What Makes a Distributed Problem Truly Local? or: why might “Coloring” just possibly be easier than “MIS”? Adrian Kosowski IRIF and Inria Paris Includes results of work with: Pierre Fraigniaud, Cyril Gavoille, Marc Heinrich, and Marcin Markiewicz SIROCCO 2016 – Helsinki, July 20, 2016
Outline • What problems do we consider local? • The LOCAL model • MIS and Coloring • A Constraint Satjsfactjon framework • What problems do others consider local? • Some insights from QCA and tjling communitjes • Non-signaling and its implicatjons • What does this all mean for us? Kosowski: Truly Local Problems 2/46
What problems do we consider local? C. Gavoille, A. Kosowski, M. Markiewicz - Round-based models of Quantum Distributed Computjng 3/41
The LOCAL model Assumptjons of the LOCAL model • The distributed system consists of a set of processors V , | V |= n. • The system operates in synchronous rounds, with no faults. • The system input is encoded as a labeled graph G = ( V , E ) • node labels (inputs) are given as x ( v ), for v V. The result of computatjons is given through local variables y ( v ), for v V. • • Messages exchanged in each round may have unbounded size. • The computatjonal capabilitjes of each node are unbounded. • As a rule, we will assume that nodes have unique identjfjers. Kosowski: Truly Local Problems 4/46
The LOCAL model Assumptjons of the LOCAL model • The distributed system consists of a set of processors V , | V |= n. • The system operates in synchronous rounds, with no faults. • The system input is encoded as a labeled graph G = ( V , E ) • node labels (inputs) are given as x ( v ), for v V. The result of computatjons is given through local variables y ( v ), for v V. • • Messages exchanged in each round may have unbounded size. • The computatjonal capabilitjes of each node are unbounded. • As a rule, we will assume that nodes have unique identjfjers. Motjvatjon? Understanding limits of locality in distributed computjng. Sandbox for running simple greedy/distributed algorithms (auctjons/pricing, load balancing, LLL,...) Kosowski: Truly Local Problems 5/46
Warm-up: A simple local settjng • The most constrained local settjng: • G has constant maximum degree • Algorithms are allowed to run for O(1) rounds • In this settjng, deterministjc approaches make the most sense. • Example: recoloring a ring to use fewer colors [Cole-Vishkin 1986] 29 29 1023 17 Kosowski: Truly Local Problems 6/46
Warm-up: A simple local settjng • The most constrained local settjng: • G has constant maximum degree • Algorithms are allowed to run for O(1) rounds • In this settjng, deterministjc approaches make the most sense. • Example: recoloring a ring to use fewer colors [Cole-Vishkin 1986] 29=(000001 1 1 0 1) 1023=(11111111 1 1) 17 = (000001 0 001) Kosowski: Truly Local Problems 7/46
Warm-up: A simple local settjng • The most constrained local settjng: • G has constant maximum degree • Algorithms are allowed to run for O(1) rounds • In this settjng, deterministjc approaches make the most sense. • Example: recoloring a ring to use fewer colors [Cole-Vishkin 1986] 29=(000001 1 1 0 1) 1023=(11111111 1 1) 17 = (000001 0 001) [(3,1),(1,0)] [(1,1),(..,..)] [(3,0),(..,..)] Kosowski: Truly Local Problems 8/46
Warm-up: A simple local settjng • The most constrained local settjng: • G has constant maximum degree • Algorithms are allowed to run for O(1) rounds • In this settjng, deterministjc approaches make the most sense. • Example: recoloring a ring to use fewer colors [Cole-Vishkin 1986] [(3,1),(1,0)] [(1,1),(..,..)] [(3,0),(..,..)] Kosowski: Truly Local Problems 9/46
Warm-up: A simple local settjng • The most constrained local settjng: • G has constant maximum degree • Algorithms are allowed to run for O(1) rounds • In this settjng, deterministjc approaches make the most sense. • Example: recoloring a ring to use fewer colors [Cole-Vishkin 1986] – We can reduce a c -coloring to a O (log c )-coloring of a ring in a single communicatjon round. – Same approach can be applied for any graph of constant maximum degree. • What can we compute in O(1) rounds? survey [Suomela, 2013] Kosowski: Truly Local Problems 10/46
Fast distributed algorithms in LOCAL • More parameters: • Number of rounds depends on the number of nodes n • Number of rounds depends on maximum degree • Randomizatjon can make a difgerence • Considered problems: local validity of a solutjon can be checked by each node by looking at the states of its neighbors (1-LCA) • Two basic benchmark problems: • “Easier” : ( )-coloring • “Harder” : Maximal Independent Set (MIS) Kosowski: Truly Local Problems 11/46
LOCAL: Coloring and MIS Deterministjc Randomized O( log + 2 O( log log n ) [HSS16] 2 O( log n ) [PS92] ( +1)-coloring: Õ ( + log* n [FHK16] (log* n ) for 2 [L92] O(log + 2 O( log log n ) [BEPS12] 2 O( log n ) [PS92] MIS: O( ) + log* n [BE09] ( log n / log log n ) [KMW04] for 2 O( log n log log n )) [Linial 1992] [Panconesi & Srinivasan 1992] [Kuhn, Moscibroda, Watuenhofger 2004] [Barenboim & Elkin 2009] [Barenboim, Elkin, Pettje, Schneider 2012 ] [Fraigniaud, Heinrich, K. 2016] [Harris, Schneider, Su 2016] Kosowski: Truly Local Problems 12/46
LOCAL: Coloring and MIS Deterministjc Randomized O( log + 2 O( log log n ) [HSS16] 2 O( log n ) [PS92] ( +1)-coloring: Õ ( + log* n [FHK16] (log* n ) for 2 [L92] O(log + 2 O( log log n ) [BEPS12] 2 O( log n ) [PS92] MIS: O( ) + log* n [BE09] ( log n / log log n ) [KMW04] for 2 O( log n log log n )) Questjon 1: Is MIS harder than coloring? • Yes, in the randomized model. Kosowski: Truly Local Problems 13/46
LOCAL: Coloring and MIS Deterministjc Randomized O( log + 2 O( log log n ) [HSS16] 2 O( log n ) [PS92] ( +1)-coloring: Õ ( + log* n [FHK16] (log* n ) for 2 [L92] O(log + 2 O( log log n ) [BEPS12] 2 O( log n) [PS92] MIS: O( ) + log* n [BE09] ( log n / log log n ) [KMW04] for 2 O( log n log log n )) Questjon 1: Is MIS harder than coloring? • Yes, in the randomized model. • Possibly, in the deterministjc model. Kosowski: Truly Local Problems 14/46
LOCAL: Coloring and MIS Deterministjc Randomized O( log + 2 O( log log n ) [HSS16] 2 O( log n ) [PS92] ( +1)-coloring: Õ ( + log* n [FHK16] (log* n ) for 2 [L92] O(log + 2 O( log log n ) [BEPS12] 2 O( log n ) [PS92] MIS: O( ) + log* n [BE09] ( log n / log log n ) [KMW04] for 2 O( log n log log n )) Questjon 2: Does randomizatjon help in the LOCAL model? (Yes, but this is not apparent from the above table.) Kosowski: Truly Local Problems 15/46
LOCAL: Time of coloring with difgerent palletues Deterministjc Randomized ( n ) 2 colors: (path) [Linial 1992] O( /log ) colors: O(log n ) (triangle-free) (roughly) [Pettje & Su, 2013] colors: (log n ) (log log n ) (tree, >54) [Chang, Kopelowitz, Pettje 2016] 2 colors: (log* n ) [Linial 1992] Kosowski: Truly Local Problems 16/46
LOCAL: Time of coloring with difgerent palletues Deterministjc Randomized ( n ) 2 colors: (path) [Linial 1992] O( /log ) colors: O(log n ) (triangle-free) (roughly) [Pettje & Su, 2013] colors: (log n ) (log log n ) (tree, >54) [Chang, Kopelowitz, Pettje 2016] 2 colors: (log* n ) [Linial 1992] Kosowski: Truly Local Problems 17/46
Constraint Satjsfactjon in the LOCAL model C. Gavoille, A. Kosowski, M. Markiewicz - Round-based models of Quantum Distributed Computjng 18/41
Constraint density vs. hardness The picture in the centralized world: Centralized SAT on random instances satjsfjability diffjculty 100% satjsfjable Contradictjon Random solutjon Random solutjon easy to fjnd works works Density = constraints / variables Kosowski: Truly Local Problems 19/46
Encoding problems through edge constraints Settjng: Settjng: • We are given a simple (1-round) algorithm or routjne which tries to do something meaningful • “obtain a partjal coloring of the graph with a given palletue” • “extend an IS towards a MIS by including new nodes” The routjne assigns an output state y ( v ) L ( v ) to each node v • • Assumptjon: for any pair of neighbors u , v, we can locally tell if the states y ( u ) and y ( v ) are compatjble just by looking at the edge { u , v } • coloring fails locally if y ( u ) = y ( v ). • Independent Set fails locally if y ( u ) = 1 and y ( v ) = 1. How constraining is the problem we are considering? Kosowski: Truly Local Problems 20/46
Constraint density vs. hardness Our predictjon for the LOCAL model diffjculty 100% feasible feasibility Random solutjon Density = constraints / “palletue size” works Kosowski: Truly Local Problems 21/46
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