randomized distributed decision pierre fraigniaud, amos korman, merav parter and david peleg yes no...
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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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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
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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