A Tight Unconditional Lower Bound on Distributed Random Walk Computation

1. A Tight Unconditional Lower Bound on Distributed Random Walk Computation DanuponNanongkai GopalPandurangan Atish Das Sarma Google Research U. of Vienna & Georgia Tech Nanyang Technological University & Brown University

2. PLAN Problem & Result Techniques Overview From bounded-round communication complexity to distributed algo. lower bound

3. Distributed network (CONGEST mode) O(log n) bits through each edge per round log n log n log n log n log n log n log n • n: number of nodes • D: diameter log n log n

4. Want a random walk of length lfrom s s

5. Trivial algorithm: Forward a token randomly for lrounds s

6. The token ends somewhere s

7. If we repeat, the token might end in a different node s

8. This process takes l rounds to send a token in a random walk manner.

9. Can we forward the token in a random walk manner fasterthan l rounds? • (Formally, we want to sample destination node according to the distribution induced by the l –step random walk.)

10. PODC’09 Exists sub-linear time algorithm when l >>D: O(l 2/3D1/3)

11. PODC’10 • Time improvement: O((l D)1/2) • W(l1/2) lower bound for a restricted class of algorithms

12. Is there an algorithm that achieves O(l 1/2-e D10) time? Is there an algorithm that achieves O(l 1/2D1/2) time?

13. PODC’11 Tight lower bound for anyalgorithm: W((l D)1/2)

14. PODC’12New speaker needed ? ?

15. Theorem For any n, D ≥ log n and l such that D ≤l≤ (n/D3log n)1/4, there exists an n-node networks of diameter D such that computing a random walk requires W((l D)1/2)rounds

16. PLAN Problem & Result Techniques Overview From bounded-round communication complexity to distributed algo. lower bound

17. Techniquebuilds on STOC’11paper “Distributed Verification and Hardness of Distributed Approximation” Amos Korman Atish Das Sarma LiahKor Stephan Holzer Roger Wattenhofer DanuponNanongkai GopalPandurangan David Peleg

18. STOC’11 General technique for proving lower bounds by reducing from communication complexity Here • General technique for proving lower bounds by reducing from bounded-round communication complexity

19. Why random walk is tougher • No clear verification version • Not optimization problem but a random process • Want D to be a multiplicative factor

20. Reductions in Das Sarma et al., STOC’11 Distributed Algorithms Communication Complexity EQALITY/DISJ/etc verification EQALITY/DISJ/etc Verification Simulatino theorem Spanning Tree verification MST Approximation

21. Approximated reduction in this paper Bounded-round Communication complexity Distributed Algorithms Communication Complexity Pointer Chasing (Search problem) Pointer Chasing NEW simulation theorem Distributed random walk

22. Why do we need bounded-round communication complexity?

23. Because the previous “connection” has nothing to do with D. (But we need W((l D)1/2)lower bound.)

24. Simulation Theorem (Das Sarma et al. STOC’11) For some graph G, If we have T-time distributed algorithm for problem P then the communication complexityversion can be solved with ≤ T communication bits and ≤ T/D rounds New Simulation Theorem (This paper)

25. PLAN Result summary Techniques Overview From bounded-round communication complexity to distributed algo. lower bound

26. Bounded-round Communication complexityof the pointer chasing problem

27. ∞ ∞ f g f g f g c d c d c d a b a b a b b b Where’s the token after 5 steps? round 1 b round 2 c d round 3 Alice Bob round 4 a round 5 b

28. We can solve the problem in 5 rounds by sending 5 nodes.

29. What if we want 4 rounds?

30. Theorem:4 rounds is impossible (have to send ∞ messages)[Nisan-Wigderson STOC’XX]

31. Distributed computing

32. Alice and Bob are connected by many paths of length 16 16 green nodes ∞ Bob Alice

33. In each step, one edge can carry one bit on each direction

34. In each step, one edge can carry one bit on each direction 16green nodes ∞ Bob Alice Write input here

35. How many steps do they need to find where the token goes?

36. A: 17steps because the network diameter is 17 16 green nodes ∞ Bob Alice

37. Let’s make the diameter smaller

38. Now the diameter is 8 How many steps do we need? 16 green nodes ∞ 4green nodes 4 green nodes Bob Alice

39. Claim: Need > 8steps.

40. Proof: Assume there is a distributed algorithm A that uses ≤ 8 steps

41. f g f g c d c d a b a b b b ? Bob Alice A ?

42. f g f g ? ? c d c d Bob’s network Alice’s network a b a b Run A Run A A A Alice Bob

43. Alice cannot run A on all machinesbecause she doesn’t know Bob’s input

44. ? ? Step 0 Alice Bob

45. ? ? Step 1 Alice Bob

46. ? ? Step 1 Alice Bob

47. ? ? ? ? ? ? a1 ? ? ? ? b1 keep this keep this Step 1 a1 b1 Alice Bob b1 = bit sent by A run on Bob’s machine

48. ? ? ? ? ? ? ? ? ? ? ? ? a2 ? ? ? ? b2 Step 2 a2 b2 Alice Bob b1 = bit sent by A run on Bob’s machine

49. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Step 3 Alice Bob b1 = bit sent by A run on Bob’s machine

50. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Step 4 Alice Bob b1 = bit sent by A run on Bob’s machine