CS 467/567: Divide and Conquer on the PRAM

Transcription

1 CS 467/567: Divide and Conquer on the PRAM Stefan D. Bruda Winter 2017

2 BINARY SEARCH Problem: Given a equence S 1..n orted in nondecreaing order and a value x, find the ubcript k uch that S i x If n proceor are available the problem can be olved in contant time: 1 All proceor read x 2 Each proceor P i compare x ith S i 3 All proceor P i (if any) that found x S i rite j into k uing min a combining operator Good running time but far from optimal Other CW model (Priority, Arbitrary, etc.) can alo be ued to break tie CS 467/567 (S. D. Bruda) Winter / 11

3 BINARY SEARCH Problem: Given a equence S 1..n orted in nondecreaing order and a value x, find the ubcript k uch that S i x If n proceor are available the problem can be olved in contant time: 1 All proceor read x 2 Each proceor P i compare x ith S i 3 All proceor P i (if any) that found x S i rite j into k uing min a combining operator Good running time but far from optimal Other CW model (Priority, Arbitrary, etc.) can alo be ued to break tie Naïve divide and conquer approach for N ă n proceor: 1 Divide the equence S into N roughly equally ized ubequence (of length Opn{Nq each) 2 Each proceor perform a equential binary earch to earch for x in one ubequence 3 Thoe proceor (if any) that found x rite the repective index into k uing min a combining operator Oplogpn{Nqq running time Ñ fater than the equential algorithm but not coniderably o CS 467/567 (S. D. Bruda) Winter / 11

4 PARALLEL BINARY SEARCH N proceor ued to perform an N ` 1-ay (rather than binary) earch Sequence of tage; in the firt tage all the equence i under conideration, in ubequent tage only a ubequence ill be under conideration At each tage the equence under conideration i plit into N ` 1 ubequence 1 Each proceor P i compare x ith the element at the right boundary of the i-th ubequence 2 If x ă then all the element in the i ` 1-t and higher ubequence can be dicarded 3 If x ą then all the element in the i-th and loer ubequence can be dicarded 4 If x then the index ha been found Thi proce reduce the equence under conidertion N time rather than jut halving it (like in the equential cae) The overall running time i thu Oplog N nq CS 467/567 (S. D. Bruda) Winter / 11

5 MERGING Problem: Given to equence of number (or more generally comparable value) A a 1, a 2,..., a r and B b 1, b 2,..., b orted in nondecreaing order, compute the equence C c 1, c 2,..., c r` uch that each c i belong to either A or B, ech a i and b i appear exactly once in C, and the equence C i orted in nondecreaing order Algorithm RAM-MERGEpA, Bq return C: 1 i Ð 1, j Ð 1 2 for k 1 to r ` do 1 if a i ă b j then c k Ð a i, i Ð i ` 1 2 ele c k Ð b j, j Ð j ` 1 Opnq running time (optimal) Requirement for the parallel algorithm: Sublinear and adaptive number of proceor Running time ubtantially maller than the equential running time, and alo adaptive Optimal CS 467/567 (S. D. Bruda) Winter / 11

6 PRAM MERGE Aume (ithout lo of generality) that r ď Algorithm PRAM-MERGEpA, Bq return C: Select N 1 element from A that divide A into N equence of approximately equal ize; call thi equence A 1 a1 1, a1 2,.... Similarly find the equence B 1 b1 1, b1 2,... that divide B into N equence of roughly the ame ize (contant time): 1 for i 1 do in parallel a 1 i Ð a i r{n, b 1 i Ð b i {N Merge A 1 and B 1 into a equence of triple V v 1, v 2,... v 2N 2, here each triple conit of an element of A 1 or B 1, it poition in A 1 or B 1, and the name of the equence of origin (A or B) (Oplog Nq time): 1 for i 1 to N do in parallel 1 Proceor P i ue binary earch on B 1 to find the mallet j uch that ai 1 ă bj 1 2 if j exit then v i`j 1 Ð pai 1, i.aq ele v i`n 1 Ð pai 1, i, Aq 2 for i 1 to N do in parallel 1 Proceor P i ue binary earch on A 1 to find the mallet j uch that bi 1 ă aj 1 2 if j exit then v i`j 1 Ð pbi 1, i.bq ele v i`n 1 Ð pbi 1, i, Aq CS 467/567 (S. D. Bruda) Winter / 11

7 PRAM MERGE (CONT D) Each proceor merge and inert into C the element of to ubequence, one from A and one from B. The indice of the to element (one in A and one in B) at hich each proceor begin merging are firt computed and tored in an array Q of pair (Opr ` {Nq time): 1 Q 1 Ð p1, 1q 2 for i 2 to N do in parallel 1 if v 2i 2 pa 1 k, k, Aq then proceor P i ue binary earch on B to find the mallet j uch that b j ą a 1 k Q i Ð pk r{n, jq 2 ele proceor P i ue binary earch on A to find the mallet j uch that a j ą b 1 k Q i Ð pj, k {N q 3 for i 1 to N do in parallel 1 Proceor P i ue RAM-MERGE and Q i px, yq to merge to ubequence beginning at a x and b y and place the reult in C beginning at index x ` y 1. The merge continue until either paq an element larger than or equal to the firt component of v 2i in each of A and B (hen i ď N 1), or pbq no element are left in either A or B (hen i N) Running time: Opn{N ` log nq Ñ optimal algorithm for N ď n{ log n CS 467/567 (S. D. Bruda) Winter / 11

8 LIGHTNING FAST PRAM SORTING Algorithm CRCW-SORTpS 1..n q return S 1 1..n : 1 for i 1 to n do in parallel for j 1 to n do in parallel 1 if i ą j _ i j ^ i ą j 2 then P ij rite 1 in c i uing ` a combining operation 3 ele P ij rite 0 in c i uing ` a combining operation 2 for i 1 to n do in parallel 1 P i1 tore S i into S 1 1`c i Method called enumeration or rank orting Contant running time Opn 2 q proceor Ñ Opn 2 q cot (not optimal) Likely of not a great practical value (number of proceor very high) CS 467/567 (S. D. Bruda) Winter / 11

9 ENUMERATION SORTING ON THE CREW PRAM What about the CREW PRAM? Can till compare one pair of value p i, j q in each proceor, but e cannot rite all the reult c i in a ingle memory location CS 467/567 (S. D. Bruda) Winter / 11

10 ENUMERATION SORTING ON THE CREW PRAM What about the CREW PRAM? Can till compare one pair of value p i, j q in each proceor, but e cannot rite all the reult c i in a ingle memory location Solution: 1 If i ą j _ i j ^ i ą j then proceor P ij rite 1 into c ij ; otherie P ij rite 0 into c ij 2 Set c i to n j 1 c ij then continue a in the CRCW algorithm Extra tep: c i Ð n j 1 c ij Keep adding (in parallel) pair of value until a ingle value remain Oplog nq time uing n proceor Overall running time: Oplog nq uing Opn 2 q proceor CS 467/567 (S. D. Bruda) Winter / 11

11 OPTIMAL SORTING ON THE CREW PRAM Algorithm CREW-SORTpS 1..n q return S 1..n : 1 Ditribute equal ize ubequence of S to the N proceor. Each proceor ill then ort it ubequence equentially (Oppn{Nq logpn{nqq time) 2 Keep merging pairie adjacent ubequence (in parallel) until one equence (of length n) i obtained (uing PRAM-MERGE) N{k ubequence (of length kn{n each) to merge in iteration k Allocate Opkq proceor per pair of ubequence for each merge Ñ Opn{N ` logpkn{nqq Opn{N ` log nq time per iteration Oplog Nq iteration Ñ Oppn{Nq log N ` log n log Nq overall time Running time: Oppn{Nq log n ` log 2 nq Cot: Opn log n ` N log 2 nq Ñ optimal for N ď n{ log n CS 467/567 (S. D. Bruda) Winter / 11

12 OPTIMAL SORTING ON THE CREW PRAM Algorithm CREW-SORTpS 1..n q return S 1..n : 1 Ditribute equal ize ubequence of S to the N proceor. Each proceor ill then ort it ubequence equentially (Oppn{Nq logpn{nqq time) 2 Keep merging pairie adjacent ubequence (in parallel) until one equence (of length n) i obtained (uing PRAM-MERGE) N{k ubequence (of length kn{n each) to merge in iteration k Allocate Opkq proceor per pair of ubequence for each merge Ñ Opn{N ` logpkn{nqq Opn{N ` log nq time per iteration Oplog Nq iteration Ñ Oppn{Nq log N ` log n log Nq overall time Running time: Oppn{Nq log n ` log 2 nq Cot: Opn log n ` N log 2 nq Ñ optimal for N ď n{ log n We can alo ort fater (Oplog nq time ith Opnq proceor, till optimal), but uch an algorithm doe not cale ell CS 467/567 (S. D. Bruda) Winter / 11

13 CONVEX HULL ON THE PRAM Algorithm PRAM-CONVEX-HULLpn, Qq: 1 Sort the point in Q according to their x coordinate 2 Partition Q into n 1{2 et Q 1, Q 2,..., Q n 1{2 of n 1{2 point each uch that the et are eparated by vertical line and Q i i to the left of Q j iff i ă j 3 for i 1 to n 1{2 do in parallel 1 if Q i ă 3 then CHpQ i q Ð Q i 2 ele CHpQ i q Ð PRAM-CONVEX-HULLpn 1{2, Q i q 4 return PRAM-MERGE-CHpCHpQ 1 q, CHpQ 2 q,..., CHpQ n 1{2qq Let the algorithm ue Opnq proceor Step 1 doable in Oplog nq time Step 2 take contant time (the et Q i are all ubequence of Q) Step 4 take Oplog nq time Overall the running time i tpnq tpn 1{2 q ` c log n and o tpnq Oplog nq Therefore cot i Opn log nq Ñ optimal (non-output enitive complexity) CS 467/567 (S. D. Bruda) Winter / 11

14 CONVEX HULL ON THE PRAM (CONT D) Algorithm PRAM-MERGE-CHpCHpQ 1 q, CHpQ 2 q,..., CHpQ n 1{2qq: Let u be the leftmot point of CHpQ 1 q and v the rightmot point of CHpQ n 1{2q Identify the upper hull: 1 Aign Opn 1{2 q proceor to each CHpQ i q 2 Each proceor aigned to CHpQ i q find the upper tangent common beteen CHpQ i q and CHpQ j q for ome i j 3 Beteen all common tangent beteen CHpQ i q and CHpQ j q, j ă i let L i (tangent ith CHpQ i q at point l i ) be the tangent ith the mallet lope 4 Beteen all common tangent beteen CHpQ i q and CHpQ j q, j ą i let R i (tangent ith CHpQ i q at point r i ) be the tangent ith the mallet lope 5 If the angle formed by L i and R i i maller than 180 degree then no point from CHpQ i q are in the upper hull; otherie include in the upper hull all the point beteen l i and r i (incluive) 6 Identify the upper hull a all the point from u to r 1, then all the point identified above, then all the point from r n 1{2 to v (incluive) Identify the loer hull (imilar to the upper hull) The loer hulli identified a above but thi time u and v are excluded Return the union of the upper and loer hull (array packing) CS 467/567 (S. D. Bruda) Winter / 11

15 CONVEX HULL ON THE PRAM (CONT D) Computing the upper tangent of CHpQ i q and CHpQ j q in Oplog nq time: Let and be the middle point in the (orted) upper hull from CHpQ i q and CHpQ j q If i the upper tangent of CHpQ i q and CHpQ j q then e are done (Cae a) Otherie repeat from Step 1 but excluding at leat half the point of at leat one upper hull (Cae b h) (a) (c) (e) (b) (d) (f) (g) (h) CS 467/567 (S. D. Bruda) Winter / 11