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Randomized Distributed Decision David Peleg Joint work with: Pierre Fraigniaud Amos Korman Merav Parter

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The goal: Solve problems collaboratively by multiple distributed processors Distributed computing Shared memory (multi-cores) Message passing (network) Communication mode:

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Point-to-point communication network The distributed network model V={v 1,…,v n } - Processors (network sites) E - bidirectional communication links

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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

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In centralized-sequential setting: Processor knows everything (inputs, intermediate results, etc.) In distributed setting: Processors have only a partial picture Incomplete knowledge

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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 X 1 =3 v1v1 V?V? X=? ? In particular, processors:

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G Intermediate topology knowledge model The KT r model: Each processor u sees B r (u), its r-neighborhood KT 2 (seeing 2-neighborhoods) (hypothetical model…) B 2 (u) u

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We assume a synchronous model (the entire system is driven by global clock) Timing and synchrony Processors machine cycle: 1. Send messages to some neighbors 2. Receive messages from neighbors 3. Perform internal computation

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The LOCAL Model Focus on impact of locality / distances Message size and internal computation are unbounded Message size and internal computation are unbounded

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The LOCAL model Complexity measure: Time (# of rounds) Complexity measure: Time (# of rounds) G r In r rounds, r each processor u can collect complete information on B r (u), its r-neighborhood r=3 B 3 (u)

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Equivalent viewpoint Instead of considering r-round algorithms – G consider 0-round (no-communication) algorithms in the KT r model

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Local distributed computation Input: Graph G(V,E) Local views of r-neighborhood Goal: Compute a global solution for a given problem G

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Language: (Decidable) collection of pairs Local distributed computation tasks as languages X 1 =3 v1v1 X 2 =6 v2v2 X 3 =8 v3v3 X 4 =2 v4v4 X 5 =3 v5v5 X 6 =3 v6v6 X 7 =3 v7v7

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Examples x (v) = color of v

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Examples At Most One Selected

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Examples Leader Election

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Examples X in =0 X out =1 X in =1 X out =1 X in =0 X out =1 X in =0 X out =1

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Examples

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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

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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]

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Decision rules: Nodes need to collectively decide whether the given instance belongs to the language. Local decision tasks [Fraigniaud, Korman, P, 11]

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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] Yes

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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

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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

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Local decision tasks Example: Consider the following very local task: Input: Binary vector x Question: Is v 1 s input x(v 1 )=1 ? X 1 =1 v1v1 X 2 =0 v2v2 X 3 =0 v3v3 X 4 =1 v4v4 X 5 =0 v5v5 X 6 =1 v6v6 X 7 =1 v7v7

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Local decision tasks Algorithm: All nodes other than v 1 say YES. Node v 1 answers according to x(v 1 ). X 1 =1 v1v1 X 2 =0 v2v2 X 3 =0 v3v3 X 4 =1 v4v4 X 5 =0 v5v5 X 6 =1 v6v6 X 7 =1 v7v7 Requiring all nodes to know the answer means no local solution is possible…

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Decision Tasks Fault tolerance Checking correctness of construction algorithm Platform for distributed complexity theory Applications:

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Class of languages that have a t-round local decider. LD(t) (Local Decision) Local analogue for the class P LOCAL Decider

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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.

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Hard to construct but easy to decide (locally)

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Deterministic decision in a single round. No

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Easy to construct but hard to decide (locally) 0-round construction: Each node marks itself non-selected ( x(u) 0 ) At Most One Selected

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Easy to construct but hard to decide (locally) Observation: AMOS is not locally decidable in o(n) rounds. At Most One Selected

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Hard to construct and hard to decide (locally) Leader Election Construction barrier due to symmetry breaking: O(diameter) rounds are required. Observation: LE is not locally decidable in o(n) rounds. Observation: LE is not locally decidable in o(n) rounds.

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Easy to construct and easy to decide (locally) Theorem [Naor, Stockmeyer, STOC 93]: Weak 2-Coloring in fixed odd degree graphs can be constructed in O(1) rounds. 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:

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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?

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Does Randomization help in local construction? Thm [Alon, Babai, Itai, 86], [Luby, 86] : randomly in O(logn ) w.h.p. So randomization does help? …

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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

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Does Randomization help in local decision? Recall: AMOS is not locally decidable in o(n) rounds.

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Does Randomization help in local decision? Use randomness to decide! Recall: AMOS is not locally decidable in o(n) rounds.

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

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Randomized LOCAL Decider Class of languages that have a t-round (p,q)-decider BPLD(p,q,t) (Bounded Probability Local Decision) BPLD(p,q,t) Local analogue for BPP

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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: p 2 +q=1 is a sharp threshold for hereditary languages* * Languages that are closed under inclusion.

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p (yes probability) q (no probability) Yes Randomization threshold No p 2 +q=1 is a sharp threshold for hereditary languages p 2 +q=1 Does randomization help in local decision? [Fraigniaud, Korman, P, 11]

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Distributed Complexity Classes No LD = class of languages decidable by t-round deterministic algorithm LD(t)LD(t)

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Distributed Complexity Classes No LD BPLD BPLD(t)BPLD(t) = class of languages decidable by t-round deterministic algorithm LD(t)LD(t)

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p 2 +q 1: randomization helps Recall: is not in LD

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p 2 +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

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Prob(every node returns yes) p Legal (YES) instance Yes Correctness of the Decider Yes

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Prob(at least one node returns no) 1-p 2 q Illegal (NO) instance Yes Correctness of the Decider

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Is my t-ball legal? p 2 +q > 1: de-randomization possible G 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 9 9 9 9 { Yes, No } (p,q)- (p,q)-decider applied by node u t

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p 2 +q > 1: de-randomization possible 9 9 8 8 3 3 7 7 4 4 5 5 6 6 1 1 2 2 G 10 u 12 13 14 15 16 17 18 19 20 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 The required radius t depends on how far p 2 +q – 1 is bounded away from zero. Is my t-ball legal? Deterministic Deterministic decider applied by node u { Yes, No } t

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Instance (G,x) t-round (p,q)-decider Sketch (on path)

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Prob(at least one node returns no) < δ 2t

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

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Proof: For every sub-path of length 2t in P, Prob(some node returns no) δ Prob(all nodes return yes) < 1-δ or

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…… 2t

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Contradiction for proper selection of constants. …… 2t

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De-randomization of (p,q)-decider for p 2 +q>1 L Given: Hereditary language L. with a t-local (p,q)-decider L A t-local (deterministic) decider for L. : t=R sec (t) Every node u inspects its radius t neighborhood B(u) B(u) L If B(u) L, then u outputs yes, else it outputs no.

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De-randomization of (p,q)-decider for p 2 +q>1 Every node u inspects its radius tneighborhood B(u) B(u) L If B(u) L, then u outputs yes, else it outputs no. Simulation correctness proof: Legal instance I L : L As L is hereditary, all neighborhoods B(u) are legal

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De-randomization of (p,q)-decider for p 2 +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. Maximal legal sub-path u

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Hence, contradiction to the fact that is the maximal legal sub-path in. De-randomization of (p,q)-decider for p 2 +q>1 Claim:

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The Gluing Lemma The union of two legal instances is legal provided their overlap is sufficiently large The required overlap size depends on the value p 2 +q-1

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The Gluing Lemma The required overlap size depends on the value p 2 +q-1

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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

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2t Proof of the Gluing Lemma Event L: yesEvent R: yes Assume towards contradiction that

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Zooming into the Randomization Region Determinism Randomization p (yes probability) q (no probability) [Fraigniaud, Korman, Parter, P, 12]

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= class of languages that have a (p,q)-decider s.t for integer k = class of languages that have a (p,q)-decider s.t for integer k The B k hierarchy BkBkBkBk BkBkBkBk

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Theorem: The B k hierarchy is strict BPLD B2B2 ALL B3B3 Determinism (B 1 ) 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

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The At most k selected Language B2B2 ALL B k+1 Determinism q p At most Kselected At most 1 selected Lemma: Integer k

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Towards Distributed Computational Complexity Theory Are there intermediate classes between B k (t) and B k+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)

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Randomi zation

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