Merging to ordered sequences. Efficient (Parallel) Sorting. Merging (cont.)

Transcription

1 Efficient (Paae) Soting One of the most fequent opeations pefomed by computes is oganising (soting) data The access to soted data is moe convenient/faste Thee is a constant need fo good soting agoithms incuding sequentia, paae and distibuted soutions Thee is a pethoa of soting agoithms. We aeady know that one can use heaps fo soting. Hee we focus on two soting pocedues incuding quick-sot and mege-sot Meging to odeed sequences The key to mege-sot is meging pocedue mege, s.t., having two input sequences A= <a 1 a 2 a m > and B= <b 1 b 2 b n > it poduces combined C= <c 1 c 2 c m+n > Exampe: A = < 3, 8, 9 > B = < 1, 5, 7 > mege(a, B) = < 1, 3, 5, 7, 8, 9 > 27/04/2015 Appied Agoithmics - week10 1 pick the minimum A: B: X: Y: and save it hee pointes

2 X: Y: X: Y: X: Y: X: 54 Y:

3 X: Y: 75 X: 54 Y: Singe un of mege pocedue poduces combined soted sequence. Thus the time compexity is inea O(m+n). Divide-and-Conque Method A vey natua ecusive appoach Divide if the input size is sma then sove the pobem diecty; othewise divide the input data into two o moe disjoint subsets Recu ecusivey sove the sub-pobems associated with the subsets Conque take the soutions to the sub-pobems and mege them into a soution to the oigina pobem Mege-Soting Divide: if input sequence S has 0 o 1 eement then etun S; othewise spit S into two sequences S 1 and S 2, each containing about ½ eements of S Recu: ecusivey sot sequences S 1 and S 2 Conque: Put the eements back into S by meging the soted sequences S 1 and S 2 into a singe soted sequence week 10 Compexity of Agoithms 11 week 10 Compexity of Agoithms 12

4 Mege-Soting (top down appoach) Mege-Soting (exampe) Divide the input sequence eveny to S 1 & S 2 Recu S 1 S 2 Conque by meging soted sequences week 10 Compexity of Agoithms 13 week 10 Compexity of Agoithms 14 Mege-Soting (anaysis) Mege-Soting (anaysis) Reca that meging two soted sequences S 1 and S 2 takes O(n 1 +n 2 ) time, whee n 1 is the size of S 1 and n 2 is the size of S 2 The depth of the ecusion is O(og n) due to the having pocess Thus mege-sot uns in O(n og n) time in the wost (and aveage) case week 4 Compexity of Agoithms 15 week 4 Compexity of Agoithms 16

5 Quick-Sot Divide if S >1, seect a pivot vaue x in S and ceate thee sequences: L, E and G, s.t., L stoes eements in S < x E stoes eements in S = x G stoes eements in S > x Recu ecusivey sot sequences L & G Conque put soted eements fom L, E and finay fom G back to S. Quick-Sot Tee Divide the sequence S using andom pivot x Recu L (<x) H(>x) Conque by concatenating soted sequences week 4 Compexity of Agoithms 17 week 4 Compexity of Agoithms 18 Quick-Sot (exampe) Quick-Sot (wost case) Let s i be the sum of the input sizes of the nodes at depth i in a quick sot tee T s i n-i (and s i = n-i when use of pivots ead aways to ony one nonempty sequence: eithe L o G) The wost-case compexity is bounded by O(n 2 ). week 4 Compexity of Agoithms 19 week 4 Compexity of Agoithms 20

6 Quick-Sot (andomised agoithm) Quick-Sot (andomised agoithm) Thm: the expected unning time of andomised (pivot is chosen in andom) quick-sot is O(n og n) Poof: The expected numbe of times that a fai coin must be fipped unti it shows heads k times is 2k. Randomy chosen pivot is ight if neithe of the goups L no G is > ¾ S The pobabiity of a success in choosing a ight pivot is ½ A path in quick-sot tee can contain at most og 4/3 n nodes with ight pivots Hence, the expected ength of each path is 2og 4/3 n week 4 Compexity of Agoithms 21 week 4 Compexity of Agoithms 22 Lowe Bound (compaison-based mode) Lowe Bound (compaison-based mode) In compaison-based mode the input eements can be compaed ony with themseves and the esut of each compaison x i x j is aways yes o no Thm: the unning time of any compaison-based soting agoithm is Ω(n og n) in the wost case Poof: Soting of n eements can be identified with ecognising a paticua pemutation of n eements Thee is n!=n (n-1) 2 1 pemutations of n eements Each compaison spits a goup of pemutations into two goups (one that satisfies the inequaity and one that doesn t) In ode to ensue that the size of each goup of pemutations is bought down to one we need og 2 (n!) > og (n/2) n/2 =n/2 og n/2 = Ω(n og n) compaisons week 4 Compexity of Agoithms 23 week 4 Compexity of Agoithms 24

7 List anking and pefix sums In the ink anking pobem one is expected to compute fo each eement its distance to the font of the ist In the pefix sum pobem one is expected to compute fo each pefix of the ist the sum of the keys stoed in this pefix Computing pefix sums with a keys of vaue 1 is equivaent to the ink anking pobem. List anking and pefix sums /04/2015 Appied Agoithmics - week /04/2015 Appied Agoithmics - week10 26 A ead at distance 2 0 & add to thei own vaues A ead at distance 2 1 & add to thei own vaues /04/2015 Appied Agoithmics - week /04/2015 Appied Agoithmics - week10 28

8 A ead at distance 2 2 & add to thei own vaues A ead at distance 2 3 & add to thei own vaues /04/2015 Appied Agoithmics - week /04/2015 Appied Agoithmics - week10 30 List anking and pefix sums List anking and pefix sums List anking and pefix sums can be computed in O(og n) time when n is the size of the input Duing evey singe ound we incease knowedge about peceding bock of 2 i positions in O(1) time. Afte O(og n) ounds od doubing the job is done We need aso anothe too that wi aow us to coect and distibute infomation to a pocessos aso in O(og n) time 27/04/2015 Appied Agoithmics - week /04/2015 Appied Agoithmics - week10 32

9 Infomation dissemination P 0 infoms neighbou at distance 2 0 m Infomation dissemination P 0, P 1 infom neighbous at distance 2 1 m m m m m m m m /04/2015 Appied Agoithmics - week /04/2015 Appied Agoithmics - week10 34 Infomation dissemination Infomation dissemination P 0, P 1 P 2, P 3 infom neighbous at distance 2 2 m m m m P 0,, P 6, P 7 infom neighbous at distance 2 3 m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m 27/04/2015 Appied Agoithmics - week /04/2015 Appied Agoithmics - week10 36

10 Infomation coection/dissemination The pocess of coection of infomation is done by evesing communication (diection of aows) used duing infomation dissemination Both pocesses take time O(og n). This means that pocessos can a agee on simpe decisions (via exchanging sma messages), e.g., is thee any wok eft to do? in time O(og n). 27/04/2015 Appied Agoithmics - week10 37 Paae Quick-Sot Sequence S[ ] is being soted S[ ] The oca size of the input n= -+1 Each P i (i=,..,) picks vaue S[i] with pob. 1/n S[ ] A unique pivot vaue p is communicated to a (if none o moe vaues ae picked the pocess is epeated) L[ ] < The vaues fom S[...] ae distibuted to L[ ] & H[ ] H[ ] < Using ist anking and pefix sums compute the anks of vaues in L and H L[ ] The numbe of vaues #L in L is communicated H[ ] The vaues ae copied back to S as foows Vaue with ank α in L is moved to S[+α-1] The pivot p is moved to S[+#L] Vaue with ank β in H is moved to S[+#L+β-1] S[ ] Sot ecusivey S[ +#L-1] & S[+#L+1 ] 27/04/2015 Appied Agoithmics - week10 38 Paae Quick-Sot Paae Mege-Sot The compexity anaysis of paae quick-sot Evey stage takes at most time O(og n) Expected numbe of stages is O(og n) The tota computation time is O(og 2 n) The numbe of pocessos needed is n The tota wok is O(nog 2 n) One can educe wok to optima using n/og n pocessos Assume two haves L & H of S[ ] ae aeady (ecusivey) soted The oca size of the input n= -+1 Using binay seach compute a ank of each L vaue in the othe haf H, and vice vesa Combine (add) the two anks (fom L and H) to find the new position in the soted sequence S[ ] S[ ] S[ ] L H 5 5 L H 10 L H 27/04/2015 Appied Agoithmics - week /04/2015 Appied Agoithmics - week10 40

11 Paae Mege-Sot The compexity anaysis of paae mege-sot Each stage (binay seach) takes at most time O(og n) The numbe of ecusive stages is O(og n) The tota computation time is O(og 2 n) The numbe of pocessos needed is n The tota wok is O(nog 2 n) One can educe wok to optima using n/og n pocessos 27/04/2015 Appied Agoithmics - week10 41