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

2. 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

3. A random walk of length lfrom s s

4. Forward a token randomly for l rounds s

5. The token ends somewhere s

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

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

8. Can we forward the token in a random walk manner faster than l rounds?

9. PODC’09 Exists sub-linear time algorithm when l >>D

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

11. 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?

12. Today Tight lower bound for any algorithms: W((l D)1/2)

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

14. Main tool comes from ... “Distributed Verification and Hardness of Distributed Approximation”, STOC’11 Amos Korman Atish Das Sarma LiahKor Stephan Holzer Roger Wattenhofer DanuponNanongkai GopalPandurangan David Peleg

15. Distributed algorithms for the above problems require W(n1/2+D)time

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

17. Reductions in Das Sarma et al., STOC’11 Distributed Algorithms Communication Complexity EQALITY/DISJ/etc verification EQALITY/DISJ/etc Verification Reduction in this paper Pointer Chasing (Search problem) Random Walk Destination Searching

18. Simulation Theorem (Das Sarma et al. STOC’11) If the distributed equality verification can be solved in T rounds, for any T ≤ b/2, then the communication complexityversion can be solved with ≤T communication G Alice Bob x = y ? x  {0, 1}b y  {0, 1}b Communication Complexity Distributed Algorithms x = y ? Alice Bob x  {0, 1}b y  {0, 1}b

19. Simulation Theorem (Das Sarma et al. STOC’11) If the distributed equality verification can be solved in T rounds, for any T ≤ b/2, then the communication complexityversion can be solved with ≤T communication and ≤ T rounds Once we know we have to reduce from the pointer chasing problem, W(l 1/2) almost follows

20. Main difficultyHaving D as multiplicative factorin the lower bound W(n1/2), W(l 1/2) W((nD)1/2), W((l D)1/2)

21. Difficulty:The connection in [STOC’11] has nothing to do with D “Connection” Theorem If the distributed equality verification can be solved in T rounds, for any T ≤ b/2, thenthe communication complexity version can be solved in ≤T rounds

22. Open problems • o((Dl )1/2)-time algorithm when l is large, say l >n ? • Lower bound on simple unweighted graphs? • Faster algorithm for generating random spanning trees and estimating mixing time? • Lower bound of computing K walks? • Random walks on graphs of low diameter? • Other problems that has D as multiplicative factor (e.g., W((nD)1/2))?

23. Thank you!