Randomized Distributed Decision

Pierre Fraigniaud, Amos Korman, Merav Parter and David Peleg

Yes

No

No

Yes

No

No

No

No

DISC 2012

The Basic Questions

What global information can be deduced from local structure?

Does randomization help?

To what extent?

Outline

The LOCAL Model

Related Work

Decision Problems

Randomized Local Decision

Contributions

Open Problems

The LOCAL model

Input:A pair (G, ) :

G connected graph vector of local inputs.*

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G

(0,1)

(0,1)

(0,1)

(0,1)

(0,1)

(0,1) (0,1)

(0,1)

(0,1)

(0,0)

(0,0)

(0,0)

(0,0)

(0,0)

(0,0)

(1,1)

(1,1)

(1,1)

(1,1)

(1,0)

(1,0)

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*To distinguish nodes, assume an ID assignment .

The LOCAL model

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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.

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(1,1)(1,1)

(1,1)

(0,0)

(0,0)

(1,0)

(1,0)

(1,0)

(1,1)

(1,1)(1,1)

Outline

The LOCAL Model

Related Work

Decision problems

Randomized local decision

Contribution

Open problems

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.

* Restricted to constant time, constant degree and constant alphabet.

The Impact of randomization in local computation

Positive Indications: (

Randomly in O(logn ) w.h.p.

Alon, Babai, Itai [J. Alg. ’86], Luby [SIAM J. Comput. ’86]

Deterministically in .

Panconesi, Srinivasan [J. Algorithms, ‘96]

Local Decision Tasks [Fraigniaud, Korman, Peleg, FOCS’11]

Distributed Complexity Theory

Locally checkable proofs.[M. GÖÖs 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]

Outline

The LOCAL Model

Related Work

Decision problems

Randomized local decision

Contribution

Open problems

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]

Def: A distributed language is a decidable collection of instances.

Coloring=.

At-Most-One-Selected={(G,x) s.t∑xi 1}.

MIS=.

Distributed Languages

Input:A pair (G, ) :

G connected graph vector of local inputs.* Language L..

Output: Yes\ No9 8

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G

(0,1)

(0,1)

(0,1)

(0,1)

(0,1)

(0,1) (0,1)

(0,1)

(0,1)

(0,0)

(0,0)

(0,0)

(0,0)

(0,0)

(0,0)

(1,1)

(1,1)

(1,1)

(1,1)

(1,0)

(1,0)

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Local Decision Tasks [FKP11]

Local Decision [FKP11]

Yes, No

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The Global Picture of Local Decision

G

(0,1)

(0,1)

(0,1)

(0,1)

(0,1)

(0,1) (0,1)

(0,1)

(0,1)

(0,0)

(0,0)

(0,0)

(0,0)

(0,0)

(0,0)

(1,1)

(1,1)

(1,1)

(1,1)

(1,0)

(1,0)

NoNo

No

No

No

Yes

Yes

Yes Yes Yes

Yes

Yes

Yes

YesYes

Yes

Yes

Yes

Yes

Yes

Yes

The final decision isthe conjunction of the output.

No

The Local Decision (LD) Class

A local decider A for language is a local alg. such that

: Everyone says yes

: At least one says no (for every Id assignment ).

Class of languages that have a t-rounds local decider.

LD(t) (Local Decision)Class Panalogue

Example: Coloring

Coloring=.

Very few languages can be decided locally

At-Most-One-Selected (AMOS-1)={(G,x) s.t ∑xi1}.

Extension: Use randomness to decide

(0) (0)(0)(0) (0) (0)(1)

Outline

The LOCAL Model

Decision problems

Randomized local decision

Related Work

Contribution

Open problems

Yes, No

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1415

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99

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Randomized Local Decision

Randomized Local Decision

A (p,q)-decider for language L is a

local 2-sided error Monte Carlo algorithm, such that:

: Everyone says yes with probability* ≥p

: At least one says no with probability* ≥q.

Class of languages that have a t-rounds (p,q)-decider.

BPLD(p,q,t) (Bounded Probability Local Decision) Class BPP

analogue

* The probabilities are taken over all coin tosses performed by the nodes.

The Question

What’s the connection between BPLD(p,q,t) classes?

Can one boost the success probability of a (p,q)-decider?

Does randomization help in local decision? [FKP11]

p (``yes” probability)

q (``

no”

prob

abili

ty)

Yes

NoRandomization threshold No

p2+q=1 is sharp threshold for hereditary languages*

* Languages that are closed under inclusion.

p 2+q=1

If p2+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

Yes w.p

YesYesYes YesYes

Yes Yes

Yes w.p

Probability that everyone says yes ≥ p

YES Instance

Yes Yes Yes

AMOS-1

At-Most-One-Selected (AMOS-1)

YesYes

Yes Yes

Yes w.p

Probability that at least one says no≥ 1-p2.

NO Instance

Yes Yes Yes

AMOS-1

At-Most-One-Selected (AMOS-1)

Yes w.p

Outline

The LOCAL Model

Decision problems

Randomized local decision

Related Work

Contribution

Open problems

(1) Contribution

p

q NoRandomization threshold

Any language

on a path topologyRandomization

Determinism

(2) Contribution

p

qDeterminismRandomization

Class of languages that have a (p,q)-decider s.t

where k is integer.

The Bk hierarchy

Bk(t)

Bk

p1+1/k+q 1

Theorem: The Bk hierarchy is strict

BPLD (~BPP)

B2

B

ALL

B3

Determinism (B1 , ~P)

p (“yes” success probability)

B1(t) ALLq

(“no

” su

cces

s pr

obab

ility

)

p 2+q>1p 3/2+q>1p 4/3+q>1

p+q>1

Determinism

At-Most-k-Selected=

At-Most-k-Selected (AMOS-k)

Lemma:Bk+1 \ Bk. B

2

B

ALL

Bk+1

Determinism q

p

AMOS-k

AMOS-1

At-Most-2-Selected (AMOS-2)

Yes Yes

Yes w.p

Probability that everyone says yes ≥ p

YES Instance

Yes w.p

B2B

3 AMOS-2

Yes Yes Yes

p 4/3+q>1

At-Most-2-Selected (AMOS-2)

Yes Yes

Probability that at least one says no (q) ≥ 1-p3/2

NO Instance

Yes w.p

Yes w.p

Yes w.p

Yes Yes

Thus p4/3 +q>1 AMOS-2

The Challenge of a (p,q)-decider

YesNoI

I’

Instance Space for language L

I’

I

If p3/2+q > 1 then

PIllegal:= probability to accept I’

Plegal:= probability to accept I

Instance (G,x)

A t-round (p,q)-decider A

Tool: -Secure Zone

probability that one says no <δ

2t

Instance (G,x)

A t-round (p,q)-decider A

Tool: -Secure Zone

Tool: -Secure Zone

2tEveryone says yes with probability Everyone says yes with

probability

and are independent.

q < Probability that one says NO <

𝑂 (𝑡 log𝑝log 1−δ ) All nodes say yes with probability >p

probability that one says no <δ

2t

Claim: Every large enough legal subpath contains a -Secure subpath.

Tool: -Secure Zone

Assume towards contradiction that there exists a t-round (p,q)- decider A s.t p3/2+q > 1.

Define 0<𝛿<12(𝑝 3/2+𝑞−1)

At-Most-2-Selected B2

NO

2t 2t

P1 P2 P3

The nodes execute the t-round (p,q) decider A.

P1 P3 P2

The probability that one says no at most

)/2

At-Most-2-Selected B2

Probability that everyone says ``yes”

NO

YES

2t 2tP1

P1

P3

P3

P2

At-Most-2-Selected B2

A is a (p,q) decider such that

NO

YES

2t 2tP1

P1

P3

P3

P2

𝑝 ≤𝑃 1×𝑃 3≤𝑃 22

Since ), contradiction!

At-Most-2-Selected B2

B∞(t) ≠ ALL for every t=o(n)

Tree=

Assume, towards contradiction the existence of

a (p,q)-decider A s.t p+q >1.

Define

0<𝛿<𝑝+𝑞−1

Tree B∞(t) for every t=o(n)

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n-2t

Yes Instances

The probability that one says no at most

The probability that everyone says yes

2t

10

1 2 3 4 57 8 99 11 1210 6

The nodes of the path execute A.

Yes Instances No instance

6 7 8 991 2 3 4 5 11 1210

Tree B∞(t) for every t=o(n)

1 2 3 4 57 8 99 11 1210 67

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Contradiction!

Prob. to say no at most

Prob. to say yes at least p

Outline

The LOCAL Model

Related Work

Decision problems

Randomized local decision

Contribution

Open problems

Towards Distributed Computational Complexity Theory

Does the class Bk+1(t) actually collapses to Bk(t) or there exist intermediate classes?

The power of a decoder:Decoder dealing with other interpretations, and more values (not only ``yes” and ``no”)

Randomization and nondeterminism:Interplay between certificate size and success guarantees.

Randomization

q

p