Order statistics. Order Statistics. Randomized divide-andconquer. Example. CS Spring 2006

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1 406 CS Sprig 006 Order Statistics Carola We Slides courtesy of Charles Leiserso with small chages by Carola We CS 5633 Aalysis of Algorithms 406 Order statistics Select the ith smallest of elemets the elemet with ra i. i : miimum; i : maximum; i + or +: media. Naive algorithm: Sort ad idex ith elemet. Worst-case ruig time Θ log Θ log, usig merge sort or heapsort ot quicsort. CS 5633 Aalysis of Algorithms 406 Radomized divide-adcoquer algorithm RAND-SLCA, p, q, i ith smallest of A[p..q] if p q the retur A[p] r RAND-PARIIONA, p, q r p + raa[r] if i the retur A[r] if i < the retur RAND-SLCA, p, r, i else retur RAND-SLCA, r +, q, i A[r] A[r] p r q CS 5633 Aalysis of Algorithms xample Select the i 7th smallest: pivot Partitio: i 7 4 Select the 7 4 3rd smallest recursively. CS 5633 Aalysis of Algorithms 4

2 Ituitio for aalysis All our aalyses today assume that all elemets are distict. Lucy: 90 Θ Ulucy: Θ Worse tha sortig! log 9 CAS arithmetic series Aalysis of expected time he aalysis follows that of radomized quicsort, but it s a little differet. Let the radom variable for the ruig time of RAND-SLC o a iput of size, assumig radom umbers are idepedet. For 0,,,, defie the idicator radom variable if PARIION geerates a : split, 0 otherwise. 406 CS 5633 Aalysis of Algorithms CS 5633 Aalysis of Algorithms Aalysis cotiued o obtai a upper boud, assume that the i th elemet always falls i the larger side of the partitio: max{0, } if 0 : split, max{, } if : split, M max{, 0} if : 0 split, 0 max{, } CS 5633 Aalysis of Algorithms Calculatig expectatio [ ] ae expectatios of both sides. CS 5633 Aalysis of Algorithms 8

3 CS 5633 Aalysis of Algorithms Calculatig expectatio Liearity of expectatio. ] [ CS 5633 Aalysis of Algorithms Calculatig expectatio Idepedece of from other radom choices. ] [ CS 5633 Aalysis of Algorithms 406 Calculatig expectatio Liearity of expectatio; [ ]. Θ + ] [ CS 5633 Aalysis of Algorithms 406 Calculatig expectatio ] [ Θ + Arithmetic series

4 Hairy recurrece But ot quite as hairy as the quicsort oe. [ ] [ ] Prove: [] c for costat c > 0. he costat c ca be chose large eough so that [] c for the base cases. 3 Use fact: 8 exercise. 406 CS 5633 Aalysis of Algorithms Substitutio method [ ] c Substitute iductive hypothesis. CS 5633 Aalysis of Algorithms 4 Substitutio method [ ] Use fact. c c 3 8 Substitutio method [ ] c c 3 8 c c Θ 4 xpress as desired residual. 406 CS 5633 Aalysis of Algorithms CS 5633 Aalysis of Algorithms 6

5 Substitutio method [ ] c c 3 8 c c Θ 4 c, if c is chose large eough so that c4 domiates the Θ. Summary of radomized order-statistic selectio Wors fast: liear expected time. xcellet algorithm i practice. But, the worst case is very bad: Θ. Q. Is there a algorithm that rus i liear time i the worst case? A. Yes, due to Blum, Floyd, Pratt, Rivest, ad arja [973]. IDA: Geerate a good pivot recursively. 406 CS 5633 Aalysis of Algorithms CS 5633 Aalysis of Algorithms 8 Worst-case liear-time order statistics SLCi,. Divide the elemets ito groups of 5. Fid the media of each 5-elemet group by rote.. Recursively SLC the media x of the 5 group medias to be the pivot. 3. Partitio aroud the pivot x. Let rax. 4. if i the retur x elseif i < the recursively SLC the ith smallest elemet i the lower part else recursively SLC the i th smallest elemet i the upper part Same as RAND- SLC Choosig the pivot 406 CS 5633 Aalysis of Algorithms CS 5633 Aalysis of Algorithms 0

6 Choosig the pivot Choosig the pivot. Divide the elemets ito groups of 5.. Divide the elemets ito groups of 5. Fid the media of each 5-elemet group by rote. 406 CS 5633 Aalysis of Algorithms 406 CS 5633 Aalysis of Algorithms Choosig the pivot Aalysis x x 406. Divide the elemets ito groups of 5. Fid the media of each 5-elemet group by rote.. Recursively SLC the media x of the 5 group medias to be the pivot. CS 5633 Aalysis of Algorithms At least half the group medias are x, which is at least 5 0 group medias. CS 5633 Aalysis of Algorithms 4

7 Aalysis Assume all elemets are distict. Aalysis Assume all elemets are distict. x x 406 At least half the group medias are x, which is at least 5 0 group medias. herefore, at least 3 0 elemets are x. CS 5633 Aalysis of Algorithms At least half the group medias are x, which is at least 5 0 group medias. herefore, at least 3 0 elemets are x. Similarly, at least 3 0 elemets are x. CS 5633 Aalysis of Algorithms 6 Aalysis Assume all elemets are distict. Need at most for worst-case rutime At least 3 0 elemets are x at most -3 0 elemets are x At least 3 0 elemets are x at most -3 0 elemets are x he recursive call to SLC i Step 4 is executed recursively o -3 0 elemets. Aalysis Assume all elemets are distict. Use fact that ab a-b-b page he recursive call to SLC i Step 4 is executed recursively o at most 70+ elemets. 406 CS 5633 Aalysis of Algorithms CS 5633 Aalysis of Algorithms 8

8 406 Θ 5 Θ 70 + Developig the recurrece SLCi,. Divide the elemets ito groups of 5. Fid the media of each 5-elemet group by rote.. Recursively SLC the media x of the 5 group medias to be the pivot. 3. Partitio aroud the pivot x. Let rax. 4. if i the retur x elseif i < the recursively SLC the ith smallest elemet i the lower part else recursively SLC the i th smallest elemet i the upper part CS 5633 Aalysis of Algorithms Solvig the recurrece d Substitutio: c + c + c + d 5 0 c 9 c + c + d 0 c c c d 0 c, if c is chose large eough to hadle d for Θ CS 5633 Aalysis of Algorithms 30 Coclusios Sice the wor at each level of recursio is basically a costat fractio 90 smaller, the wor per level is a geometric series domiated by the liear wor at the root. I practice, this algorithm rus slowly, because the costat i frot of is large. he radomized algorithm is far more practical. xercise: ry to divide ito groups of 3 or CS 5633 Aalysis of Algorithms 3