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Distributed coloring in O( √ log n) bit rounds
 In International Parallel & Distributed Processing Symposium (IPDPS
, 2006
"... We consider the wellknown vertex coloring problem: given a graph G, find a coloring of the vertices so that no two neighbors in G have the same color. It is trivial to see that every graph of maximum degree ∆ can be colored with ∆+1 colors, and distributed algorithms that find a (∆+1)coloring in a ..."
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Cited by 10 (1 self)
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We consider the wellknown vertex coloring problem: given a graph G, find a coloring of the vertices so that no two neighbors in G have the same color. It is trivial to see that every graph of maximum degree ∆ can be colored with ∆+1 colors, and distributed algorithms that find a (∆+1)coloring in a logarithmic number of communication rounds, with high probability, are known since more than a decade. This is in general the best possible if only a constant number of bits can be sent along every edge in each round. In fact, we show that for the nnode cycle the bit complexity of the coloring problem is Ω(log n). More precisely, if only one bit can be sent along each edge in a round, then every distributed coloring algorithm (i.e., algorithms in which every node has the same initial state and initially only knows its own edges) needs at least Ω(log n) rounds, with high probability, to color the cycle, for any finite number of colors. But what if the edges have orientations, i.e., the endpoints of an edge agree on its orientation (while bits may still flow in both directions)? Does this allow one to provide faster coloring algorithms? Interestingly, for the cycle in which all edges have the same orientation, we show that a simple randomized algorithm can achieve a 3coloring with only O ( √ log n) rounds of bit transmissions, with high probability (w.h.p.). This re∗ Supported by NSF grant CCR0311121. † Partially supported by the DFGSonderforschungsbereich 376 and by the EU within 6th Framework Programme under contract 001907 Dynamically
Graph Coloring on Coarse Grained Multicomputers
, 2002
"... We present an efficient and scalable Coarse Grained Multicomputer (CGM) coloring algorithm that colors a graph G with at most D+ 1 colors where D is the maximum degree in G. This algorithm is given in two variants: randomized and deterministic. We show that on a pprocessor CGM model the proposed al ..."
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Cited by 6 (1 self)
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We present an efficient and scalable Coarse Grained Multicomputer (CGM) coloring algorithm that colors a graph G with at most D+ 1 colors where D is the maximum degree in G. This algorithm is given in two variants: randomized and deterministic. We show that on a pprocessor CGM model the proposed algorithms require a parallel time of O( G p ) and a total work and overall communication cost of O(G). These bounds correspond to the average case for the randomized version and to the worstcase for the deterministic variant. Key words: graph algorithms, parallel algorithms, graph coloring, Coarse Grained Multicomputers 1
Distributed Coloring Depending on the Chromatic Number or the Neighborhood Growth
"... Abstract 1 We deterministically compute a ∆+1 coloring in time O(∆5c+2·(∆5) 2/c /(∆1) ɛ + (∆1) ɛ + log ∗ n) and O(∆5c+2 · (∆5) 1/c / ∆ ɛ + ∆ ɛ + (∆5) d log ∆5 log n) for arbitrary constants d, ɛ and arbitrary constant integer c, where ∆i is defined as the maximal number of nodes within distance i fo ..."
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Cited by 2 (2 self)
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Abstract 1 We deterministically compute a ∆+1 coloring in time O(∆5c+2·(∆5) 2/c /(∆1) ɛ + (∆1) ɛ + log ∗ n) and O(∆5c+2 · (∆5) 1/c / ∆ ɛ + ∆ ɛ + (∆5) d log ∆5 log n) for arbitrary constants d, ɛ and arbitrary constant integer c, where ∆i is defined as the maximal number of nodes within distance i for a node and ∆: = ∆1. Our greedy algorithm improves the stateoftheart ∆+1 coloring algorithms for a large class of graphs, e.g. graphs of moderate neighborhood growth. We also state and analyze a randomized coloring algorithm in terms of the chromatic number, the run time and the used colors. If ∆ ∈ Ω(log 1+1 / log ∗ n n) and χ ∈ O(∆ / log 1+1 / log ∗ n n) then our algorithm executes in time O(log χ + log ∗ n) with high probability. For graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest ∆ + 1 coloring algorithm running in time O(log ∆ + √ log n). The algorithm works without knowledge of χ and uses less than ∆ colors, i.e., (1 − 1/O(χ)) ∆ with high probability. To the best of our knowledge this is the first distributed algorithm for (such) general graphs taking the chromatic number χ into account. 1
The Power of Orientation in SymmetryBreaking
"... distributedcomputing.Ithasapplicationstoimportantproblems such as graph vertex and edge coloring, maximal independentsets,andthelike.Deterministicalgorithmsforsymmetry breakingthatruninapolylogarithmicnumberofroundsare notknown.However,randomizedalgorithmsthatruninpolylogarithmicnumberofroundsarekno ..."
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distributedcomputing.Ithasapplicationstoimportantproblems such as graph vertex and edge coloring, maximal independentsets,andthelike.Deterministicalgorithmsforsymmetry breakingthatruninapolylogarithmicnumberofroundsare notknown.However,randomizedalgorithmsthatruninpolylogarithmicnumberofroundsareknownstartingfromLuby’s algorithm[17]. Recently,orientationonedgeswasconsideredanditwasshown thatan O(∆)coloringoftheverticesofagivenorientedgraph can be arrived at using essentially O(log ∆ + √ log n) bits of communication. Inthispaperwefurtherdemonstratethepoweroforientation onedgesinsymmetrybreaking.Wepresentefficientalgorithms toconstructfractionalindependentsetsinconstantdegreegraphs usingverylowordercommunicationbetweenthevertices.For instance,weshowthatinboundeddegreegraphsandplanar graphs,itispossibletoconstructafractionalindependentsetby exchanging O(1)bits.Further,wepresentalgorithmstoconstruct maximalindependentsetsinboundeddegreegraphsandoriented trees.OuralgorithmforconstructinganMISofanorientedtree usesonly O(log n)bitsofcommunication.
Evolving, Large Scale Information Systems (DELIS).
"... We consider the wellknown vertex coloring problem: given a graph G, find a coloring of the vertices so that no two neighbors in G have the same color. It is trivial to see that every graph of maximum degree ∆ can be colored with ∆+1 colors, and distributed algorithms that find a (∆+1)coloring in a ..."
Abstract
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We consider the wellknown vertex coloring problem: given a graph G, find a coloring of the vertices so that no two neighbors in G have the same color. It is trivial to see that every graph of maximum degree ∆ can be colored with ∆+1 colors, and distributed algorithms that find a (∆+1)coloring in a logarithmic number of communication rounds, with high probability, are known since more than a decade. This is in general the best possible if only a constant number of bits can be sent along every edge in each round. In fact, we show that for the nnode cycle the bit complexity of the coloring problem is Ω(log n). More precisely, if only one bit can be sent along each edge in a round, then every distributed coloring algorithm (i.e., algorithms in which every node has the same initial state and initially only knows its own edges) needs at least Ω(log n) rounds, with high probability, to color the cycle, for any finite number of colors. But what if the edges have orientations, i.e., the endpoints of an edge agree on its orientation (while bits may still flow in both directions)? Does this allow one to provide faster coloring algorithms? Interestingly, for the cycle in which all edges have the same orientation, we show that a simple randomized algorithm can achieve a 3coloring with only O ( √ log n) rounds of bit transmissions, with high probability (w.h.p.). This re∗ Supported by NSF grant CCR0311121. † Partially supported by the DFGSonderforschungsbereich 376 and by the EU within 6th Framework Programme under contract 001907 Dynamically
Abstract Graph Coloring on Coarse Grained
"... We present an efficient and scalable Coarse Grained Multicomputer (CGM) coloring algorithm that colors a graph G with at most ∆+1 colors where ∆ is the maximum degree in G. This algorithm is given in two variants: a randomized and a deterministic. We show that on a pprocessor CGM model the proposed ..."
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We present an efficient and scalable Coarse Grained Multicomputer (CGM) coloring algorithm that colors a graph G with at most ∆+1 colors where ∆ is the maximum degree in G. This algorithm is given in two variants: a randomized and a deterministic. We show that on a pprocessor CGM model the proposed algorithms require a parallel time of O ( G p) and a total work and overall communication cost of O(G). These bounds correspond to the average case for the randomized version and to the worst case for the deterministic variant.
Topology and Routing in Overlay Networks by
, 2006
"... In this age of information, new models of information exchange methodologies based on overlay networks are gaining popular attention. Overlay networks provide a logical interconnection topology over an existing physical network. Overlay networks offer benefits such as ease of implementation, flexib ..."
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In this age of information, new models of information exchange methodologies based on overlay networks are gaining popular attention. Overlay networks provide a logical interconnection topology over an existing physical network. Overlay networks offer benefits such as ease of implementation, flexibility, adaptability, and incremental deployability. Due to the wide range of applications and advantages, formal study of overlay networks is required to understand the various research challenges in this context. In this thesis, we study two classes of overlay networks namely peertopeer networks and wireless ad hoc networks. Our focus will be along two central issues in overlay networks: how to arrive at efficient topologies and how to provide efficient routing strategies. Peertopeer networks have gained a lot of research attention in recent years for various reasons. Despite many advances however, fundamental questions such as how to design deterministic constructions, and how to organize peers of nonuniform bandwidth have remained open. In this thesis, we answer these questions by providing a deterministic overlay topology, Pagoda, that can be used for efficient routing, data management and
This document in subdirectoryRS/97/37/ Fast Distributed Algorithms for BrooksVizing Colourings
, 909
"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS ..."
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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS