Local Distributed Decision Pierre Fraigniaud Amos Korman David Peleg CNRS and University Paris Diderot Workshop on Sublinear Algorithms, Bertinoro, May 23-27, 2011 1 / 33
Decision problems Does randomization helps? Nondeterminism Power of oracles Further works 2 / 33
Outline Decision problems Does randomization helps? Nondeterminism Power of oracles Further works 3 / 33
Decide coloring 4 / 33
Computational model LOCAL model In each round during the execution of a distributed algorithm, every processor: 1. sends messages to its neighbors, 2. receives messages from its neighbors, and 3. computes, i.e., performs individual computations. 5 / 33
Computational model LOCAL model In each round during the execution of a distributed algorithm, every processor: 1. sends messages to its neighbors, 2. receives messages from its neighbors, and 3. computes, i.e., performs individual computations. Input An input configuration is a pair ( G , x ) where G is a connected graph, and every node v ∈ V ( G ) is assigned as its local input a binary string x ( v ) ∈ { 0 , 1 } ∗ . 5 / 33
Computational model LOCAL model In each round during the execution of a distributed algorithm, every processor: 1. sends messages to its neighbors, 2. receives messages from its neighbors, and 3. computes, i.e., performs individual computations. Input An input configuration is a pair ( G , x ) where G is a connected graph, and every node v ∈ V ( G ) is assigned as its local input a binary string x ( v ) ∈ { 0 , 1 } ∗ . Output The output of node v performing Algorithm A running in G with input x and identity assignment Id: out A ( G , x , Id , v ) 5 / 33
Languages A distributed language is a decidable collection of configurations. 6 / 33
Languages A distributed language is a decidable collection of configurations. ◮ Coloring = { ( G , x ) s.t. ∀ v ∈ V ( G ) , ∀ w ∈ N ( v ) , x ( v ) � = x ( w ) } . 6 / 33
Languages A distributed language is a decidable collection of configurations. ◮ Coloring = { ( G , x ) s.t. ∀ v ∈ V ( G ) , ∀ w ∈ N ( v ) , x ( v ) � = x ( w ) } . ◮ At-Most-One-Marked = { ( G , x ) s.t. � x � 1 ≤ 1 } . 6 / 33
Languages A distributed language is a decidable collection of configurations. ◮ Coloring = { ( G , x ) s.t. ∀ v ∈ V ( G ) , ∀ w ∈ N ( v ) , x ( v ) � = x ( w ) } . ◮ At-Most-One-Marked = { ( G , x ) s.t. � x � 1 ≤ 1 } . ◮ Consensus = { ( G , ( x 1 , x 2 )) s.t. ∃ u ∈ V ( G ) , ∀ v ∈ V ( G ) , x 2 ( v ) = x 1 ( u ) } . 6 / 33
Languages A distributed language is a decidable collection of configurations. ◮ Coloring = { ( G , x ) s.t. ∀ v ∈ V ( G ) , ∀ w ∈ N ( v ) , x ( v ) � = x ( w ) } . ◮ At-Most-One-Marked = { ( G , x ) s.t. � x � 1 ≤ 1 } . ◮ Consensus = { ( G , ( x 1 , x 2 )) s.t. ∃ u ∈ V ( G ) , ∀ v ∈ V ( G ) , x 2 ( v ) = x 1 ( u ) } . ◮ MIS = { ( G , x ) s.t. S = { v ∈ V ( G ) | x ( v ) = 1 } is a MIS } . 6 / 33
Decision Let L be a distributed language. Algorithm A decides L ⇐ ⇒ for every configuration ( G , x ) : ◮ If ( G , x ) ∈ L , then for every identity assignment Id, out A ( G , x , Id , v ) = “yes” for every node v ∈ V ( G ) ; ◮ If ( G , x ) / ∈ L , then for every identity assignment Id, out A ( G , x , Id , v ) = “no” for at least one node v ∈ V ( G ) . 7 / 33
Local decision Let t be a function of triplets ( G , x , Id ) . Definition LD ( t ) is the class of all distributed languages that can be decided by a distributed algorithm that runs in at most t communication rounds. 8 / 33
Local decision Let t be a function of triplets ( G , x , Id ) . Definition LD ( t ) is the class of all distributed languages that can be decided by a distributed algorithm that runs in at most t communication rounds. ◮ Coloring ∈ LD ( 1 ) and MIS ∈ LD ( 1 ) . ◮ AMOM , Consensus , and SpanningTree are not in LD ( t ) , for any t = o ( n ) . 8 / 33
Outline Decision problems Does randomization helps? Nondeterminism Power of oracles Further works 9 / 33
Related work What can be computed locally? Define LCL as LD ( O ( 1 )) involving ◮ solely graphs of constant maximum degree ◮ inputs taken from a set of constant size Theorem ( Naor and Stockmeyer [STOC ’93] ) If there exists a randomized algorithm that constructs a solution for a problem in LCL in O ( 1 ) rounds, then there is also a deterministic algorithm constructing a solution for that problem in O ( 1 ) rounds. Proof uses Ramsey theory. Not clearly extendable to languages in LD ( O ( 1 )) \ LCL. 10 / 33
(∆ + 1 ) -coloring Arbitrary graphs ◮ can be randomly computed in expected #rounds O ( log n ) (Alon, Babai, Itai [J. Alg. 1986]) (Luby [SIAM J. Comput. 1986]) ◮ best known deterministic algorithm performs in 2 O ( √ log n ) rounds (Panconesi, Srinivasan [J. Algorithms, 1996]) Bounded degree graphs ◮ Randomization does not help for 3-coloring the ring (Naor [SIAM Disc. Maths 1991]) ◮ can be randomly computed in expected #rounds � O ( log ∆ + log n ) (Schneider, Wattenhofer [PODC 2010]) ◮ best known deterministic algorithm performs in O (∆ + log ∗ n ) rounds (Barenboim, Elkin [STOC 2009]) (Kuhn [SPAA 2009]) 11 / 33
2-sided error Monte Carlo algorithms Focus on distributed algorithms that use randomization but whose running time are deterministic. 12 / 33
2-sided error Monte Carlo algorithms Focus on distributed algorithms that use randomization but whose running time are deterministic. ( p , q ) -decider ◮ If ( G , x ) ∈ L then, for every identity assignment Id, Pr [ out A ( G , x , Id , v ) = “yes” for every node v ∈ V ( G )] ≥ p ◮ If ( G , x ) / ∈ L then, for every identity assignment Id, Pr [ out A ( G , x , Id , v ) = “no” for at least one node v ∈ V ( G )] ≥ q 12 / 33
Example: AMOM 13 / 33
Example: AMOM Randomized algorithm ◮ every unmarked node says “yes” with probability 1; ◮ every marked node says “yes” with probability p . 13 / 33
Example: AMOM Randomized algorithm ◮ every unmarked node says “yes” with probability 1; ◮ every marked node says “yes” with probability p . Remarks: ◮ Runs in zero time; ◮ If the configuration has at most one marked node then correct with probability at least p . ◮ If there are at least k ≥ 2 marked nodes, correct with probability at least 1 − p k ≥ 1 − p 2 ◮ Thus there exists a ( p , q ) -decider for q + p 2 ≤ 1. 13 / 33
Bounded-probability error local decision Definition BPLD ( t , p , q ) is the class of all distributed languages that have a randomized distributed ( p , q ) -decider running in time at most t . I.e., can be decided in time at most t by a randomized distributed algorithm with “yes” success probability p and “no” success probability q . 14 / 33
Bounded-probability error local decision Definition BPLD ( t , p , q ) is the class of all distributed languages that have a randomized distributed ( p , q ) -decider running in time at most t . I.e., can be decided in time at most t by a randomized distributed algorithm with “yes” success probability p and “no” success probability q . Remark For p and q such that p 2 + q ≤ 1, there exists a language L ∈ BPLD ( 0 , p , q ) , such that L / ∈ LD ( t ) , for any t = o ( n ) . 14 / 33
A sharp threshold for hereditary languages A prefix of a configuration ( G , x ) is a configuration ( G [ U ] , x [ U ]) , where U ⊆ V ( G ) Hereditary languages A language L is hereditary if every prefix of every configuration ( G , x ) ∈ L is also in L . ◮ Coloring and AMOM are hereditary languages. ◮ Every language { ( G , ǫ ) | G ∈ G} where G is hereditary is... hereditary. (Examples of hereditary graph families are planar graphs, interval graphs, forests, chordal graphs, cographs, perfect graphs, etc.) 15 / 33
A sharp threshold for hereditary languages A prefix of a configuration ( G , x ) is a configuration ( G [ U ] , x [ U ]) , where U ⊆ V ( G ) Hereditary languages A language L is hereditary if every prefix of every configuration ( G , x ) ∈ L is also in L . ◮ Coloring and AMOM are hereditary languages. ◮ Every language { ( G , ǫ ) | G ∈ G} where G is hereditary is... hereditary. (Examples of hereditary graph families are planar graphs, interval graphs, forests, chordal graphs, cographs, perfect graphs, etc.) Theorem Let L be an hereditary language and let t be a function of triples ( G , x , Id ) . If L ∈ BPLD ( t , p , q ) for constants p , q ∈ ( 0 , 1 ] such that p 2 + q > 1 , then L ∈ LD ( O ( t )) . 15 / 33
One ingredient in the proof Let 0 < δ < p 2 + q − 1, and define λ = 11 · ⌈ log p / log ( 1 − δ ) ⌉ . Separating partition A separating partition of ( G , x , Id ) is a triplet ( S , U 1 , U 2 ) of pairwise disjoint subsets of nodes such that S ∪ U 1 ∪ U 2 = V , and dist G ( U 1 , U 2 ) ≥ λ · t . U S U 2 1 16 / 33
Glueing lemma Given a separating partition ( S , U 1 , U 2 ) of ( G , x , Id ) , let G k = G [ U k ∪ S ] , and let x k be the input x restricted to nodes in G k , for k = 1 , 2. S S U U 2 1 Lemma ( ⋆ ) For every instance ( G , x ) with identity assignment Id, and every separating partition ( S , U 1 , U 2 ) of ( G , x , Id ) , we have: � � ( G 1 , x 1 ) ∈ L and ( G 2 , x 2 ) ∈ L ⇒ ( G , x ) ∈ L . Remark. Lemma ( ⋆ ) does not use the fact that L is hereditary, but uses p 2 + q > 1. 17 / 33
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