David W. Matula

The maximum concurrent flow problem

The maximum concurrent flow problem (MCFP) is a multicommodity flow problem in which every pair of entities can send and receive flow concurrently. The ratio of the flow supplied between a pair of entities to the predefined demand for that pair is called throughput and must be the same for all pairs of entities for a concurrent flow. The MCFP objective is to maximize the throughput, subject to the capacity constraints. We develop a fully polynomial-time approximation scheme for the MCFP for the case of arbitrary demands and uniform capacity. Computational results are presented. It is shown that the problem of associating costs (distances) to the edges so as to maximize the minimum-cost of routing the concurrent flow is the dual of the MCFP. A path-cut type duality theorem to expose the combinatorial structure of the MCFP is also derived. Our duality theorems are proved constructively for arbitrary demands and uniform capacity using the algorithm. Applications include packet-switched networks [1, 4, 8], and cluster analysis [16].

Smallest-last ordering and clustering and graph coloring algorithms

Smallest-last vertex ordering and prlonty search are utdlzed to show for any graph G = (IT, E) that the set of all connected subgraphs maxunal with respect to their minimum degree can be determined in O(I EI + I VI) time and 21El + O(I VI) space It is further noted that the smallest-last graph coloring algonthrn can be unplemented in O(I E I + I V[) tune, and particularly effective aspects of the resulting coloring are discussed.

GRAPH COLORING ALGORITHMS

Publisher Summary This chapter focuses on sequential vertex colorings, where vertices are sequentially added to the portion of the graph already colored, and the new colorings are determined to include each newly adjoined vertex. Considerable literature in the field of graph theory has dealt with the coloring of graphs. The majority of this effort has been devoted to the theory of graph coloring, and relatively little study has been directed toward the design of efficient graph coloring procedures. Because numerous proofs of properties relevant to graph coloring are constructive, many coloring procedures are at least implicit in the theoretical development. The chapter describes the concept of sequential colorings is formalized and certain upper bounds on the minimum number of colors needed to color a graph, the chromatic number x(G). The chief results show that the recursive-smallest-vertex-degree-last-ordering-with-interchange coloring algorithm will color any planar graph in five or fewer colors. The algorithm is evidently quite efficient even on large planar graphs.

Faithful bipartite ROM reciprocal tables

We describe bipartite reciprocal tables that employ separate table lookup of the positive and negative portions of a borrow-save reciprocal value. The fusion of the parts includes a rounding so the output reciprocals are guaranteed correct to a unit in the last place, and typically provide a round-to-nearest reciprocal for over 90% of input arguments. The output rounding can be accomplished in conjunction with multiplier recoding yielding practically no cost in logic complexity or time in employing bipartite tables. We demonstrate these tables to be 2 to 4 times smaller than conventional 4-bit reciprocal tables. For 10-16 bit reciprocal table lookup the compression grows from a factor of 4 to over 16, making possible the use of larger seed reciprocals than previously considered cost effective.<<ETX>>

k-Components, Clusters and Slicings in Graphs

Utilizing the edge-connectivity of graphs, certain maximally connected subgraphs of a graph termed k-components and clusters are characterized and their interrelations are investigated. The cohesiveness function is described and shown to be a useful measure of the local intensity of connectivity within a graph. Sequences of cuts totally separating a graph, termed slicings, are shown to be intimately related to the k-components and clusters of a graph. An efficient algorithm is presented for determining k-components and clusters. Applications of these notions to graph coloring and numerical taxonomy are discussed.

Graph Theoretic Techniques for Cluster Analysis Algorithms

Publisher Summary The output of a cluster analysis method is a collection of subsets of the object set termed clusters characterized in some manner by relative internal coherence and/or external isolation, along with a natural stratification of these identified clusters by levels of cohesive intensity. In formalizing a model of the cluster analysis methods, it is essential to consider the nature and inherent reliability of the proximity data that constitutes the input in substantive clustering applications. The proximity value scales are dichotomous. It is the practice of most authors of cluster methods to assume that the proximity values are available in the form of a real symmetric matrix, where any unjustified structure implicit in the real values is either to be ignored or axiomatically disallowed. The most desirable cluster analysis models for substantive applications should have the input proximity data expressible in a manner faithfully representing only the reliable information content of the empirically measured data.

Subtree Isomorphism in O(n5/2)

The problem of determining if the tree S (unrooted) on n , vertices is isomorphic to any subtree of the tree T on n , t ≥ n s vertices is shown to be solvable in O( n t 3/2 n s ) steps. The method involves the solution of an ( n t ,-1) by 2( n t ,-1) array of maximum bipartite matching problems where some of these subproblems are solved in groups. Recognition of isomorphic subproblems yields a compacted data structure reducing practical storage requirements with no increase in the order of time complexity.

Sparsest cuts and bottlenecks in graphs

Abstract The problem of determining a sparsest cut in a graph is characterized and its computation shown to be NP-hard. A class of sparsest cuts, termed bottlenecks, is characterized by a dual relation to a particular polynomial time computable multicommodity flow problem. Efficient computational techniques for determining bottlenecks in a broad class of instances are presented.

Basic digit sets for radix representation

Abstract : The use of a negative base did not appear until the 1950s when several authors independently introduced the concept. Complement representation also became much discussed in this period as an alternative to sign magnitude for designing the arithmetic unit of a computer. The arithmetic of numbers represented in positional notation has a firm foundation derived from the theory of polynomial arithmetic that readily allows these extensions to negative bases and/or negative digit values, complement representation, and digit values in excess of the base. Our primary concern in this paper is the characterization and computation of those integral valued base and digit set pairs that provide complete and unique finite radix representation of the integers.

Improving Goldschmidt division, square root, and square root reciprocal

The aim of this paper is to accelerate division, square root, and square root reciprocal computations when the Goldschmidt method is used on a pipelined multiplier. This is done by replacing the last iteration by the addition of a correcting term that can be looked up during the early iterations. We describe several variants of the Goldschmidt algorithm, assuming 4-cycle pipelined multiplier, and discuss obtained number of cycles and error achieved. Extensions to other than 4-cycle multipliers are given. If we call G/sub m/ the Goldschmidt algorithm with m iterations, our variants allow us to reach an accuracy that is between that of G/sub 3/ and that of G/sub 4/, with a number of cycle equal to that of G/sub 3/.