Distributed Set Reachability

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1 Dstt St Rty S Gj Mt T Mx-P Isttt Its, Usty U Gy SIGMOD 2016, S Fs, USA

2 Dstt St Rty Dstt St Rty (DSR) s zt ty xt t sts stt stt Dstt St Rty 2

3 Dstt St Rty Dstt St Rty (DSR) s zt ty xt t sts stt stt Dstt St Rty 2

4 Dstt St Rty Dstt St Rty (DSR) s zt ty xt t sts stt stt : T G G Rty. s t, t xsts t s t t G [SABW13, YCZ10] Dstt St Rty 2

5 Dstt St Rty Dstt St Rty (DSR) s zt ty xt t sts stt stt R s,,,,,,,,, G G Rty. s t, t xsts t s t t G [SABW13, YCZ10] St Rty. S T, s s s, t, s tt s S, t T s t G [TKC + 14, GA13] Dstt St Rty 2

6 Dstt St Rty Dstt St Rty (DSR) s zt ty xt t sts stt stt G 1 G 2 G 3 R s,,,,,,,,, G G s tt t G 1, G 2, G 3 Dstt Rty. s t, t xsts t s t t G [FWW12] Dstt St Rty. S T, s s s, t, s tt s S, t T s t G [?] Dstt St Rty 2

7 Ats At 1: SPARQL 1.1 ty ts ss w s Dstt St Rty 3

8 Ats At 1: SPARQL 1.1 ty ts ss w s Ex: F E w w N Pz SELECT?s WHERE {?s <I>?ty.?s <w> N Pz.?ty <ti>*?ty.?ty <to> E.} Dstt St Rty 3

9 Ats At 1: SPARQL 1.1 ty ts ss w s Ex: F E w w N Pz SELECT?s WHERE {?s <I>?ty.?s <w> N Pz.?ty <ti>*?ty.?ty <to> E.} Atw Cts = Sü U B ti Cts = F Gy Dstt St Rty 3

10 Ats At 1: SPARQL 1.1 ty ts ss w s Ex: F E w w N Pz SELECT?s WHERE {?s <I>?ty.?s <w> N Pz.?ty <ti>*?ty.?ty <to> E.} Atw Cts = Sü U B ti Cts = F Gy St Rty Qy: Cts Cts Dstt St Rty 3

11 Ats At 2: Cty tss s tws Dstt St Rty 4

12 Ats At 2: Cty tss s tws Ex: Hw s t zts t B Gts Bs = W Bt D J. T T G P Ozts = F Ft M Gts Ft Dstt St Rty 4

13 Ats At 2: Cty tss s tws Ex: Hw s t zts t B Gts Bs = W Bt D J. T T G P Ozts = F Ft M Gts Ft St Rty Qy: Bs Ozts Dstt St Rty 4

14 Hw t s DSR y? Qy: {,, } {, } Ssts G 1 G 2 G 3 Dstt St Rty 5

15 Hw t s DSR y? Vtx-Ct s: L P, G,... Ct( ) T Vtx sss Msss tx Msss ssss Sst ( 1) Sst Sst ( + 1) Qy: {,, } {, } Ssts G 1 G 2 G 3 Dstt St Rty 5

16 Hw t s DSR y? Vtx-Ct s: L P, G,... Ct( ) T Vtx sss Msss tx Msss ssss Sst ( 1) Sst Sst ( + 1) Qy: {,, } {, } Ssts 1 [] [] [] G 1 G 2 G 3 Dstt St Rty 5

17 Hw t s DSR y? Vtx-Ct s: L P, G,... Ct( ) T Vtx sss Msss tx Msss ssss Sst ( 1) Sst Sst ( + 1) Qy: {,, } {, } Ssts 1 2 [] [, ] [] [, ] [] [] [] [] G 1 [] G 2 G 3 Dstt St Rty 5

18 Hw t s DSR y? Vtx-Ct s: L P, G,... Ct( ) T Vtx sss Msss tx Msss ssss Sst ( 1) Sst Sst ( + 1) Qy: {,, } {, } Ssts 1 2. [,, ] [,, ] G 1 G 2 G 3 Dstt St Rty 5

19 Hw t s DSR y? Vtx-Ct s: L P, G,... Ct( ) T Vtx sss Msss tx Msss ssss Sst ( 1) Sst Sst ( + 1) Qy: {,, } {, } Ssts 1 2. [,, ] [,, ] G 1 G 2 G 3 P: O s s ( 10M s) y wt S = 10 T = 10 Dtst T( s) Ssts C. Sz(MB) NtD St Dstt St Rty 5

20 Hw t s DSR y? Vtx-Ct s: L P, G,... Ct( ) T Vtx sss Msss tx Msss ssss Sst ( 1) Sst Sst ( + 1) Qy: {,, } {, } Ssts 1 2. [,, ] [,, ] G 1 G 2 G 3 Cs: s tts & st xs t y tts ( t) ts t sts y ss ts Dstt St Rty 5

21 A t st DSR? Ojts: 1. z t sz sss x 2. t s & s tts s s ss s 3. s t s Dstt St Rty 6

22 A t st DSR? Ojts: 1. z t sz sss x ty s t tw ssts t & t t t ty.., ty y s By 2. t s & s tts s s ss s 3. s t s Dstt St Rty 6

23 A t st DSR? Ojts: 1. z t sz sss x ty s t tw ssts t & t t t ty.., ty y s By 2. t s & s tts s s ss s + y C t, xs tz (st) ty s [YCZ10, SABW13, GA13, TKC + 14] 3. s t s Dstt St Rty 6

24 A t st DSR? Ojts: 1. z t sz sss x ty s t tw ssts t & t t t ty.., ty y s By 2. t s & s tts s s ss s + y C t, xs tz (st) ty s [YCZ10, SABW13, GA13, TKC + 14] 3. s t s y ss sts s s y t SCCs Dstt St Rty 6

25 Ptt G1 G2 G3 Dstt St Rty 7

26 Ptt G1 G2 G3 I 1 = {} I 2 = {,, } I 3 = {, } Dstt St Rty 7

27 Ptt G1 G2 G3 I 1 = {} I 2 = {,, } I 3 = {, } O 1 = {, } O 2 = {, } O 3 = {} Dstt St Rty 7

28 Ptt G1 G2 G3 I 1 = {} I 2 = {,, } I 3 = {, } O 1 = {, } O 2 = {, } O 3 = {} Dstt St Rty 7

29 Ptt G1 G2 G3 I 1 = {} I 2 = {,, } I 3 = {, } O 1 = {, } O 2 = {, } O 3 = {} St 1: Ct ty I O (st ty) (By G) By G Dstt St Rty 7

30 Ptt G1 G2 G3 I 1 = {} I 2 = {,, } I 3 = {, } O 1 = {, } O 2 = {, } O 3 = {} St 1: Ct ty I O (st ty) (By G) G B 1 G B 2 G B 3 Dstt St Rty 7

31 Ptt G1 G2 G3 I 1 = {} I 2 = {,, } I 3 = {, } O 1 = {, } O 2 = {, } O 3 = {} St 1: Ct ty I O (st ty) (By G) G B 1 G B 2 G B 3 Dstt St Rty 7

32 Ptt G1 G2 G3 I 1 = {} I 2 = {,, } I 3 = {, } O 1 = {, } O 2 = {, } O 3 = {} St 1: Ct ty I O (st ty) (By G) St 2: M y (C G) Dstt St Rty 7

33 Ptt G1 G2 G3 I 1 = {} I 2 = {,, } I 3 = {, } O 1 = {, } O 2 = {, } O 3 = {} St 1: Ct ty I O (st ty) (By G) St 2: M y (C G) St 3: Oty, xs t s tts s y ss Dstt St Rty 7

34 Qy ss Dstt St Rty 8

35 Qy ss Fw Lst: F 1 = {,,,, } Bw Lst: B 1 = {,, } F 2 = {,, } B 2 = {,, } F 3 = {,,, } B 3 = {,,, } Dstt St Rty 8

36 Qy ss Qy: Fw Lst: F 1 = {,,,, } Bw Lst: B 1 = {,, } F 2 = {,, } B 2 = {,, } F 3 = {,,, } B 3 = {,,, } I s t t, t y ss y Dstt St Rty 8

37 Qy ss Qy: Fw Lst: F 1 = {,,,, } Bw Lst: B 1 = {,, } F 2 = {,, } B 2 = {,, } F 3 = {,,, } B 3 = {,,, } Dstt St Rty 8

38 Qy ss Qy: Fw Lst: F 1 = {,,,, } Bw Lst: B 1 = {,, } F 2 = {,, } B 2 = {,, } F 3 = {,,, } B 3 = {,,, } I s t - st t, s tst tw tts Dstt St Rty 8

39 Qy ss Qy: {,, } {, } Fw Lst: F 1 = {,,,, } Bw Lst: B 1 = {,, } St1: F 2 = {,, } B 2 = {,, } {, } F 3 = {,,, } B 3 = {,,, } Dstt St Rty 8

40 Qy ss Qy: {,, } {, } Fw Lst: F 1 = {,,,, } Bw Lst: B 1 = {,, } St1: St2: F 2 = {,, } B 2 = {,, } {, } {,,, } F 3 = {,,, } B 3 = {,,, } {,,,, } Dstt St Rty 8

41 S t L Gs O t s sty s t sz y Sz: O( =1 ( I O ) + E C ) G: 43.6M (8.5 E ) LJ-20M: 861.4M (43.07 E ) By G Dstt St Rty 9

42 S t L Gs O t s sty s t sz y Sz: O( =1 ( I O ) + E C ) G: 43.6M (8.5 E ) LJ-20M: 861.4M (43.07 E ) By G F tt G = {G 1, G 2, G 3 }, w t y s ws. G1 G2 G3 Fw sts (-t s) Dstt St Rty 9

43 S t L Gs O t s sty s t sz y Sz: O( =1 ( I O ) + E C ) G: 43.6M (8.5 E ) LJ-20M: 861.4M (43.07 E ) By G F tt G = {G 1, G 2, G 3 }, w t y s ws. G1 G2 G3 Fw sts (-t s) -s, s st ts G 3 w t Dstt St Rty 9

44 S t L Gs O t s sty s t sz y Sz: O( =1 ( I O ) + E C ) G: 43.6M (8.5 E ) LJ-20M: 861.4M (43.07 E ) By G F tt G = {G 1, G 2, G 3 }, w t y s ws. G1 G2 G3 Fw sts (-t s) -s, s st ts G 3 w t t-s, s st ts G 1 w t Dstt St Rty 9

45 S t L Gs O t s sty s t sz y Sz: O( =1 ( I O ) + E C ) G: 43.6M (8.5 E ) LJ-20M: 861.4M (43.07 E ) By G F tt G = {G 1, G 2, G 3 }, w t y s ws. G1 G2 G3 Fw sts (-t s) -s, s st ts G 3 w t t-s, s st ts G 1 w t Dstt St Rty 9

46 S t L Gs O t s sty s t sz y Sz: O( =1 ( I O ) + E C ) G: 43.6M (8.5 E ) LJ-20M: 861.4M (43.07 E ) By G F tt G = {G 1, G 2, G 3 }, w t y s ws. G1 G2 G3 Fw sts (-t s) I1 = {} O1 = {, } I2 = {,, } O2 = {, } I3 = {, } O3 = {} Dstt St Rty 9

47 S t L Gs O t s sty s t sz y Sz: O( =1 ( I O ) + E C ) G: 43.6M (8.5 E ) LJ-20M: 861.4M (43.07 E ) By G F tt G = {G 1, G 2, G 3 }, w t y s ws. G1 G2 G3 Fw sts (-t s) I1 = {} I2 = {,, } I3 = {, } O1 = {, } O2 = {, } O3 = {} υ1 = {} υ2 = {, }, υ3 = {} υ4 = {, } Fw sts (-t s) ν1 = {, } ν2 = {}, ν3 = {} ν4 = {} Bw sts (t-t s) Dstt St Rty 9

48 S t L Gs O t s sty s t sz y ν1, υ2 ν3 υ4 υ3 ν2 ν4 Sz: O( =1 ( N Υ ) + E C ) N : st -t s t Υ : st t-t s t N I, Υ O, E C E C By G Css By G F tt G = {G 1, G 2, G 3 }, w t y s ws. G1 G2 G3 Fw sts (-t s) I1 = {} I2 = {,, } I3 = {, } O1 = {, } O2 = {, } O3 = {} υ1 = {} υ2 = {, }, υ3 = {} υ4 = {, } Fw sts (-t s) ν1 = {, } ν2 = {}, ν3 = {} ν4 = {} Bw sts (t-t s) Dstt St Rty 9

49 S t L Gs O t s sty s t sz y ν1, υ2 ν3 υ4 υ3 ν2 ν4 Sz: O( =1 ( N Υ ) + E C ) N : st -t s t Υ : st t-t s t N I, Υ O, E C E C By G Css By G F tt G = {G 1, G 2, G 3 }, w t y s ws. υ2 υ3 ν3 ν2 υ4 ν1 υ4, ν1 υ2 ν2, υ1 υ3 ν3 ν4 υ1 ν4 w t sts, s t y s Dstt St Rty 9

50 Dtsts: Et S L Gs V E Gs V E Az 403,394 3,387,388 LJ-68M 4,847,571 68,993,773 BSt 685,230 7,600,595 Twtt-1.4B 41,652,230 1,468,364,884 G 875,713 5,105,039 Fs-500M 97,290, ,982,284 NtD 325,729 1,497,134 Fs-1B 156,595, ,965,047 St 281,903 2,312,497 LUBM-500M 115,561, ,002,176 LJ-20M 2,545,981 20,000,000 LUBM-1B 222,213, ,394,352 G tsts St: Mst:1, Ss:9, My:64 GB Dstt St Rty 10

51 Dtsts: Et S L Gs V E Gs V E Az 403,394 3,387,388 LJ-68M 4,847,571 68,993,773 BSt 685,230 7,600,595 Twtt-1.4B 41,652,230 1,468,364,884 G 875,713 5,105,039 Fs-500M 97,290, ,982,284 NtD 325,729 1,497,134 Fs-1B 156,595, ,965,047 St 281,903 2,312,497 LUBM-500M 115,561, ,002,176 LJ-20M 2,545,981 20,000,000 LUBM-1B 222,213, ,394,352 G tsts St: Mst:1, Ss:9, My:64 GB P: DSR DSR-F G G++ T (s) x x x x x Az BSt G NtD St LJ-20M LJ-68M Fs DSR s s t st t G, G++, DSR-F Dstt St Rty 10 Twtt LUBM

52 Dtsts: Et S L Gs V E Gs V E Az 403,394 3,387,388 LJ-68M 4,847,571 68,993,773 BSt 685,230 7,600,595 Twtt-1.4B 41,652,230 1,468,364,884 G 875,713 5,105,039 Fs-500M 97,290, ,982,284 NtD 325,729 1,497,134 Fs-1B 156,595, ,965,047 St 281,903 2,312,497 LUBM-500M 115,561, ,002,176 LJ-20M 2,545,981 20,000,000 LUBM-1B 222,213, ,394,352 G tsts St: Mst:1, Ss:9, My:64 GB Sty: DSR G++wE G++ G Qy T ( s) # Ptts # Ptts LJ-68 Fs-1B Dstt St Rty 10

53 Dtsts: Et S L Gs V E Gs V E Az 403,394 3,387,388 LJ-68M 4,847,571 68,993,773 BSt 685,230 7,600,595 Twtt-1.4B 41,652,230 1,468,364,884 G 875,713 5,105,039 Fs-500M 97,290, ,982,284 NtD 325,729 1,497,134 Fs-1B 156,595, ,965,047 St 281,903 2,312,497 LUBM-500M 115,561, ,002,176 LJ-20M 2,545,981 20,000,000 LUBM-1B 222,213, ,394,352 G tsts St: Mst:1, Ss:9, My:64 GB Sty: DSR G++wE G++ G DSR G++wE G++ G Qy T ( s) # Ptts # Ptts Qy T ( s) x10 50x50 100x100 Qy Szs x10 50x50 100x100 Qy Szs LJ-68 Fs-1B LJ-68 Fs-1B Dstt St Rty 10

54 Css W t stt t Dstt St Rty Dstt St Rty 11

55 Css W t stt t Dstt St Rty Ptt t ty t s xs sty t wtt t sty Dstt St Rty 11

56 Css W t stt t Dstt St Rty Ptt t ty t s xs sty t wtt t sty Css t sts s s t s Dstt St Rty 11

57 Css W t stt t Dstt St Rty Ptt t ty t s xs sty t wtt t sty Css t sts s s t s Exts t y s ( y tsts) Dstt St Rty 11

58 Css W t stt t Dstt St Rty Ptt t ty t s xs sty t wtt t sty Css t sts s s t s Exts t y s ( y tsts) T y y ttt Dstt St Rty 11

59 Rs I W F, X W, Y W, P ts stt ty s, PVLDB 5 (2012),. 11, S G K Ayw, PxS: ty s t-s t-stt t s RDF s y s sx tts, WWW, 2013, St St, As A, St J. Bt, G W, FERRARI: x t ty sst x, ICDE, 2013, M T, Mtz K, F Ct, T-A H-V, K P, As K, Ts N, Hy T. V, T t : Et t-s ts, PVLDB 8 (2014),. 4, H Y, Vt Cj, M J. Z, GRAIL: S Rty Ix L Gs, PVLDB 3 (2010),. 1-2, Qsts? F ts, s st y st: 84 Dstt St Rty 12

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