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 n-node, m-edge 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...

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 high-performance 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...

A linear-time algorithm for the minimum-ratio 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...

We give linear-time algorithms for a class of parametric search problems on weighted graphs of bounded tree-width. We also discuss the implications of our results to approximate parametric search on planar graphs.

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 rectangle-based 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 m-way jagged partitions, propose new optimal algorithms for m-way 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 state-of-the-art algorithms. 1.

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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 non-negative integers) into rectangles, such that the load of the most loaded rectangle (processor) is minimized. Since finding the optimal arbitrary rectangle-based partition is an NP-hard problem, we investigate particular classes of solutions: rectilinear, jagged and hierarchical. We present a new class of solutions called m-way jagged partitions, propose new optimal algorithms for m-way 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 state-of-the-art algorithms. We also show how to design a two-phase algorithm that reaches different time/quality tradeoff.

www.elsevier.com/locate/tcs Approximations for the general block distribution of a matrix �