High Performance Issues on Parallel Architectures

In an effort to reduce communication latency in mesh-type architectures, these architectures have been augmented by various types of global and reconfigurable bus structures. The static bus structures provide excellent performance in many areas of computation especially structured numerical computations, but they lack the flexibility required of many large numerical and non-numerical applications. Reconfigurable bus systems have the dynamic adaptability to handle a much wider range of applications. While reconfigurable meshes can often yield constant time results for many problems, the cost of this performance is paid in the number of processors required. While in actuality the majority of these processors are employed as switching elements for the bus system and often do little actual computation. In an effort to reduce the processor cost while maintaining performance and communication flexibility, we present a new hybrid parallel array architecture with the goal of optimizing the best features of arrays with global buses and arrays with reconfigurable bus systems. The result is an architecture of n processing elements and a bus interconnection network which requires very basic circuitry to construct and control. This architecture allows prefix computations, such as prefix sum, prefix maximum(minimum) to be accomplished in O(log n) time. These functions then form the building blocks for complex procedures, which more fully exploit the communication flexibility of the architecture. Application of the architecture to graph theory produces optimal algorithms for graph properties such as spanning forest bipartiteness, fundamental cycles, bridges and biconnected components. Other optimal algorithms for the more complex least common ancestor and the connected component problems are also presented. By design, all algorithms maintain optimality for very large sparse graphs. We further examine the architecture\u27s ability to handle basic image processing tasks as well as its potential to simulate other parallel architectures and theoretic models

Optimal Greedy Algorithms for Indifference Graphs

A fundamental problem in social sciences and management is understanding and predicting decisions made by individuals, various groups, or the society as a whole. In this context, one important concept is the notion of indifference. We characterize the class of indifference graphs, that is, graphs which arise in the process of quantifying indifference relations. In particular, we show that these graphs are characterized by the existence of a special ordering of their vertices. As it turns out, this ordering leads naturally to optimal greedy algorithms for a number of computational problems, including coloring, finding a shortest path between two vertices, computing a maximum matching, the center, and a Hamiltonian path

Subclasses of Normal Helly Circular-Arc Graphs

A Helly circular-arc model M = (C,A) is a circle C together with a Helly family \A of arcs of C. If no arc is contained in any other, then M is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly circular-arc model, and if there are no two arcs covering the circle, then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc graph is the intersection graph of the arcs of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model. In this article we study these subclasses of Helly circular-arc graphs. We show natural generalizations of several properties of (proper) interval graphs that hold for some of these Helly circular-arc subclasses. Next, we describe characterizations for the subclasses of Helly circular-arc graphs, including forbidden induced subgraphs characterizations. These characterizations lead to efficient algorithms for recognizing graphs within these classes. Finally, we show how do these classes of graphs relate with straight and round digraphs.Comment: 39 pages, 13 figures. A previous version of the paper (entitled Proper Helly Circular-Arc Graphs) appeared at WG'0

Indifference Graphs and the Single Row Routing Problem

This thesis investigates the subclass of interval graphs known as indifference graphs. New optimal algorithms for recognition, center, diameter, maximum matching, Hamiltonian path and domination in indifference graphs are presented. The recognition algorithm produces a linear order with properties which allow the solution of the other problems in linear time. Indifference graphs are further applied to the single row routing problem which results in both sequential,. and parallel routing algorithms

A subexponential parameterized algorithm for proper interval completion

In the Proper Interval Completion problem we are given a graph G and an integer k, and the task is to turn G using at most k edge additions into a proper interval graph, i.e., a graph admitting an intersection model of equal-length intervals on a line. The study of Proper Interval Completion from the viewpoint of parameterized complexity has been initiated by Kaplan, Shamir and Tarjan [FOCS 1994; SIAM J. Comput. 1999], who showed an algorithm for the problem working in O(16k⋅(n+m)) time. In this paper we present an algorithm with running time kO(k2/3)+O(nm(kn+m)), which is the first subexponential parameterized algorithm for Proper Interval Completion

Sitting closer to friends than enemies, revisited

Signed graphs, i.e., undirected graphs with edges labelled with a plus or minus sign, are commonly used to model relationships in social networks. Recently, Kermarrec and Thraves (2011) initiated the study of the problem of appropriately visualising the network: They asked whether any signed graph can be embedded into the metric space TeX in such a manner that every vertex is closer to all its friends (neighbours via positive edges) than to all its enemies (neighbours via negative edges). Interestingly, embeddability into TeX can be expressed as a purely combinatorial problem. In this paper we pursue a deeper study of this case, answering several questions posed by Kermarrec and Thraves. First, we refine the approach of Kermarrec and Thraves for the case of complete signed graphs by showing that the problem is closely related to the recognition of proper interval graphs. Second, we prove that the general case, whose polynomial-time tractability remained open, is in fact NP-complete. Finally, we provide lower and upper bounds for the time complexity of the general case: we prove that the existence of a subexponential time (in the number of vertices and edges of the input signed graph) algorithm would violate the Exponential Time Hypothesis, whereas a simple dynamic programming approach gives a running time single-exponential in the number of vertices