Randomized Distributed Decision

David Peleg

Joint work with:Pierre Fraigniaud

Amos KormanMerav Parter

The goal: Solve problems collaboratively by multiple distributed processors

Distributed computing

• Shared memory (multi-cores)• Message passing (network)

Communication mode:

Point-to-point communication network

The distributed network model

V={v1,…,vn} - Processors (network sites)E - bidirectional communication links

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

In centralized-sequential setting: Processor knows everything (inputs, intermediate results, etc.)

In distributed setting:Processors have only a partial picture

Incomplete knowledge

• 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

X1=3

v1

V?

X=??

In particular, processors:

GIntermediate topology knowledge model

The KTr model:Each processor u sees Br(u), itsr-neighborhood

KT2

(seeing 2-neighborhoods)

(hypothetical model…) B2(u)

u

We assume a synchronous model(the entire system is driven by global clock)

Timing and synchrony

Processors’ machine cycle: 1. Send messages to some neighbors2. Receive messages from neighbors3. Perform internal computation

The LOCAL Model

Focus on impact of locality / distances

Message size and internal computationare unbounded

The LOCAL model

Complexity measure:Time (# of rounds)

G

In r rounds,each processor u can collect complete information on Br(u), its r-neighborhood⇒ r=3 B3(u)

Equivalent viewpoint

Instead of considering r-round algorithms –

G

consider 0-round (no-communication) algorithmsin the KTr model

Local distributed computation

Input:• Graph G(V,E)• Local views of

r-neighborhood

Goal:Compute a global solution for a given problem

G

*i ,G(V,E)G 10 , | , xxL

Language: (Decidable) collection of pairs

Local distributed computation tasks as languages

X1=3v1

X2=6v2

X3=8v3

X4=2v4

X5=3v5

X6=3v6

X7=3v7

)()( , | , vuE(G)(u,v)Goloring xxx C

oloringColoringC

Examples

x(v) = color of v

Examples

1)( ,1,0)( | ,Gu

uuG xxxAMOS

AMOS AMOS

At Most One Selected

Examples

1)(,1,0)( | ,Gu

uuG xxxLE

LE

Leader Election

Examples

)()(),(

),( | ),(,

0

0

uvGVv

GVuGonsensus

inout

outin

xx

xxC

Xin=0Xout=1

Xin=1Xout=1

Xin=0Xout=1

Xin=0Xout=1

onsensusC

Examples

.a MIS is 1)( | )( | , vGVvSGMIS xx

MIS

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

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]

Decision rules: Nodes need to collectively decide whether the given instance belongs to the language.

Local decision tasks[Fraigniaud, Korman, P, 11]

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]

YesYes

Yes

Yes

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

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

Local decision tasks

Example: Consider the following “very local” task:Input: Binary vector xQuestion: Is v1‘s input x(v1)=1 ?

X1=1v1

X2=0v2

X3=0v3

X4=1v4

X5=0v5

X6=1v6

X7=1v7

Local decision tasks

Algorithm: • All nodes other than v1 say YES.• Node v1 answers according to x(v1).

X1=1v1

X2=0v2

X3=0v3

X4=1v4

X5=0v5

X6=1v6

X7=1v7

Requiring all nodes to know the answer means no local solution is possible…

Decision Tasks

Fault tolerance

Checking correctness of construction algorithm

Platform for distributed complexity theory

Applications:

A t-round distributed algorithm is a LOCAL Decider for if:

: Every processor says “yes”

: At least one says “no”

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

LD(t) (Local Decision)

Local analogue for the class P

LOCAL Decider

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.

Hard to construct but easy to decide (locally)

)()( , | , vuE(G)(u,v)Goloring xxx C

Deterministic construction of ( in rounds

[Panconesi, Srinivasan, 96]

Hard to construct but easy to decide (locally)

)()( , | , vuE(G)(u,v)Goloring xxx C

Deterministic decision in a single round.

No

Easy to construct but hard to decide (locally)

1)( | ,Gu

uG xxAMOS

0-round construction: Each node marks itself non-selected ( x(u)← 0 )

At Most One Selected

Easy to construct but hard to decide (locally)

1)( | ,Gu

uG xxAMOS

Observation: AMOS is not locally decidable in o(n) rounds.

At Most One Selected

Hard to construct and hard to decide (locally)

Leader Election

1)( | ,Gu

uG xxLE

Construction barrier due to symmetry breaking:O(diameter) rounds are required.

Observation:LE is not locally decidable in o(n) rounds.

Easy to construct and easy to decide (locally)

(v)(u)s.t

neighbor v hasu every | ,_

xx

xGColoringeakW

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:

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?

Does Randomization help in local construction?

Large-Degree Graphs( can be computed

Thm [Alon, Babai, Itai, 86], [Luby, 86] : randomly in O(logn ) w.h.p.

Thm [Panconesi, Srinivasan , 96] : deterministically in time .

So randomization does help?

…

Yes, No

9 8

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Randomized local decision

Does Randomization help in local decision?

1)( | ,Gu

uG xxAMOS

Recall: AMOS is not locally decidable in o(n) rounds.

Does Randomization help in local decision?

1)( | ,Gu

uG xxAMOS

Use randomness to decide!

Recall: AMOS is not locally decidable in o(n) rounds.

Randomized LOCAL Decider

A (p,q)-decider for language L is a local, 2-sided error Monte Carlo algorithm, such that:

: With probability* ≥p, every node returns “yes”

: With probability* ≥q, at least one node returns “no”

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

Randomized LOCAL Decider

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

BPLD(p,q,t) (Bounded Probability

Local Decision) Local analogue for BPP

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

* Languages that are closed under inclusion.

p (“yes” probability)

q (“

no”

prob

abili

ty)

Yes

Randomization threshold No

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

p 2+q=1

Does randomization help in local decision?

[Fraigniaud, Korman, P, 11]

Distributed Complexity Classes

NoLD

= class of languages decidable by t-round deterministic algorithm

LD(t)

If problem is here (randomly decidable

with high p and q)

t is in LD (can be decided

deterministically)

Distributed Complexity Classes

NoLD

BPLD

= class of languages with t-round (p,q)-decider s.t. p2+q 1BPLD(t)

∃ problems here (randomly decidable

with low p and q)

t do not belong to LD

= class of languages decidable by t-round deterministic algorithm

LD(t)

p2+q ≤ 1: randomization helps

1)( | ,Gu

uG xxAMOS

Recall: is not in LD AMOS

p2+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” w/prob

YesYesYesYes YesYes

1)( | ,Gu

uG xxAMOS

Yes Yes

“Yes” w/prob

Prob(every node returns “yes”) ≥ p

Legal (“YES”) instance

Yes Yes Yes

Correctness of the Decider

YesYes

Yes Yes

“Yes” w/prob

Prob(at least one node returns “no”) ≥ 1-p2 ≥ q

Illegal (“NO”) instance

Yes Yes Yes

“Yes” w/prob

Correctness of the Decider

Is my t-ball legal?

p2+q > 1: de-randomization possible

G

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{Yes, No}

(p,q)-deciderapplied by node u

t

p2+q > 1: de-randomization possible

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The required radius t’ depends on how far p2+q – 1 is bounded away from zero.

Is my t’-ball legal?

Deterministic decider applied by node u

{Yes, No} t’

Instance (G,x)

t-round (p,q)-decider

Sketch (on path)

Prob(at least one node returns “no”) < δ2t

-Secure Zone

-Secure zone: a “safe-separator” (of width 2t) with low rejection probability

A -Secure zone divides the instance into two independent sub-instances carrying the mass of rejection probability

-Secure Zones

tp

R

)1log(

log11sec

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

Claim: Every legal sub-path of length contains a secure zone.

Define

𝑹𝒔𝒆𝒄

By contradiction: consider a legal sub-path P of length ,suppose P does not contain a secure zone

Proof:

-Secure Zones

For every sub-path of length 2t in P, Prob(some node returns “no”) ≥ δ

⇒

Prob(all nodes return “yes”) < 1-δ

or

-Secure Zones

… …2t 2t

P has K= ``independent” sub-paths of length 2t.

In each of these K sub-paths, Prob(all nodes return “yes”

< )K

𝐾 <log𝑝

log(1−𝛿)Contradiction for proper selection of constants.

-Secure Zones

… …2t 2t

-Secure Zones

Claim: Every legal sub-path of length contains a secure zone.

De-randomization of (p,q)-decider for p2+q>1

Given: Hereditary language L. with a t-local (p,q)-decider

A t-local (deterministic) decider for L. :

t’=Rsec (t)

Every node u inspects its radius t’ neighborhood B(u)

If B(u) ∈ L, then u outputs ”yes”, else it outputs “no”.

De-randomization of (p,q)-decider for p2+q>1

Every node u inspects its radius t’neighborhood B(u)

If B(u) ∈ L, then u outputs ”yes”, else it outputs “no”.

Simulation correctness proof:

Legal instance I ∈ L:As L is hereditary, all neighborhoods B(u) are legal

De-randomization of (p,q)-decider for p2+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.

LI

Maximal legal sub-pathL'I

L)(uBu

sec2R

L'IL)(uB

secR

L''I

Hence, contradiction to the fact that is the maximal legal sub-path in .

'II

De-randomization of (p,q)-decider for p2+q>1

Claim:

The Gluing Lemma

The union of two legal instances is legal provided their overlap is sufficiently large

L1I

L2I

L3I

The required overlap size depends on the value p2+q-1

tp

R

)1log(

log11sec

The Gluing Lemma

The required overlap size depends on the value p2+q-1

Fsatisfying < p2+q-1

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”

The overlap section contains a secure sub-path

3I

2t

q ≤ Prob(at least one node returns “no”) ≤ +

Proof of the Gluing Lemma

Event L: “yes” Event R: “yes”

Contradiction to the definition of

Assume towards contradiction that

3I

The overlap section contains a secure sub-path

Zooming into the Randomization Region

DeterminismRandomization

p (“yes” probability)

q (“

no”

prob

abili

ty)

[Fraigniaud, Korman, Parter, P, 12]

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

for integer k

The Bk hierarchy

Bk(t)

Bk

p1+1/k + q 1

Theorem: The Bk hierarchy is strict

BPLD

B2

B

ALL

B3

Determinism (B1)

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

The At most k selected Language

B2

B

ALL

Bk+1

Determinism q

p

At most Kselected

At most 1 selected

kuGGu

)( | , xxselected- k mostAt

Lemma:

kk BB \1 selected k mostAt

Integer k

Towards Distributed Computational Complexity Theory

Are there intermediate classes between Bk(t) and Bk+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”)

Randomization

Thank you for your attention