Introduction to Algorithms 6.046J/18.401J/SMA5503

Transcription

1 Itroductio to Algorithms 6.046J/8.40J/SMA5503 Lecture 4 Prof. Charles E. Leiserso

2 Quicsort Proposed by C.A.R. Hoare i 962. Divide-ad-coquer algorithm. Sorts i place lie isertio sort, but ot lie merge sort. Very practical with tuig. Day 6 Itroductio to Algorithms L4.2

3 Divide ad coquer Quicsort a -elemet array:. Divide: Partitio the array ito two subarrays aroud a pivot x such that elemets i lower subarray x elemets i upper subarray. x xx x 2. Coquer: Recursively sort the two subarrays. 3. Combie: Trivial. Key: Liear-time partitioig subroutie. Day 6 Itroductio to Algorithms L4.3

4 Partitioig subroutie PARTITIONA, p, q A[p.. q] x A[p] pivot A[p] i p for j p + to q do if A[ j] x the i i + exchage A[i] A[ j] exchage A[p] A[i] retur i Ivariat: Ruig time O for elemets. xx x x?? p i j q Day 6 Itroductio to Algorithms L4.4

5 Example of partitioig i j Day 6 Itroductio to Algorithms L4.5

6 Example of partitioig i j Day 6 Itroductio to Algorithms L4.6

7 Example of partitioig i j Day 6 Itroductio to Algorithms L4.7

8 Example of partitioig i j Day 6 Itroductio to Algorithms L4.8

9 Example of partitioig i j Day 6 Itroductio to Algorithms L4.9

10 Example of partitioig i j Day 6 Itroductio to Algorithms L4.0

11 Example of partitioig i j Day 6 Itroductio to Algorithms L4.

12 Example of partitioig i j Day 6 Itroductio to Algorithms L4.2

13 Example of partitioig i j Day 6 Itroductio to Algorithms L4.3

14 Example of partitioig i j Day 6 Itroductio to Algorithms L4.4

15 Example of partitioig i j Day 6 Itroductio to Algorithms L4.5

16 Example of partitioig i Day 6 Itroductio to Algorithms L4.6

17 Pseudocode for quicsort QUICKSORTA, p, r if p < r the q PARTITIONA, p, r QUICKSORTA, p, q QUICKSORTA, q+, r Iitial call: QUICKSORTA,, Day 6 Itroductio to Algorithms L4.7

18 Aalysis of quicsort Assume all iput elemets are distict. I practice, there are better partitioig algorithms for whe duplicate iput elemets may exist. Let T worst-case ruig time o a array of elemets. Day 6 Itroductio to Algorithms L4.8

19 Worst-case of quicsort Iput sorted or reverse sorted. Partitio aroud mi or max elemet. Oe side of partitio always has o elemets. T T 0 + T Θ + T T Θ 2 + Θ + Θ + Θ arithmetic series Day 6 Itroductio to Algorithms L4.9

20 Worst-case recursio tree T T0 + T + c Day 6 Itroductio to Algorithms L4.20

21 Worst-case recursio tree T T T0 + T + c Day 6 Itroductio to Algorithms L4.2

22 Worst-case recursio tree T T0 + T + c c T0 T Day 6 Itroductio to Algorithms L4.22

23 Worst-case recursio tree T T0 + T + c c T0 c T0 T 2 Day 6 Itroductio to Algorithms L4.23

24 Worst-case recursio tree T T0 + T + c c T0 c T0 c 2 T0 O Θ Day 6 Itroductio to Algorithms L4.24

25 Worst-case recursio tree T T0 + T + c c T0 c T0 c 2 T0 Θ O Θ Θ 2 Day 6 Itroductio to Algorithms L4.25

26 h Worst-case recursio tree c Θ c T T0 + T + c Θ c 2 Θ O Θ Θ Θ 2 T Θ + Θ 2 Θ 2 Day 6 Itroductio to Algorithms L4.26

27 Best-case aalysis For ituitio oly! If we re lucy, PARTITION splits the array evely: T 2T/2 + Θ Θ lg same as merge sort What if the split is always 9 :? T + Θ T T 0 0 What is the solutio to this recurrece? Day 6 Itroductio to Algorithms L4.27

28 Aalysis of almost-best case T Day 6 Itroductio to Algorithms L4.28

29 Aalysis of almost-best case c T 9 0 T 0 Day 6 Itroductio to Algorithms L4.29

30 Aalysis of almost-best case c c 0 9 c T 00 T 00 T 00 T 00 Day 6 Itroductio to Algorithms L4.30

31 Aalysis of almost-best case 00 c c 0 c 9 c 0 log 0/9 9 c 9 c 8 c c c c Θ O leaves Θ Day 6 Itroductio to Algorithms L4.3

32 Aalysis of almost-best case 00 c c 0 c 9 c 0 log 0 log 0/9 9 c 9 c 8 c c c c Θ Θ lg Lucy! O leaves c log 0 Θ T c log 0/9 + Ο Day 6 Itroductio to Algorithms L4.32

33 More ituitio Suppose we alterate lucy, ulucy, lucy, ulucy, lucy,. L 2U/2 + Θ lucy U L + Θ ulucy Solvig: L 2L/2 + Θ/2 + Θ 2L/2 + Θ Θ lg Lucy! How ca we mae sure we are usually lucy? Day 6 Itroductio to Algorithms L4.33

34 Radomized quicsort IDEA: Partitio aroud a radom elemet. Ruig time is idepedet of the iput order. No assumptios eed to be made about the iput distributio. No specific iput elicits the worst-case behavior. The worst case is determied oly by the output of a radom-umber geerator. Day 6 Itroductio to Algorithms L4.34

35 Radomized quicsort aalysis Let T the radom variable for the ruig time of radomized quicsort o a iput of size, assumig radom umbers are idepedet. For 0,,,, defie the idicator radom variable X if PARTITION geerates a : split, 0 otherwise. E[X ] Pr{X } /, sice all splits are equally liely, assumig elemets are distict. Day 6 Itroductio to Algorithms L4.35

36 Aalysis cotiued T T0 + T + Θ if 0: split, T + T 2 + Θ if : 2 split, M T + T0 + Θ if : 0 split, 0 X T + T + Θ. Day 6 Itroductio to Algorithms L4.36

37 Calculatig expectatio E[ T ] E 0 X T + T + Θ Tae expectatios of both sides. Day 6 Itroductio to Algorithms L4.37

38 Day 6 Itroductio to Algorithms L4.38 Calculatig expectatio [ ] + Θ + + Θ ] [ T T X E T T X E E T Liearity of expectatio.

39 Day 6 Itroductio to Algorithms L4.39 Calculatig expectatio [ ] [ ] [ ] + Θ + + Θ + + Θ ] [ T E T X E T T X E T T X E E T Idepedece of X from other radom choices.

40 Day 6 Itroductio to Algorithms L4.40 Calculatig expectatio [ ] [ ] [ ] [ ] [ ] Θ Θ + + Θ + + Θ ] [ E T E T T E T X E T T X E T T X E E T Liearity of expectatio; E[X ] /.

41 Day 6 Itroductio to Algorithms L4.4 Calculatig expectatio [ ] [ ] [ ] [ ] [ ] [ ] 2 ] [ E T E T E T T E T X E T T X E T T X E E T + Θ Θ Θ + + Θ + + Θ + Summatios have idetical terms.

42 Hairy recurrece [ ] 2 E T 2 [ ] + Θ E T The 0, terms ca be absorbed i the Θ. Prove: E[T] alg for costat a > 0. Choose a large eough so that alg domiates E[T] for sufficietly small 2. Use fact: lg 2 lg Day 6 Itroductio to Algorithms L exercise.

43 Substitutio method 2 [ ] a lg + Θ E T 2 Substitute iductive hypothesis. Day 6 Itroductio to Algorithms L4.43

44 Substitutio method [ ] E T 2 Use fact. 2a 2 2 a lg 2 lg + Θ Θ Day 6 Itroductio to Algorithms L4.44

45 Substitutio method [ ] E T 2 2a 2 2 a lg lg + Θ a lg a Θ 4 Express as desired residual Θ Day 6 Itroductio to Algorithms L4.45

46 Substitutio method [ ] E T 2 2a 2 a lg a lg 2 a, a 4 + Θ + Θ Day 6 Itroductio to Algorithms L4.46 lg 2 lg 8 2 Θ if a is chose large eough so that a/4 domiates the Θ.

47 Quicsort i practice Quicsort is a great geeral-purpose sortig algorithm. Quicsort is typically over twice as fast as merge sort. Quicsort ca beefit substatially from code tuig. Quicsort behaves well eve with cachig ad virtual memory. Day 6 Itroductio to Algorithms L4.47