Randomized Distributed Decision
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Transcript of Randomized Distributed Decision
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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
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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
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)
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(1,1)
(1,1)
(1,0)
(1,0)
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*To distinguish nodes, assume an ID assignment .
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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)
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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.
* Restricted to constant time, constant degree and constant alphabet.
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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]
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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]
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Outline
The LOCAL Model
Related Work
Decision problems
Randomized local decision
Contribution
Open problems
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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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Def: A distributed language is a decidable collection of instances.
Coloring=.
At-Most-One-Selected={(G,x) s.t∑xi 1}.
MIS=.
Distributed Languages
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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)
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(0,0)
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(1,1)
(1,1)
(1,1)
(1,1)
(1,0)
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Local Decision Tasks [FKP11]
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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)
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(0,0)
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(1,0)
NoNo
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Yes
Yes
Yes Yes Yes
Yes
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YesYes
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Yes
Yes
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Yes
The final decision isthe conjunction of the output.
No
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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
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Example: Coloring
Coloring=.
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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)
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Outline
The LOCAL Model
Decision problems
Randomized local decision
Related Work
Contribution
Open problems
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Yes, No
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Randomized Local Decision
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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.
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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”
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
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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
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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
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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
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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 NoRandomization threshold
Any language
on a path topologyRandomization
Determinism
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(2) Contribution
p
qDeterminismRandomization
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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
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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
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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
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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
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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
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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
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Instance (G,x)
A t-round (p,q)-decider A
Tool: -Secure Zone
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probability that one says no <δ
2t
Instance (G,x)
A t-round (p,q)-decider A
Tool: -Secure Zone
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Tool: -Secure Zone
2tEveryone says yes with probability Everyone says yes with
probability
and are independent.
q < Probability that one says NO <
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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
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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
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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”
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NO
YES
2t 2tP1
P1
P3
P3
P2
At-Most-2-Selected B2
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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
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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
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Tree B∞(t) for every t=o(n)
6 7 8 991 2 3 4 5 11 12
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.
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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
6
12
3
5
12
4
8
9
10
11
Contradiction!
Prob. to say no at most
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 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