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Approximations for the General Block Distribution of a Matrix
, 1997
"... The general block distribution of a matrix is a rectilinear partition of the matrix into orthogonal blocks such that the maximum sum of the elements within a single block is minimized. This corresponds to partitioning the matrix onto parallel processors so as to minimize processor load while maintai ..."
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Cited by 7 (0 self)
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The general block distribution of a matrix is a rectilinear partition of the matrix into orthogonal blocks such that the maximum sum of the elements within a single block is minimized. This corresponds to partitioning the matrix onto parallel processors so as to minimize processor load while maintaining regular communication patterns. Applications of the problem include various parallel sparse matrix computations, compilers for highperformance languages, particle in cell computations, video and image compression, and simulations associated with a communication network. We analyze the performance guarantee of a natural and practical heuristic based on iterative refinement, which has previously been shown to give good empirical results. When p 2 is the number of blocks, we show that the tight performance ratio is `( p p). When the matrix has rows of large cost, the details of the objective function of the algorithm are shown to be important, since a naive implementation can lead to...
Using Sparsification for Parametric Minimum Spanning Tree Problems
 Nordic J. Computing
, 1996
"... Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning t ..."
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Cited by 7 (2 self)
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Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning tree problem, as well as other search problems, on dense graphs. 1 Introduction In the parametric minimum spanning tree problem, one is given an nnode, medge undirected graph G where each edge e has a linear weight function w e (#)=a e +#b e . Let Z(#) denote the weight of the minimum spanning tree relative to the weights w e (#). It can be shown that Z(#) is a piecewise linear concave function of # [Gus80]; the points at which the slope of Z changes are called breakpoints. We shall present two results regarding parametric minimum spanning trees. First, we show that Z(#) can be constructed in O(min{nm log n, TMST (2n, n) # Department of Computer Science, Iowa State University, Ames, IA...
LinearTime Algorithms for Parametric Minimum Spanning Tree Problems on Planar Graphs
, 1995
"... A lineartime algorithm for the minimumratio spanning tree problem on planar graphs is presented. The algorithm is based on a new planar minimum spanning tree algorithm. The approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having s ..."
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Cited by 3 (2 self)
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A lineartime algorithm for the minimumratio spanning tree problem on planar graphs is presented. The algorithm is based on a new planar minimum spanning tree algorithm. The approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having small separators. 1 Introduction Suppose we are given an undirected graph G where each edge e has two weights a e and b e ; the b e 's are assumed to be either all negative or all positive. The minimum ratio spanning tree problem (MRST) [Cha77] is to find a spanning tree T of G such that the ratio P e2T a e = P e2T b e is minimized. One application of MRST arises in the design of communication networks. The number a e represents the cost of building link e, while b e represents the time required to build that link. The goal is to find a tree that minimizes the ratio of total cost over construction time. Other applications of MRST are given elsewhere [CMV89, Meg83]. The main result of thi...
Optimal Parametric Search on Graphs of Bounded Treewidth
, 1994
"... We give lineartime algorithms for a class of parametric search problems on weighted graphs of bounded treewidth. We also discuss the implications of our results to approximate parametric search on planar graphs. ..."
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Cited by 2 (2 self)
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We give lineartime algorithms for a class of parametric search problems on weighted graphs of bounded treewidth. We also discuss the implications of our results to approximate parametric search on planar graphs.
Partitioning Spatially Located Computations using Rectangles
 IPDPS
, 2011
"... The ideal distribution of spatially located heterogeneous workloads is an important problem to address in parallel scientific computing. We investigate the problem of partitioning such workloads (represented as a matrix of positive integers) into rectangles, such that the load of the most loaded rec ..."
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Cited by 2 (1 self)
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The ideal distribution of spatially located heterogeneous workloads is an important problem to address in parallel scientific computing. We investigate the problem of partitioning such workloads (represented as a matrix of positive integers) into rectangles, such that the load of the most loaded rectangle (processor) is minimized. Since finding the optimal arbitrary rectanglebased partition is an NPhard problem, we investigate particular classes of solutions, namely, rectilinear partitions, jagged partitions and hierarchical partitions. We present a new class of solutions called mway jagged partitions, propose new optimal algorithms for mway jagged partitions and hierarchical partitions, propose new heuristic algorithms, and provide worst case performance analyses for some existing and new heuristics. Moreover, the algorithms are tested in simulation on a wide set of instances. Results show that two of the algorithms we introduce lead to a much better load balance than the stateoftheart algorithms. 1.
LoadBalancing Spatially Located Computations using Rectangular Partitions
 ARXIV
, 2011
"... Distributing spatially located heterogeneous workloads is an important problem in parallel scientific computing. We investigate the problem of partitioning such workloads (represented as a matrix of nonnegative integers) into rectangles, such that the load of the most loaded rectangle (processor) i ..."
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Cited by 2 (1 self)
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Distributing spatially located heterogeneous workloads is an important problem in parallel scientific computing. We investigate the problem of partitioning such workloads (represented as a matrix of nonnegative integers) into rectangles, such that the load of the most loaded rectangle (processor) is minimized. Since finding the optimal arbitrary rectanglebased partition is an NPhard problem, we investigate particular classes of solutions: rectilinear, jagged and hierarchical. We present a new class of solutions called mway jagged partitions, propose new optimal algorithms for mway jagged partitions and hierarchical partitions, propose new heuristic algorithms, and provide worst case performance analyses for some existing and new heuristics. Moreover, the algorithms are tested in simulation on a wide set of instances. Results show that two of the algorithms we introduce lead to a much better load balance than the stateoftheart algorithms. We also show how to design a twophase algorithm that reaches different time/quality tradeoff.
Exploiting SelfCanceling Demand Point Aggregation Error for Some Planar Rectilinear Median Location Problems
"... Abstract: When solving location problems in practice it is quite common to aggregate demand points into centroids. Solving a location problem with aggregated demand data is computationally easier, but the aggregation process introduces error. We develop theory and algorithms for certain types of cen ..."
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Abstract: When solving location problems in practice it is quite common to aggregate demand points into centroids. Solving a location problem with aggregated demand data is computationally easier, but the aggregation process introduces error. We develop theory and algorithms for certain types of centroid aggregations for rectilinear 1median problems. The objective is to construct an aggregation that minimizes the maximum aggregation error. We focus on rowcolumn aggregations, and make use of aggregation results for 1median problems on the line to do aggregation for 1median problems in the plane. The aggregations developed for the 1median problem are then used to construct approximate nmedian problems. We test the theory computationally on nmedian problems (n � 1) using both randomly generated, as well as real, data. Every error measure we consider can be well approximated by some power function in the number of aggregate demand points. Each such function exhibits decreasing returns to scale.
Reading, Mass.: AddisonWesley Publishing Company, 1985. Efficient Partitioning of Sequences
"... checkers for optimal tunidirectional error detecting codes,” ..."
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"... www.elsevier.com/locate/tcs Approximations for the general block distribution of a matrix � ..."
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www.elsevier.com/locate/tcs Approximations for the general block distribution of a matrix �
LoadBalancing Spatially Located Computations using Rectangular Partitions ✩
"... Distributing spatially located heterogeneous workloads is an important problem in parallel scientific computing. We investigate the problem of partitioning such workloads (represented as a matrix of nonnegative integers) into rectangles, such that the load of the most loaded rectangle (processor) i ..."
Abstract
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Distributing spatially located heterogeneous workloads is an important problem in parallel scientific computing. We investigate the problem of partitioning such workloads (represented as a matrix of nonnegative integers) into rectangles, such that the load of the most loaded rectangle (processor) is minimized. Since finding the optimal arbitrary rectanglebased partition is an NPhard problem, we investigate particular classes of solutions: rectilinear, jagged and hierarchical. We present a new class of solutions called mway jagged partitions, propose new optimal algorithms for mway jagged partitions and hierarchical partitions, propose new heuristic algorithms, and provide worst case performance analyses for some existing and new heuristics. Moreover, the algorithms are tested in simulation on a wide set of instances. Results show that two of the algorithms we introduce lead to a much better load balance than the stateoftheart algorithms. We also show how to design a twophase algorithm that reaches different time/quality tradeoff.