksort with 3-way partitioning after
lo lt gt hi
3-way partitioning
33 34
Duplicate keys: lower bound
Sorting lower bound. If there are n distinct keys and the ith one occurs
xi times, any compare-based sorting algorithm must use at least
n N! xi lg x lg N lg N when all distinct; x ! x ! x ! ∼− i N 1 2 n i=1 linear when only a constant number of distinct keys ··· compares in the worst case.
‣ selection Proposition. [Sedgewick-Bentley, 1997] proportional to lower bound ‣ duplicate keys Quicksort with 3-way partitioning is entropy-optimal. ‣ comparators Pf. [beyond scope of course] ‣ system sorts
Bottom line. Randomized quicksort with 3-way partitioning reduces running time from linearithmic to linear in broad class of applications.
35 36 Sorting applications Java system sorts
Sorting algorithms are essential in a broad variety of applications: Java uses both mergesort and quicksort. • Sort a list of names. • Arrays.sort() sorts array of Comparable or any primitive type. • Organize an MP3 library. • Uses quicksort for primitive types; mergesort for objects. • Display Google PageRank results. obvious applications • List RSS news items in reverse chronological order. import java.util.Arrays;
• Find the median. public class StringSort • Find the closest pair. { public static void main(String[] args) Binary search in a database. • problems become easy once items { • Identify statistical outliers. are in sorted order String[] a = StdIn.readAll().split("\\s+"); Find duplicates in a mailing list. Arrays.sort(a); • for (int i = 0; i < N; i++) StdOut.println(a[i]); • Data compression. } } • Computer graphics. • Computational biology. non-obvious applications • Supply chain management. Load balancing on a parallel computer. • Q. Why use different algorithms, depending on type? . . . Every system needs (and has) a system sort! 37 38
Java system sort for primitive types Achilles heel in Bentley-McIlroy implementation (Java system sort)
Engineering a sort function. [Bentley-McIlroy, 1993] Based on all this research, Java’s system sort is solid, right? • Original motivation: improve qsort(). more disastrous consequences in C • Basic algorithm = 3-way quicksort with cutoff to insertion sort. A killer input. • Partition on Tukey's ninther: median of the medians of 3 samples, • Blows function call stack in Java and crashes program. each of 3 elements. Would take quadratic time if it didn’t crash first. approximate median-of-9 •
nine evenly % more 250000.txt % java IntegerSort < 250000.txt R L A P M C G A X Z K R B R J J E spaced elements 0 Exception in thread "main" 218750 java.lang.StackOverflowError groups of 3 R A M G X K B J E 222662 at java.util.Arrays.sort1(Arrays.java:562) medians M K E 11 at java.util.Arrays.sort1(Arrays.java:606) 166672 at java.util.Arrays.sort1(Arrays.java:608) ninther K 247070 at java.util.Arrays.sort1(Arrays.java:608) 83339 at java.util.Arrays.sort1(Arrays.java:608) ......
Why use Tukey's ninther? 250,000 integers Java's sorting library crashes, even if • Better partitioning than random shuffle. between 0 and 250,000 you give it as much stack space as Windows allows • Less costly than random shuffle.
39 40 Achilles heel in Bentley-McIlroy implementation (Java system sort) System sort: Which algorithm to use?
McIlroy's devious idea. [A Killer Adversary for Quicksort] Many sorting algorithms to choose from: • Construct malicious input while running system quicksort, in response to elements compared. Internal sorts. • If v is partitioning element, commit to (v < a[i]) and (v < a[j]), but don't • Insertion sort, selection sort, bubblesort, shaker sort. commit to (a[i] < a[j]) or (a[j] > a[i]) until a[i] and a[j] are compared. • Quicksort, mergesort, heapsort, samplesort, shellsort. • Solitaire sort, red-black sort, splaysort, Dobosiewicz sort, psort, ... Consequences. • Confirms theoretical possibility. External sorts. Poly-phase mergesort, cascade-merge, oscillating sort. • Algorithmic complexity attack: you enter linear amount of data; server performs quadratic amount of work. Radix sorts. Distribution, MSD, LSD, 3-way radix quicksort.
Remark. Attack is not effective if array is shuffled before sort. Parallel sorts. • Bitonic sort, Batcher even-odd sort. • Smooth sort, cube sort, column sort. • GPUsort. Q. Why do you think system sort is deterministic?
41 42
System sort: Which algorithm to use? Sorting summary
Applications have diverse attributes. • Stable? • Parallel? inplace? stable? worst average best remarks • Deterministic? selection x N 2 / 2 N 2 / 2 N 2 / 2 N exchanges • Keys all distinct? • Multiple key types? insertion x x N 2 / 2 N 2 / 4 N use for small N or partially ordered • Linked list or arrays? shell x ? ? N tight code, subquadratic • Large or small records? N log N probabilistic guarantee Is your array randomly ordered? quick x N 2 / 2 2 N ln N N lg N • fastest in practice Need guaranteed performance? many more combinations of • improves quicksort in presence of attributes than algorithms 3-way quick x N 2 / 2 2 N ln N N duplicate keys
Elementary sort may be method of choice for some combination. merge x N lg N N lg N N lg N N log N guarantee, stable Cannot cover all combinations of attributes. ??? x x N lg N N lg N N lg N holy sorting grail
Q. Is the system sort good enough? A. Usually.
43 44 Which sorting algorithm?
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original ? ? ? ? ? ? sorted
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