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Randomized Distributed Decision Pierre Fraigniaud, Amos Korman, Merav Parter and David Peleg Yes No Yes No DISC 2012

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The Basic Questions What global information can be deduced from local structure? Does randomization help? To what extent?

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Outline The LOCAL Model Related Work Decision Problems Randomized Local Decision Contributions Open Problems

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The LOCAL model 9 9 8 8 3 3 7 7 4 4 5 5 6 6 1 1 2 2 G (0,1) (0,0) (1,1) (1,0) 10 11 12 13 14 15 16 17 18 19 20 *To distinguish nodes, assume an ID assignment.

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The LOCAL model 8 8 7 7 4 4 6 6 3 3 2 2 5 5 1 1 9 9 Simultaneous wakeup, fault-free synchronous communication. Computation: In each round, every processor: 1.Receives messages from neighbors. 2.Computes (internally). 3.Sends messages to its neighbors. Complexity measure: number of communication rounds. No restriction on memory, local computation and message size. 10 11 12 (1,1) (0,0) (1,0) (1,1)

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Outline The LOCAL Model Related Work Decision problems Randomized local decision Contribution Open problems

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The Impact of randomization in local computation Negative Indications: Naor and Stockmeyer [STOC ’93] : Define the LCL* class. Every constant time algorithm for constructing LCL can be derandomized. Naor [SIAM Disc. Maths ‘96] Randomization does not help for 3-coloring the ring. Negative Indications: Naor and Stockmeyer [STOC ’93] : Define the LCL* class. Every constant time algorithm for constructing LCL can be derandomized. Naor [SIAM Disc. Maths ‘96] Randomization does not help for 3-coloring the ring. * Restricted to constant time, constant degree and constant alphabet.

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The Impact of randomization in local computation Local Decision Tasks [Fraigniaud, Korman, Peleg, FOCS’11]

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Distributed Complexity Theory Locally checkable proofs. [M. G ÖÖ s and J. Suomela. PODC’11.] Decidability Classes for Mobile Agents Computing. [P. Fraigniaud and A. Pelc. Proc. 10th LATIN, 2012.] Locality and Checkability in Wait-free Computing. [P. Fraigniaud, S. Rajsbaum, and C. Travers. DISC’11.] Local Distributed Decision. [P. Fraigniaud, A. Korman, and D. Peleg. FOCS’11]

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Outline The LOCAL Model Related Work Decision problems Randomized local decision Contribution Open problems

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language Goal: nodes need to collectively decide whether the instance they live in belongs to a given distributed language. Local Decision Tasks [Fraigniaud, Korman, Peleg FOCS’11]

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Distributed Languages

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9 9 8 8 3 3 7 7 4 4 5 5 6 6 1 1 2 2 G (0,1) (0,0) (1,1) (1,0) 10 11 12 13 14 15 16 17 18 19 20 Local Decision Tasks [FKP11]

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Local Decision [FKP11] 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

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The Global Picture of Local Decision G (0,1) (0,0) (1,1) (1,0) No Yes The final decision is the conjunction of the output. No

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The Local Decision (LD) Class Class of languages that have a t-rounds local decider. LD(t) (Local Decision) Class P analogue

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Very few languages can be decided locally Extension: Use randomness to decide (0) (1)

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Outline The LOCAL Model Decision problems Randomized local decision Related Work Contribution Open problems

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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

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Class of languages that have a t-rounds (p,q)-decider. BPLD(p,q,t) (Bounded Probability Local Decision) BPLD(p,q,t) Class BPP analogue * The probabilities are taken over all coin tosses performed by the nodes.

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The Question What’s the connection between BPLD(p,q,t) classes? Can one boost the success probability of a (p,q)-decider?

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Does randomization help in local decision? [FKP11] p (``yes” probability) q (``no” probability) Yes No Randomization threshold No p 2 +q=1 is sharp threshold for hereditary languages* * Languages that are closed under inclusion. p 2 +q=1

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If p 2 +q 1 randomization helps! [FKP11] 0-round (p,q)-decider every unmarked node says “yes” with probability 1; every marked node says “yes” with probability p. At-Most-One-Selected (AMOS-1) Yes

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Probability that everyone says yes ≥ p YES Instance Yes At-Most-One-Selected (AMOS-1) Yes

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Probability that at least one says no≥ 1-p 2. NO Instance Yes At-Most-One-Selected (AMOS-1)

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Outline The LOCAL Model Decision problems Randomized local decision Related Work Contribution Open problems

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(1) Contribution p q No Randomization threshold Any language path on a path topology Randomization Determinism

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(2) Contribution p q Determinism Randomization

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Class of languages that have a (p,q)-decider s.t where k is integer. Class of languages that have a (p,q)-decider s.t where k is integer. The B k hierarchy BkBkBkBk BkBkBkBk

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Theorem: The B k hierarchy is strict BPLD (~BPP) B2B2 ALL B3B3 Determinism (B 1, ~P) 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

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At-Most-k-Selected (AMOS-k) B2B2 ALL B k+1 Determinism q p AMOS-k AMOS-1

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Yes Probability that everyone says yes ≥ p YES Instance B2B2 B3B3 AMOS-2 Yes p 4/3 +q>1

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At-Most-2-Selected (AMOS-2) Yes Probability that at least one says no (q) ≥ 1-p 3/2 NO Instance Yes

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The Challenge of a (p,q)-decider Yes No I I’ Instance Space for language L I’ I P Illegal := probability to accept I’ P legal := probability to accept I

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Instance (G,x) A t-round (p,q)-decider A

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probability that one says no <δ 2t Instance (G,x) A t-round (p,q)-decider A

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2t

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All nodes say yes with probability >p probability that one says no <δ 2t

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Assume towards contradiction p 3/2 +q > 1 that there exists a t-round (p,q)- decider A s.t p 3/2 +q > 1. Define

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NO 2t P1P1 P2P2 P3P3 The nodes execute the t-round (p,q) decider A. P 1 P 3 P 2 Probability that everyone says ``yes”

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NO YES 2t P1P1 P1P1 P3P3 P3P3 P2P2

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NO YES 2t P1P1 P1P1 P3P3 P3P3 P2P2

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B ∞ (t) ≠ ALL for every t=o(n) Assume, towards contradiction the existence of a (p,q)-decider A s.t p+q >1. Define

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6 6 7 7 8 8 9 9 9 9 1 1 2 2 3 3 4 4 5 5 11 12 n-2t Yes Instances 2t 10 1 1 2 2 3 3 4 4 5 5 7 7 8 8 9 9 9 9 11 12 10 6 6 The nodes of the path execute A.

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Yes Instances No instance No instance 6 6 7 7 8 8 9 9 9 9 1 1 2 2 3 3 4 4 5 5 11 12 10 1 1 2 2 3 3 4 4 5 5 7 7 8 8 9 9 9 9 11 12 10 6 6 7 7 6 6 1 1 2 2 3 3 5 5 12 4 4 8 8 9 9 10 11 Contradiction! Prob. to say yes at least p

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Outline The LOCAL Model Related Work Decision problems Randomized local decision Contribution Open problems

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Towards Distributed Computational Complexity Theory Does the class B k+1 (t) actually collapses to B k (t) or there exist intermediate classes? The power of a decoder: interpretations Decoder dealing with other interpretations, and more values (not only ``yes” and ``no”) Randomization and nondeterminism: Interplay between certificate size and success guarantees. Randomi zation q p

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