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50
Randomized Distributed Edge Coloring via an Extension of the ChernoffHoeffding Bounds
 SIAM J. Comput
, 1997
"... . Certain types of routing, scheduling, and resourceallocation problems in a distributed setting can be modeled as edgecoloring problems. We present fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed pointtopoint model of computation. Our algorithms co ..."
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Cited by 56 (9 self)
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. Certain types of routing, scheduling, and resourceallocation problems in a distributed setting can be modeled as edgecoloring problems. We present fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed pointtopoint model of computation. Our algorithms compute an edge coloring of a graph G with n nodes and maximum degree # with at most 1.6# +O(log 1+# n) colors with high probability (arbitrarily close to 1) for any fixed #>0; they run in polylogarithmic time. The upper bound on the number of colors improves upon the (2#  1)coloring achievable by a simple reduction to vertex coloring. To analyze the performance of our algorithms, we introduce new techniques for proving upper bounds on the tail probabilities of certain random variables. The Cherno#Hoe#ding bounds are fundamental tools that are used very frequently in estimating tail probabilities. However, they assume stochastic independence among certain random variables, which may n...
On Algorithms for Efficient Data Migration
, 2001
"... The data migration problem is the problem of computing an efficient plan for moving data stored on devices in a network from one configuration to another. Load balancing or changing usage patterns could necessitate such a rearrangement of data. In this paper, we consider the case where the objects a ..."
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Cited by 44 (3 self)
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The data migration problem is the problem of computing an efficient plan for moving data stored on devices in a network from one configuration to another. Load balancing or changing usage patterns could necessitate such a rearrangement of data. In this paper, we consider the case where the objects are fixedsize and the network is complete. The direct migration problem is closely related to edgecoloring. However, because there are space constraints on the devices, the problem is more complex. Our main results are polynomial time algorithms for finding a nearoptimal migration plan in the presence of space constraints when a certain number of additional nodes is available as temporary storage, and a 3/2approximation for the case where data must be migrated directly to its destination.
Algorithms for Data Migration with Cloning
, 2003
"... Our work is motivated by the problem of managing data on storage devices, typically a set of disks. Such high demand storage servers are used as web servers, or multimedia servers for handling high demand for data. As the system is running, it needs to dynamically respond to changes in demand for di ..."
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Cited by 32 (4 self)
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Our work is motivated by the problem of managing data on storage devices, typically a set of disks. Such high demand storage servers are used as web servers, or multimedia servers for handling high demand for data. As the system is running, it needs to dynamically respond to changes in demand for di#erent data items. In this work we study the data migration problem, which arises when we need to quickly change one storage configuration into another. We show that this problem is NPhard. In addition, we develop polynomialtime approximation algorithms for this problem and prove a worst case bound of 9.5 on the approximation factor achieved by our algorithm. We also compare the algorithm to several heuristics for this problem.
Coloring Graphs With Sparse Neighborhoods
, 1998
"... It is shown that the chromatic number of any graph with maximum degree d in which the number of edges in the induced subgraph on the set of all neighbors of any vertex does not exceed d²/f is at most O(d/log f). This is tight (up to a constant factor) for all admissible values of d and f. ..."
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Cited by 30 (15 self)
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It is shown that the chromatic number of any graph with maximum degree d in which the number of edges in the induced subgraph on the set of all neighbors of any vertex does not exceed d²/f is at most O(d/log f). This is tight (up to a constant factor) for all admissible values of d and f.
Star Coloring of Graphs
, 2001
"... A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored. The star ..."
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Cited by 25 (1 self)
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A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored. The star
Representations of graphs modulo n
 J. Graph Theory
, 1994
"... A graph is said to be representable modulo n if its vertices can be labelled with distinct integers between 0 and n − 1 inclusive such that two vertices are adjacent if and only if their labels are relatively prime to n. The representation number of graph G is the smallest n representing G. We revie ..."
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Cited by 7 (0 self)
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A graph is said to be representable modulo n if its vertices can be labelled with distinct integers between 0 and n − 1 inclusive such that two vertices are adjacent if and only if their labels are relatively prime to n. The representation number of graph G is the smallest n representing G. We review known results and investigate representation numbers for several new classes. In particular, we relate the representation number of the disjoint union of complete graphs to the existence of complete families of mutually orthogonal Latin squares. 1
DimensionExchange Algorithms for Load Balancing on Trees
 Procs. of 9th Int. Colloquium on Structural Information and Communication Complexity
, 2002
"... This paper considers dimensionexchange algorithms for load balancing on trees with finitelydivisible loads (token distribution). We present improved analysis of an existing protocol, and in particular, establish a logarithmic upper bound on the discrepancy of the final distribution. Our second con ..."
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Cited by 7 (0 self)
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This paper considers dimensionexchange algorithms for load balancing on trees with finitelydivisible loads (token distribution). We present improved analysis of an existing protocol, and in particular, establish a logarithmic upper bound on the discrepancy of the final distribution. Our second contribution is a new algorithm, which assuming each node has knowledge of the total number of nodes, determines a perfectly balanced distribution.
On Fans in Multigraphs
 JOURNAL OF GRAPH THEORY
, 2005
"... We introduce a unifying framework for studying edgecolouring problems on multigraphs. This is defined in terms of a rooted directed multigraph F which is naturally associated to the set of fans based at a given vertex u in a multigraph G. We call F the “Fan Digraph”. We show that fans in G based at ..."
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Cited by 7 (5 self)
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We introduce a unifying framework for studying edgecolouring problems on multigraphs. This is defined in terms of a rooted directed multigraph F which is naturally associated to the set of fans based at a given vertex u in a multigraph G. We call F the “Fan Digraph”. We show that fans in G based at u are in onetoone correspondence with directed trails in F starting at the root of F. We state and prove a central theorem about the fan digraph, which embodies many edgecolouring results and expresses them at a higher level of abstraction. Using this result, we derive short proofs of classical theorems. We conclude with a new, generalized version of Vizing’s Adjacency Lemma for multigraphs, which is stronger than all those known to the author.
Hadwiger’s conjecture for line graphs
, 2003
"... We prove that Hadwiger’s conjecture holds for line graphs. Equivalently, we show that for every loopless graph G (possibly with parallel edges) and every integer k ≥ 0, either G is kedgecolourable, or there are k + 1 connected subgraphs A1,..., Ak+1 of G, each with at least one edge, such that E(A ..."
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Cited by 7 (0 self)
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We prove that Hadwiger’s conjecture holds for line graphs. Equivalently, we show that for every loopless graph G (possibly with parallel edges) and every integer k ≥ 0, either G is kedgecolourable, or there are k + 1 connected subgraphs A1,..., Ak+1 of G, each with at least one edge, such that E(Ai ∩ Aj) = ∅ and V (Ai ∩ Aj) � = ∅ for 1 ≤ i < j ≤ k.