In this paper we introduce a direct motivation for solving the minimum 2-sum problem, for which we present a linear-time algorithm inspired by the Algebraic Multigrid approach which is based on weighted edge contraction. Our results turned out to be better than previous results, while the short running time of the algorithm enabled experiments with very large graphs. We thus introduce a new benchmark for the minimum 2-sum problem which contains 66 graphs of various characteristics. In addition, we propose the straightforward use of a part of our algorithm as a powerful local reordering method for any other (than multilevel) framework.

partitioning

In this paper we generalize and improve the multiscale organization of graphs by introducing a new measure that quantifies the “closeness” between two nodes. The calculation of the measure is linear in the number of edges in the graph and involves just a small number of relaxation sweeps. A similar notion of distance is then calculated and used at each coarser level. We demonstrate the use of this measure in multiscale methods for several important combinatorial optimization problems and discuss the multiscale graph organization.

Abstract — Random walk simulation is employed in many experimental algorithmic applications. Efficient execution on modern computer architectures demands that the random walk be implemented to exploit data locality for improving the cache performance. In this research, we demonstrate how different one-dimensional data reordering functionals can be used as a preprocessing step for speeding the random walk runtime.

The Multiscale method is a class of algorithmic techniques for solving efficiently and effectively large-scale computational and optimization problems. This method was originally invented for solving elliptic partial differential equations and up to now it represents the most effective class of numerical algorithms for them. During the last two decades, there were many successful attempts to adapt the multiscale method for combinatorial optimization problems. Whereas the variety of continuous systems’ multiscale algorithms turned into a separate field of applied mathematics, for combinatorial optimization problems they still have not reached an advanced stage of development, consisting in practice of a very limited number of multiscale techniques. The main goal of this dissertation is to extend the knowledge of multiscale techniques for the combinatorial optimization problems. In the first part of this dissertation we formulate the principles of designing the multilevel algorithms for combinatorial optimization problems defined on a simple graph (or matrix) model. We present the results for a variety of linear ordering

Abstract. We summarize recent advances in partitioning, load balancing, and matrix ordering for scientific computing performed by members of the CSCAPES SciDAC institute. 1.

Graphs are often used to encapsulate relationship between objects. Graph drawing enables visualization of such relationships. The usefulness of this visual representation is dependent on whether the drawing is aesthetic. While there are no strict criteria for aesthetics of a drawing, it is generally agreed, for example,

Combinatorial scientific computing plays an important enabling role in computational science, particularly in high performance scientific computing. In this thesis, we will describe our work on optimizing matrix-vector multiplication using combinatorial techniques. Our research has focused on two different problems in combinatorial scientific computing, both involving matrix-vector multiplication, and both are solved using hypergraph models. For both of these problems, the cost of the combinatorial optimization process can be effectively amortized over many matrix-vector products. The first problem we address is optimization of serial matrix-vector multiplication for relatively small, dense matrices that arise in finite element assembly. Previous work showed that combinatorial optimization of matrix-vector multiplication can lead to faster assembly of finite element stiffness matrices by eliminating redundant operations. Based on a graph model characterizing row relationships, a more efficient set of operations can be generated to perform matrix-vector multiplication. We improved this graph model by extending the