Download presentation

Presentation is loading. Please wait.

Published byAlex Ridler Modified over 2 years ago

1
Randomized Distributed Decision Pierre Fraigniaud, Amos Korman, Merav Parter and David Peleg Yes No Yes No DISC 2012

2
The Basic Questions What global information can be deduced from local structure? Does randomization help? To what extent?

3
Outline The LOCAL Model Related Work Decision Problems Randomized Local Decision Contributions Open Problems

4
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.

5
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)

6
Outline The LOCAL Model Related Work Decision problems Randomized local decision Contribution Open problems

7
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.

8
The Impact of randomization in local computation Local Decision Tasks [Fraigniaud, Korman, Peleg, FOCS’11]

9
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]

10
Outline The LOCAL Model Related Work Decision problems Randomized local decision Contribution Open problems

11
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]

12
Distributed Languages

13
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]

14
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

15
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

16
The Local Decision (LD) Class Class of languages that have a t-rounds local decider. LD(t) (Local Decision) Class P analogue

18
Very few languages can be decided locally Extension: Use randomness to decide (0) (1)

19
Outline The LOCAL Model Decision problems Randomized local decision Related Work Contribution Open problems

20
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

21
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.

22
The Question What’s the connection between BPLD(p,q,t) classes? Can one boost the success probability of a (p,q)-decider?

23
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

24
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

25
Probability that everyone says yes ≥ p YES Instance Yes At-Most-One-Selected (AMOS-1) Yes

26
Probability that at least one says no≥ 1-p 2. NO Instance Yes At-Most-One-Selected (AMOS-1)

27
Outline The LOCAL Model Decision problems Randomized local decision Related Work Contribution Open problems

28
(1) Contribution p q No Randomization threshold Any language path on a path topology Randomization Determinism

29
(2) Contribution p q Determinism Randomization

30
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

31
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

32
At-Most-k-Selected (AMOS-k) B2B2 ALL B k+1 Determinism q p AMOS-k AMOS-1

33
Yes Probability that everyone says yes ≥ p YES Instance B2B2 B3B3 AMOS-2 Yes p 4/3 +q>1

34
At-Most-2-Selected (AMOS-2) Yes Probability that at least one says no (q) ≥ 1-p 3/2 NO Instance Yes

35
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

36
Instance (G,x) A t-round (p,q)-decider A

37
probability that one says no <δ 2t Instance (G,x) A t-round (p,q)-decider A

38
2t

39
All nodes say yes with probability >p probability that one says no <δ 2t

40
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

41
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”

42
NO YES 2t P1P1 P1P1 P3P3 P3P3 P2P2

43
NO YES 2t P1P1 P1P1 P3P3 P3P3 P2P2

44
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

45
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.

46
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

47
Outline The LOCAL Model Related Work Decision problems Randomized local decision Contribution Open problems

48
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

Similar presentations

OK

MULTIPLYING MONOMIALS TIMES POLYNOMIALS (DISTRIBUTIVE PROPERTY)

MULTIPLYING MONOMIALS TIMES POLYNOMIALS (DISTRIBUTIVE PROPERTY)

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on storage devices of computer Ppt on data collection methods and instruments Ppt on trial and error dvd Ppt on asian continental Ppt on different types of dance forms in africa Ppt on water pollution and conservation Ppt on business communication skills Heat energy for kids ppt on batteries Ppt online viewer free Ppt on motivation for students