# Research supported in part by National Science Foundation and Army Research Office - PowerPoint PPT Presentation Download Presentation Research supported in part by National Science Foundation and Army Research Office

Research supported in part by National Science Foundation and Army Research Office Download Presentation ## Research supported in part by National Science Foundation and Army Research Office

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1. Equality Function Computation(How to make simple things complicated)Nitin VaidyaUniversity of Illinois at Urbana-ChampaignJoint work with Guanfeng Liang Research supported in part by National Science Foundation and Army Research Office

2. Background

3. Equality Function B A K-valued input K-valued input Determine whether the two inputs are identical

4. Communication cost of an algorithm: # bits of communication required in the worst case(over all possible inputs)

5. Communication cost of an algorithm: # bits of communication required in the worst case (over all possible inputs) • Communication complexity of a problem: Minimum communication cost over all algorithms to solve the problem [Andrew Yao, STOC 1979]

6. Equality Function B A K-valued input K-valued input

7. Upper Bound log K Proof by construction B A

8. Lower Bound log K Proof by fooling set argument B A

9. Generalization to n parties

10. The MEQ(n,K) Problem • n nodes each given xi from {1,…,K},to check if all xi are equal • Each node i computes EQi “Everyone detects” (MEQ-ED)

11. The MEQ(n,K) Problem • n nodes each given xi from {1,…,K},to check if all xi are equal • Each node i computes EQi “Anyone detects” (MEQ-AD)

12. n-Node Equality Problem B A Network C

13. Number-in-Hand Model K-valuedinput B A Broadcast Channel C Node i initially knows Xi

14. n-Party Equality : Complexity • Broadcast channel + Number-in-hand model log K bits

15. Number-on-Forehead Model B A Broadcast Channel C Node i initially knows everything except Xi

16. n-Party Equality : Complexity • Broadcast channel + Number-on-forehead model 2 bits when n > 2

17. Point-to-Point Networks Private channels & number-in-hand B A n = 3 C

19. Upper Bound = 2 log K K-valued input log K bits B A log K bits C

20. Lower Bound = log K K-valued input B A cut log K by 2-nodelower bound identical valueat B and C C

21. 1.5 log K Complexity 2 log K B A C Neither bound tightin general

22. 1.5 log K Not Tight B A C K = 2 Requires at least 2 bits

23. 2 log K Not Tight B A Proof by construction for K = 6 2 log K = log 36 C

24. K = 6 x y A B C z

25. Example 3 4 A B C 3

26. Example 3 4 2 A B C 3

27. Example 3 4 2 A B 2 C 3

28. Example 3 4 2 A B 1 2 C 3

29. Example 3 4 2 A B AB(4) = 2 AC(2) = 3 BC(3) = 1 1 2 C 3

30. Communication Cost 3 log 3 = log 27 <log 36 = 2 log K Can be generalized to large K and n to yield communication cost approximately 0.92 (n-1) log K

31. Communication Cost 3 log 3 = log 27 <log 36 = 2 log K Can be generalized to large K and n to yield communication cost approximately 0.92 (n-1) log K Cost of informing outcome to each other negligible for large K

32. Reduce Search Space “Static” Algorithms • Node transmitting in round R &its output function in round Rpre-determined • Output… function of initial input, and history

33. Fixed Algorithm: Directed Graph Representation y B 1, f1 4, f4 6, f6 2, f2 5, f5 A 3, f3 x C z Round number R , function f used in round R

34. Fixed Algorithm: Directed Graph Representation y B 1, f1 4, f4 6, f6 2, f2 5, f5 A 3, f3 x C z Round number R , function f used in round R

35. Equivalent Algorithm: Directed AcyclicGraph y B 1, f1 4, f4 6, f6 2, f2 5, f5 A 3, f3 x C z Round number R , function f used in round R

36. Equivalent Algorithm • Acyclic graph • Output depends only on initial input B FBC FAB FAC A C

37. Mapping to a Bipartite Graph • Each such algorithm can be mapped to a bipartite graph representation B FBC FAB FAC A C

38. Example 1,2 3,4 5,6 AB

39. Example 1,2 6,1 3,4 2,3 5,6 4,5 AC AB

40. Example 1 1,2 6,1 2 3 3,4 2,3 6 4 5 5,6 4,5 AC AB

41. Example 1 1,2 6,1 2 3 3,4 2,3 6 4 5 5,6 4,5 BC 3 2 1 AC BC AB

42. BC assigns colors to edges 1 1,2 6,1 2 3 3,4 2,3 6 4 5 5,6 4,5 BC 3 2 1 AC BC AB

43. U : # nodes on left V : # colors W : # nodes on right U V W 3 3 3 1 1,2 6,1 2 3 3,4 2,3 6 4 5 5,6 4,5 BC 3 2 1 AC BC AB

44. Equality  Bipartite Graph A colored bipartite graph corresponds to a fixed algorithm for 3-node equality withcostlog UVW if and only if • distance-2 colored (strong edge coloring) • Number of edges = K • U x V ≥ K • U x W ≥ K • V x W ≥ K

45. Fixed Algorithm Design • Find a suitable bipartite graph • Our algorithm  6-cycle

46. Lower Bounds • The mapping can be used to prove lower bounds for small K For K = 6 • Least cost over all fixed algorithms is log 27

47. Detour … an open conjecture A bipartite graph with • D1 = maximum degree on left • D2 = maximum degree on right can be distance-2 colored with D1 * D2 colors

48. Why is equality interesting ?

49. Lower Bound on Consensus • Mapping between Byzantine broadcastand multiple instances of equality