All-Pairs shortest paths via fast matrix multiplication

Transcription

1 All-Pars shortest paths va fast matrx multplato Ur Zw Tel Avv Uversty Summer Shool o Shortest Paths (PATH0) DIKU, Uversty of Copehage 1. Algebra matrx multplato a. Strasse s algorthm b. Retagular matrx multplato. M-Plus matrx multplato a. Equvalee to the APSP problem b. Expesve reduto to algebra produts. Fredma s tr 3. APSP udreted graphs a. A O(.38 ) algorthm for uweghted graphs (Sedel) b. A O(M.38 ) algorthm for weghted graphs (Shosha-Zw) 4. APSP dreted graphs 1. A O(M ) algorthm (Zw). A O(M.38 ) preproessg / O() query aswerg alg. (Yuster-Z) 3. A O(.38 logm) (1+ )-approxmato algorthm. Summary ad ope problems 1. Algebra matrx multplato a. Strasse s algorthm b. Retagular matrx multplato. M-Plus matrx multplato a. Equvalee to the APSP problem b. Expesve reduto to algebra produts. Fredma s tr 3. APSP udreted graphs a. A O(.38 ) algorthm for uweghted graphs (Sedel) b. A O(M.38 ) algorthm for weghted graphs (Shosha-Zw) 4. APSP dreted graphs 1. A O(M ) algorthm (Zw). A O(M.38 ) preproessg / O() query aswerg alg. (Yuster-Z) 3. A O(.38 logm) (1+ )-approxmato algorthm. Summary ad ope problems Matrx multplato A ( a j ) B ( b j ) C ( j ) j abj 1 j Ca be omputed avely O( 3 ) tme. Matrx multplato Complexty (by defto) Strasse (1969) Coppersmth, Wograd (1990) Cojeture/Ope problem: +o(1)??? Multplyg matres C 11 C1 A11 A1 B 11 B1 C1 C A1 A B1 B multplatos 4 addtos

2 Strasse s algorthm Strasse s algorthm C M M M M C M M 1 3 C M M 1 4 C M M M M M ( A A )( B B ) M ( A A ) B 1 11 M A ( B B ) M A ( B B ) M ( A A ) B 11 1 M ( A A )( B B ) M ( A A )( B B ) multplatos 18 addtos/subtratos Vew eah matrx as a matrx whose elemets are // matres. Apply the algorthm reursvely. T() = 7 T(/) + O( ) T() = O( log7/log )=O(.81 ) Retagular Matrx multplato p A p B C p j abj 1 Coppersmth (1997): Complexty 1.8 p o(1) For p 0.9, omplexty = +o(1)!!! 1. Algebra matrx multplato a. Strasse s algorthm b. Retagular matrx multplato. M-Plus matrx multplato a. Equvalee to the APSP problem b. Expesve reduto to algebra produts. Fredma s tr 3. APSP udreted graphs a. A O(.38 ) algorthm for uweghted graphs (Sedel) b. A O(M.38 ) algorthm for weghted graphs (Shosha-Zw) 4. APSP dreted graphs 1. A O(M ) algorthm (Zw). A O(M.38 ) preproessg / O() query aswerg alg. (Yuster-Z) 3. A O(.38 logm) (1+ )-approxmato algorthm. Summary ad ope problems A terestg speal ase of the APSP problem M-Plus Produts A B C AB m{ a b j j M-Plus produt C AB 8 3 m { a b j j

3 Solvg APSP by repeated squarg If W s a by matrx otag the edge weghts of a graph. The W s the dstae matrx. By duto, W gves the dstaes realzed by paths that use edges. Thus: for 1 to log do DD*D APSP() MPP() log Atually: APSP() = O(MPP()) B A B A D X = C D C E F (ABD*C)* EBD* X * = = G H D*CE D*GBD* APSP() APSP(/) + 6 MPP(/) + O( ) Algebra Produt C AB a b j j O(.38 ) M-Plus Produt C AB m{ a b j j? The fast algebra algorthms aot be used, as the m operato has o verse 1. Algebra matrx multplato a. Strasse s algorthm b. Retagular matrx multplato. M-Plus matrx multplato a. Equvalee to the APSP problem b. Expesve reduto to algebra produts. Fredma s tr 3. APSP udreted graphs a. A O(.38 ) algorthm for uweghted graphs (Sedel) b. A O(M.38 ) algorthm for weghted graphs (Shosha-Zw) 4. APSP dreted graphs 1. A O(M ) algorthm (Zw). A O(M.38 ) preproessg / O() query aswerg alg. (Yuster-Z) 3. A O(.38 logm) (1+ )-approxmato algorthm. Summary ad ope problems Usg matrx multplato to ompute m-plus produts 11 1 a11 a1 b11 b1 1 a1 a b1 b m{ a b ' j j j x x x x x x x x a11 a1 b11 b a1 a b1 b 1 a bj x j frst( ) j Usg matrx multplato to ompute m-plus produts x x x x x x x x a11 a1 b11 b a1 a b1 b 1.38 polyomal produts Assume: 0 a j, b j M M operatos per polyomal produt M.38 operatos per max-plus produt

4 Tryg to mplemet the repeated squarg algorthm for 1 to log do DD*D Cosder a easy ase: all weghts are 1. After the -th terato, the fte elemets D are the rage {1,,. The ost of the m-plus produt s.38 The ost of the last produt s 3.38!!! 1. Algebra matrx multplato a. Strasse s algorthm b. Retagular matrx multplato. M-Plus matrx multplato a. Equvalee to the APSP problem b. Expesve reduto to algebra produts. Fredma s tr 3. APSP udreted graphs a. A O(.38 ) algorthm for uweghted graphs (Sedel) b. A O(M.38 ) algorthm for weghted graphs (Shosha-Zw) 4. APSP dreted graphs 1. A O(M ) algorthm (Zw). A O(M.38 ) preproessg / O() query aswerg alg. (Yuster-Z) 3. A O(.38 logm) (1+ )-approxmato algorthm. Summary ad ope problems Fredma s tr The m-plus produt of two matres a be dedued after oly O(. ) addtos ad omparsos. Breag a square produt to several retagular produts m A 1 A B 1 B A* B m A * B MPP() (/m) (MPP(,m,) + ) m Fredma s tr Fredma s tr (ot.) a r +b rj a s +b sj a r - a s b sj - b rj A B m a r +b rj a s +b sj a r - a s b sj - b rj Geerate all the dfferees a r - a s ad b sj - b rj. Sort them usg O(m ) omparsos. (No-trval!) Merge the two sorted lsts usg O(m ) omparsos. Naïve alulato requres m operatos Fredma observed that the result a be ferred after performg oly O(m ) operatos The orderg of the elemets the sorted lst determes the result of the m-plus produt!!!

5 . 11 =a 17 +b 71 1 =a 14 +b 4... Deso Tree Complexty a 17 -a 19 b 9 -b 7 yes o 11 =a 13 +b 31 1 =a 1 +b =a 18 +b 81 1 =a 16 +b =a 1 +b 1 1 =a 13 +b 3... All-Pars Shortest Paths dreted graphs wth real edge weghts Rug tme [Floyd 6] [Warshall 6] 3 3 (log log / log ) 1/3 [Fredma 76] 3 (log log / log ) 1/ [Taaoa 9] 3 / (log ) 1/ [Dobosewz 90] 3 (log log / log ) /7 [Ha 04] 3 log log / log 3 (log log ) 1/ / log 3 / log [Taaoa 04] [Zw 04] [Cha 0] 1. Algebra matrx multplato a. Strasse s algorthm b. Retagular matrx multplato. M-Plus matrx multplato a. Equvalee to the APSP problem b. Expesve reduto to algebra produts. Fredma s tr 3. APSP udreted graphs a. A O(.38 ) algorthm for uweghted graphs (Sedel) b. A O(M.38 ) algorthm for weghted graphs (Shosha-Zw) 4. APSP dreted graphs 1. A O(M ) algorthm (Zw). A O(M.38 ) preproessg / O() query aswerg alg. (Yuster-Z) 3. A O(.38 logm) (1+ )-approxmato algorthm. Summary ad ope problems Dreted versus udreted graphs y x z (x,z) (x,y) + (y,z) y x z (x,z) (x,y) + (y,z) (x,z) (x,y) (y,z) Dstaes G ad ts square G Let G=(V,E). The G =(V,E ), where (u,v)e f ad oly f (u,v)e or there exsts wv suh that (u,w),(w,v)e Let (u,v) be the dstae from u to v G. Let (u,v) be the dstae from u to v G. Lemma: (u,v)= (u,v)/, for every u,vv. Thus: (u,v)= (u,v) or (u,v)= (u,v)-1 Dstaes G ad ts square G (ot.) Lemma: If (u,v)= (u,v) the for every eghbor w of v we have (u,w) (u,v). Lemma: If (u,v)= (u,v) 1 the for every eghbor w of v we have (u,w) (u,v) ad for at least oe eghbor (u,w) < (u,v). ( v, w) E Let A be the adjaey matrx of the G. Let C be the dstae matrx of G a ( CA) : deg( v) u, w u, w w, v u, v u, v w

6 1. If A s a all oe matrx, the all dstaes are 1.. Compute A, the adjaey matrx of the squared graph. Sedel s algorthm 3. Fd, reursvely, the dstaes the square. 4. Dede, usg matrx multplato, for every two vertes u,v, whether ther dstae s twe the dstae the square, or twe mus 1. Algorthm APD(A) f A=J the retur J-I else C APD(A ) X CA, deg Ae 1 d j j -[x j < j deg j ] retur D ed Complexty: O(.38 log ) 1. Algebra matrx multplato a. Strasse s algorthm b. Retagular matrx multplatobv. M-Plus matrx multplato a. Equvalee to the APSP problem b. Expesve reduto to algebra produts. Fredma s tr 3. APSP udreted graphs a. A O(.38 ) algorthm for uweghted graphs (Sedel) b. A O(M.38 ) algorthm for weghted graphs (Shosha-Zw) 4. APSP dreted graphs 1. A O(M ) algorthm (Zw). A O(M.38 ) preproessg / O() query aswerg alg. (Yuster-Z) 3. A O(.38 logm) (1+ )-approxmato algorthm. Summary ad ope problems Sampled Repeated Squarg (Z 98) Choose a subset of V for 1 to log 3/ do { of sze (9l)/s s(3/) +1 Brad( V, (9l)/s ) Dm{ D, D[V,B]*D[B,V] Selet the olums Selet the rows The s also of D Wth a slghtly whose hgh probablty, more omplated of D whose des all determst are dstaes B are algorthm orret! des are B Sampled Dstae Produts (Z 98) I the -th terato, the set B s of sze l / s, where s = (3/) +1 The matres get smaller ad smaller but the elemets get larger ad larger Sampled Repeated Squarg - Corretess for 1 to log 3/ do { s(3/) +1 Brad(V,(9 l )/s) Dm{ D, D[V,B]*D[B,V] Ivarat: After the -th terato, dstaes that are attaed usg (3/) edges are orret. Cosder a shortest path that uses (3/) +1 edges s /3 Falure 9l Let s = (3/) + (1 ) probablty : s Retagular Matrx multplato p p Naïve omplexty: p [Coppersmth 97]: 1.8 p o(1) For p 0.9, omplexty = +o(1)!!!

7 Complexty of APSP algorthm The -th terato: l / s l / s m{ Ms, s s s=(3/) +1 The elemets are of absolute value Ms M Algebra matrx multplato a. Strasse s algorthm b. Retagular matrx multplato. M-Plus matrx multplato a. Equvalee to the APSP problem b. Expesve reduto to algebra produts. Fredma s tr 3. APSP udreted graphs a. A O(.38 ) algorthm for uweghted graphs (Sedel) b. A O(M.38 ) algorthm for weghted graphs (Shosha-Zw) 4. APSP dreted graphs 1. A O(M ) algorthm (Zw). A O(M.38 ) preproessg / O() query aswerg alg. (Yuster-Z) 3. A O(.38 logm) (1+ )-approxmato algorthm. Summary ad ope problems The preproessg algorthm (YZ 0) ; B V for 1 to log 3/ do { s(3/) +1 Brad(B,(9l)/s) D[V,B] m{d[v,b], D[V,B]*D[B,B] D[B,V] m{d[b,v], D[B,B]*D[B,V] The APSP algorthm for 1 to log 3/ do { s(3/) +1 Brad(V,(9l)/s) Dm{ D, D[V,B]*D[B,V] Twe Sampled Dstae Produts The query aswerg algorthm (u,v) D[{u,V]*D[V,{v] v u Query tme: O()

8 The preproessg algorthm: Corretess Let B be the -th sample. B 1 B B 3 Ivarat: After the -th terato, f u B or vb ad there s a shortest path from u to v that uses at most (3/) edges, the D(u,v)= (u,v). The query aswerg algorthm: Corretess Suppose that the shortest path from u to v uses betwee (3/) ad (3/) +1 edges Cosder a shortest path that uses (3/) +1 edges u v 1. Algebra matrx multplato a. Strasse s algorthm b. Retagular matrx multplato. M-Plus matrx multplato a. Equvalee to the APSP problem b. Expesve reduto to algebra produts. Fredma s tr 3. APSP udreted graphs a. A O(.38 ) algorthm for uweghted graphs (Sedel) b. A O(M.38 ) algorthm for weghted graphs (Shosha-Zw) 4. APSP dreted graphs 1. A O(M ) algorthm (Zw). A O(M.38 ) preproessg / O() query aswerg alg. (Yuster-Z) 3. A O(.38 logm) (1+ )-approxmato algorthm. Summary ad ope problems Approxmate m-plus produts SCALE(A,M,R): Obvous dea: salg APX-MPP(A,B,M,R) : A SCALE(A,M,R) B SCALE(B,M,R) retur MPP(A,B ) Ra j / M, f 0 aj M aj! " #, otherwse $ Complexty s R.38, stead of M.38, but small values a be greatly dstorted. Addaptve Salg APX-MPP(A,B,M,R) : C for r log R to log M do A SCALE(A, r,r) B SCALE(B, r,r) C m{c,mpp(a,b ) ed Complexty s R.38 logm Streth 1+4/R 1. Algebra matrx multplato a. Strasse s algorthm b. Retagular matrx multplato. M-Plus matrx multplato a. Equvalee to the APSP problem b. Expesve reduto to algebra produts. Fredma s tr 3. APSP udreted graphs a. A O(.38 ) algorthm for uweghted graphs (Sedel) b. A O(M.38 ) algorthm for weghted graphs (Shosha-Zw) 4. APSP dreted graphs 1. A O(M ) algorthm (Zw). A O(M.38 ) preproessg / O() query aswerg alg. (Yuster-Z) 3. A O(.38 logm) (1+ )-approxmato algorthm. Summary ad ope problems

9 All-Pars Shortest Paths graphs wth small teger weghts Udreted graphs. Edge weghts {0,1, M Rug tme M.38 [Shosha-Zw 99] Improves results of [Alo-Gall-Margalt 91] [Sedel 9] All-Pars Shortest Paths graphs wth small teger weghts Dreted graphs. Edge weghts { M,,0, M Rug tme M [Zw 98] Improves results of [Alo-Gall-Margalt 91] [Taaoa 98] Aswerg dstae queres Dreted graphs. Edge weghts { M,,0, M Preproessg tme M.38 Query tme [Yuster-Zw 0] Approxmate All-Pars Shortest Paths graphs wth o-egatve teger weghts Dreted graphs. Edge weghts {0,1, M (1+ )-approxmate dstaes I partular, ay M 1.38 dstaes a be omputed M.38 tme. For dese eough graphs wth small eough edge weghts, ths mproves o Goldberg s SSSP algorthm. M.38 vs. m 0. logm Rug tme (.38 log M)/ [Zw 98] Ope problems A O(.38 ) algorthm for the dreted uweghted APSP problem? A O( 3- ) algorthm for the APSP problem wth edge weghts {1,,,? A O(.- ) algorthm for the SSSP problem wth edge weghts {0,±1, ±,, ±?