1. Randomized Distributed Decision David Peleg Joint work with: Pierre Fraigniaud Amos Korman Merav Parter

2. The goal: Solve problems collaboratively by multiple distributed processors Distributed computing Shared memory (multi-cores) Message passing (network) Communication mode:

3. Point-to-point communication network The distributed network model V={v 1,…,v n } - Processors (network sites) E - bidirectional communication links

4. Why is it interesting? Theoretical viewpoint: Several inherent differences between the distributed and the traditional centralized- sequential computational models Practical viewpoint: Extremely wide applicability on all levels

5. In centralized-sequential setting: Processor knows everything (inputs, intermediate results, etc.) In distributed setting: Processors have only a partial picture Incomplete knowledge

6. Do not know the network topology Know only their local portion of the input Do not know who else participates Do not know current stage of others Incomplete knowledge X 1 =3 v1v1 V?V? X=? ? In particular, processors:

7. G Intermediate topology knowledge model The KT r model: Each processor u sees B r (u), its r-neighborhood KT 2 (seeing 2-neighborhoods) (hypothetical model…) B 2 (u) u

8. We assume a synchronous model (the entire system is driven by global clock) Timing and synchrony Processors machine cycle: 1. Send messages to some neighbors 2. Receive messages from neighbors 3. Perform internal computation

9. The LOCAL Model Focus on impact of locality / distances Message size and internal computation are unbounded Message size and internal computation are unbounded

10. The LOCAL model Complexity measure: Time (# of rounds) Complexity measure: Time (# of rounds) G r In r rounds, r each processor u can collect complete information on B r (u), its r-neighborhood r=3 B 3 (u)

11. Equivalent viewpoint Instead of considering r-round algorithms – G consider 0-round (no-communication) algorithms in the KT r model

12. Local distributed computation Input: Graph G(V,E) Local views of r-neighborhood Goal: Compute a global solution for a given problem G

13. Language: (Decidable) collection of pairs Local distributed computation tasks as languages X 1 =3 v1v1 X 2 =6 v2v2 X 3 =8 v3v3 X 4 =2 v4v4 X 5 =3 v5v5 X 6 =3 v6v6 X 7 =3 v7v7

14. Examples x (v) = color of v

15. Examples At Most One Selected

16. Examples Leader Election

17. Examples X in =0 X out =1 X in =1 X out =1 X in =0 X out =1 X in =0 X out =1

18. Examples

19. Local computational tasks Local Construction (LC) Tasks: Given a global problem π on a graph G, construct a solution x locally Local Decision (LD) Tasks: Given a global problem π on a graph G and a proposed solution x, decide (or verify) locally that x solves π on G

20. Distributed Complexity Theory Proof Labeling Schemes [Korman, Fraigniaud, P, 05] Locally Checkable Proofs [Goos, Suomela, 11] Decidability Classes for Mobile Agent Computing [Fraigniaud, Pelc, 2012] Locality & Checkability in Wait-free Computing [Fraigniaud, Rajsbaum, Travers, 11] Local Distributed Decision [Fraigniaud, Korman, P, 11] On the Impact of Identifiers on Local Decision [Fraigniaud, Halldórsson, Korman, 12] [Fraigniaud, Goos, Korman, Suomela, 13]

21. Decision rules: Nodes need to collectively decide whether the given instance belongs to the language. Local decision tasks [Fraigniaud, Korman, P, 11]

22. YesNo Local decision tasks Decision rules: In a legal (YES) instance: all nodes output YES In an illegal (NO) instance: some nodes output NO [Fraigniaud, Korman, P, 11] Yes

23. Local decision tasks [Fraigniaud, Korman, P, 11] The asymmetry between the two cases looks odd, but… Decision rules: In a legal (YES) instance: all nodes output YES In an illegal (NO) instance: some nodes output NO

24. Local decision tasks [Fraigniaud, Korman, P, 11] Note: If every node is required to know the correct answer on both legal and illegal instances, then no local solutions are possible! Decision rules: In a legal (YES) instance: all nodes output YES In an illegal (NO) instance: some nodes output NO

25. Local decision tasks Example: Consider the following very local task: Input: Binary vector x Question: Is v 1 s input x(v 1 )=1 ? X 1 =1 v1v1 X 2 =0 v2v2 X 3 =0 v3v3 X 4 =1 v4v4 X 5 =0 v5v5 X 6 =1 v6v6 X 7 =1 v7v7

26. Local decision tasks Algorithm: All nodes other than v 1 say YES. Node v 1 answers according to x(v 1 ). X 1 =1 v1v1 X 2 =0 v2v2 X 3 =0 v3v3 X 4 =1 v4v4 X 5 =0 v5v5 X 6 =1 v6v6 X 7 =1 v7v7 Requiring all nodes to know the answer means no local solution is possible…

27. Decision Tasks Fault tolerance Checking correctness of construction algorithm Platform for distributed complexity theory Applications:

28. Class of languages that have a t-round local decider. LD(t) (Local Decision) Local analogue for the class P LOCAL Decider

29. Local Computation Tasks The class of all languages is divided into 4 classes: 1.Hard to construct and hard to decide. 2. Easy to construct and easy to decide. 3.Hard to construct but easy to decide. 4. Easy to construct but hard to decide.

30. Hard to construct but easy to decide (locally)

31. Deterministic decision in a single round. No

32. Easy to construct but hard to decide (locally) 0-round construction: Each node marks itself non-selected ( x(u) 0 ) At Most One Selected

33. Easy to construct but hard to decide (locally) Observation: AMOS is not locally decidable in o(n) rounds. At Most One Selected

34. Hard to construct and hard to decide (locally) Leader Election Construction barrier due to symmetry breaking: O(diameter) rounds are required. Observation: LE is not locally decidable in o(n) rounds. Observation: LE is not locally decidable in o(n) rounds.

35. Easy to construct and easy to decide (locally) Theorem [Naor, Stockmeyer, STOC 93]: Weak 2-Coloring in fixed odd degree graphs can be constructed in O(1) rounds. Theorem [Naor, Stockmeyer, STOC 93]: Weak 2-Coloring in fixed odd degree graphs can be constructed in O(1) rounds. Local decision: in single round No Local construction:

36. Thm [Naor, 96] : Randomization does not help for 3-coloring a ring: Randomized lower bound = deterministic upper bound = Θ(log*n) rounds. Thm [Naor, Stockmeyer, 93] : For constant degree graphs, certain randomized labeling algorithms can be de-randomized. Does Randomization help in local construction? Low-Degree Graphs So randomization fails to help?

37. Does Randomization help in local construction? Thm [Alon, Babai, Itai, 86], [Luby, 86] : randomly in O(logn ) w.h.p. So randomization does help? …

38. Yes, No 9 9 8 8 3 3 7 7 4 4 5 5 6 6 1 1 2 2 10 u 12 13 14 15 16 17 18 19 20 9 9 9 9 9 9 9 9 9 9 9 9 23 Randomized local decision

39. Does Randomization help in local decision? Recall: AMOS is not locally decidable in o(n) rounds.

40. Does Randomization help in local decision? Use randomness to decide! Recall: AMOS is not locally decidable in o(n) rounds.

41. Randomized LOCAL Decider * The probabilities are taken over all coin tosses performed by the nodes

42. Randomized LOCAL Decider Class of languages that have a t-round (p,q)-decider BPLD(p,q,t) (Bounded Probability Local Decision) BPLD(p,q,t) Local analogue for BPP

43. Does randomization help in local decision? [Fraigniaud, Korman, P, 11] Partial answer: Yes, for: - some families of languages, - some values of p and q Thm: p 2 +q=1 is a sharp threshold for hereditary languages* * Languages that are closed under inclusion.

44. p (yes probability) q (no probability) Yes Randomization threshold No p 2 +q=1 is a sharp threshold for hereditary languages p 2 +q=1 Does randomization help in local decision? [Fraigniaud, Korman, P, 11]

45. Distributed Complexity Classes No LD = class of languages decidable by t-round deterministic algorithm LD(t)LD(t)

46. Distributed Complexity Classes No LD BPLD BPLD(t)BPLD(t) = class of languages decidable by t-round deterministic algorithm LD(t)LD(t)

47. p 2 +q 1: randomization helps Recall: is not in LD

48. p 2 +q 1: randomization helps 0-round (p,q)-decider (code for node u) If unselected, return yes with probability 1 If selected, return yes with probability p Yes

49. Prob(every node returns yes) p Legal (YES) instance Yes Correctness of the Decider Yes

50. Prob(at least one node returns no) 1-p 2 q Illegal (NO) instance Yes Correctness of the Decider

51. Is my t-ball legal? p 2 +q > 1: de-randomization possible G 9 9 8 8 3 3 7 7 4 4 5 5 6 6 1 1 2 2 10 u 12 13 14 15 16 17 18 19 20 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 { Yes, No } (p,q)- (p,q)-decider applied by node u t

52. p 2 +q > 1: de-randomization possible 9 9 8 8 3 3 7 7 4 4 5 5 6 6 1 1 2 2 G 10 u 12 13 14 15 16 17 18 19 20 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 The required radius t depends on how far p 2 +q – 1 is bounded away from zero. Is my t-ball legal? Deterministic Deterministic decider applied by node u { Yes, No } t

53. Instance (G,x) t-round (p,q)-decider Sketch (on path)

54. Prob(at least one node returns no) < δ 2t

55. t=Run time of the (p,q)-decider Define

56. Proof: For every sub-path of length 2t in P, Prob(some node returns no) δ Prob(all nodes return yes) < 1-δ or

57. …… 2t

58. Contradiction for proper selection of constants. …… 2t

60. De-randomization of (p,q)-decider for p 2 +q>1 L Given: Hereditary language L. with a t-local (p,q)-decider L A t-local (deterministic) decider for L. : t=R sec (t) Every node u inspects its radius t neighborhood B(u) B(u) L If B(u) L, then u outputs yes, else it outputs no.

61. De-randomization of (p,q)-decider for p 2 +q>1 Every node u inspects its radius tneighborhood B(u) B(u) L If B(u) L, then u outputs yes, else it outputs no. Simulation correctness proof: Legal instance I L : L As L is hereditary, all neighborhoods B(u) are legal

62. De-randomization of (p,q)-decider for p 2 +q>1 IlLegal instance I L: Need to show that at least one ball B(u) is illegal. Towards contradiction assume all balls are legal. Maximal legal sub-path u

63. Hence, contradiction to the fact that is the maximal legal sub-path in. De-randomization of (p,q)-decider for p 2 +q>1 Claim:

64. The Gluing Lemma The union of two legal instances is legal provided their overlap is sufficiently large The required overlap size depends on the value p 2 +q-1

65. The Gluing Lemma The required overlap size depends on the value p 2 +q-1

66. 2t Since the (p,q) decider is a t-round algorithm, L and R are independent! Proof of the Gluing Lemma Event L: every node on the left returns yes Event R: every node on the right returns yes

67. 2t Proof of the Gluing Lemma Event L: yesEvent R: yes Assume towards contradiction that

68. Zooming into the Randomization Region Determinism Randomization p (yes probability) q (no probability) [Fraigniaud, Korman, Parter, P, 12]

69. = class of languages that have a (p,q)-decider s.t for integer k = class of languages that have a (p,q)-decider s.t for integer k The B k hierarchy BkBkBkBk BkBkBkBk

70. Theorem: The B k hierarchy is strict BPLD B2B2 ALL B3B3 Determinism (B 1 ) p (yes success probability) q (no success probability) p 2 +q>1 p 3/2 +q>1 p 4/3 +q>1 p+q>1 Determinism

71. The At most k selected Language B2B2 ALL B k+1 Determinism q p At most Kselected At most 1 selected Lemma: Integer k

72. Towards Distributed Computational Complexity Theory Are there intermediate classes between B k (t) and B k+1 (t)? Hardness/ completeness: Notions of reductions and complete problems for locality classes Randomization and non-determinism: Interplay between certificate size and success guarantees. The role of identifiers [Fraigniaud, Goos, Korman, Suomela, 13] Complexity theory for the CONGEST model Other combining rules for local decision (instead of logical and)

73. Randomi zation