Phase-based algorithms for file migration

Transcription

1 Phase-based algorithms for file migration Marcin Bieńkowski Jarek Byrka Marcin Mucha University of Wrocław University of Warsaw HALG 2018 (previously on ICALP 2017)

2 File migration Weighted graph "2

3 File migration Weighted graph One shared file of size D "2

4 File migration Weighted graph One shared file of size D One step of input: "2

5 File migration Weighted graph One shared file of size D I want to access a part of the file One step of input: Request to file at v i. "2

6 File migration Request cost = distance Weighted graph One shared file of size D I want to access a part of the file One step of input: Request to file at v i. "2

7 File migration Request cost = distance Weighted graph One shared file of size D Migration cost = D * distance I want to access a part of the file One step of input: Request to file at v i. Optional migration. "2

8 File migration Request cost = distance One step of input: Request to file at vi. Optional migration. Weighted graph One shared file of size D I want to access a part of the file Migration cost = D * distance "2 Online problem Competitive ratio: maxi ALG(I) / OPT(I)

9 What is known (deterministic algorithms and large D) Trees 3-competitive algorithm exists It is optimal "3

10 What is known (deterministic algorithms and large D) Trees 3-competitive algorithm exists It is optimal General graphs Best lower bound = Best algorithm (Move-To-Local-Min) is competitive [Bartal, Charikar, Indyk 97] "3

11 Move-To-Local-Min (4.086-competitive) Works in phases of length c D = Θ(D). A 0 "4

12 Move-To-Local-Min (4.086-competitive) Works in phases of length c D = Θ(D). Within a phase: pay for requests r 1, r 2,,r cd A 0 "4

13 Move-To-Local-Min (4.086-competitive) Works in phases of length c D = Θ(D). Within a phase: pay for requests r 1, r 2,,r cd A 0 "4

14 Move-To-Local-Min (4.086-competitive) Works in phases of length c D = Θ(D). Within a phase: pay for requests r 1, r 2,,r cd A 0 "4

15 Move-To-Local-Min (4.086-competitive) Works in phases of length c D = Θ(D). Within a phase: pay for requests r 1, r 2,,r cd A 0 "4

16 Move-To-Local-Min (4.086-competitive) Works in phases of length c D = Θ(D). Within a phase: pay for requests r 1, r 2,,r cd A 0 "4

17 Move-To-Local-Min (4.086-competitive) Works in phases of length c D = Θ(D). Within a phase: pay for requests r 1, r 2,,r cd A 0 "4

18 Move-To-Local-Min (4.086-competitive) Works in phases of length c D = Θ(D). Within a phase: pay for requests r 1, r 2,,r cd At the end of a phase: migrate the file to node x minimizing D d(a 0, x) + X c D i=1 d(x, r i) A 0 "4

19 Move-To-Local-Min (4.086-competitive) Works in phases of length c D = Θ(D). Within a phase: pay for requests r 1, r 2,,r cd At the end of a phase: migrate the file to node x minimizing D d(a 0, x) + X c D i=1 d(x, r i) A 0 Make close migrations Migrate towards requests "4

20 Proof for MTLM? A piece of proof: Creative applications of TRIANGLE INEQUALITY "5

21 Can we do better?

22 Can we do better? > "6

23 Can we do better? > "6

24 Recreating MTLM proof Parameters c and α for MTLM "7

25 Recreating MTLM proof Parameters c and α for MTLM LP black box "7

26 Recreating MTLM proof Parameters c and α for MTLM LP black box instance maximizing the comp. ratio "7

27 Recreating MTLM proof Parameters c and α for MTLM LP black box Not an exact description! instance maximizing the comp. ratio "7

28 Recreating MTLM proof Parameters c and α for MTLM LP black box instance maximizing the comp. ratio "7

29 Recreating MTLM proof Parameters c and α for MTLM Optimize numerically LP black box instance maximizing the comp. ratio "7

30 Recreating MTLM proof Parameters c and α for MTLM Optimize numerically LP black box instance maximizing the comp. ratio Comp. ratio = "7

31 Our approach Parameters c and α for MTLM LP black box "8

32 Our approach Parameters c and α for MTLM LP black box "8

33 Our approach Parameters c and α for MTLM LP black box "8

34 Our approach Parameters c and α for MTLM LP black box "8

35 Our approach Parameters c and α for MTLM LP black box "8

36 Our approach Parameters c and α for MTLM LP black box "8

37 Our approach Parameters α, β, γ, δ, ε, λ, for variable-length phase algorithms LP black box "8

38 Our approach Parameters α, β, γ, δ, ε, λ, for variable-length phase algorithms LP black box instance maximizing the comp. ratio "8

39 Our approach Parameters α, β, γ, δ, ε, λ, for variable-length phase algorithms LP black box Optimize by a local search instance maximizing the comp. ratio "8

40 Our approach Parameters α, β, γ, δ, ε, λ, for variable-length phase algorithms LP black box Optimize by a local search instance maximizing the comp. ratio Competitive ratio = 4 "8

41 Thank you! Icons made by Freepik from