A parallel matching algorithm for convex bipartite graphs and applications to scheduling

Motivated by an application in controllability we consider maximum matchings in random bipartite graphs G = (A, B). First we analyse Karp–Sipser's algorithm to determine the asymptotic size of maximum matchings in random bipartite graphs with a fixed degree distribution. We then allow an adversary to delete one edge adjacent to every vertex in A in the more restricted model where each vertex in A chooses d neighbours uniformly at random from B.

An efficient parallel algorithm for scheduling interval ordered tasks

This paper addresses the multi-attributed graph matching problem considering multiple attributes jointly while preserving the characteristics of each attribute. Since most of conventional graph matching algorithms integrate multiple attributes to construct a single attribute in an oversimplified way, the information from multiple attributes are not often fully exploited. In order to solve this problem, we propose a novel multi-layer graph structure that can preserve the particularities of each attribute in separated layers. Then, we also propose a multi-attributed graph matching algorithm based on the random walk centrality for the proposed multi-layer graph structure. We compare the proposed algorithm with other state-of-the-art graph matching algorithms based on the single-layer structure using synthetic and real datasets, and prove the superior performance of the proposed multi-layer graph structure and matching algorithm.

Efficient parallel algorithms are presented, on the CREW PRAM model, for generating a succinct encoding of all pairs shortest path information in a directed planar graphG with real-valued edge costs but no negative cycles. We assume that a planar embedding ofG is given, together with a set ofq faces that cover all the vertices. Then our algorithm runs inO(log2 n) time and employsO(nq+M(q)) processors (whereM(t) is the number of processors required to multiply twot×t matrices inO(logt) time). Let us note here that wheneverq<n then our processor bound is better than the best previous one (M(n)).O(log2 n) time,n-processor algorithms are presented for various subproblems, including that of generating all pairs shortest path information in a directedouterplanar graph. Our work is based on the fundamental hammock-decomposition technique of G. Frederickson. We achieve this decomposition inO(logn log*n) parallel time by usingO(n) processors. The hammock-decomposition seems to be a fundamental operation that may help in improving efficiency of many parallel (and sequential) graph algorithms.

PARALLEL AND LARGE SCALE MATRIX COMPUTATIONS IN CONTROL: SOME IDEAS

We consider the problem of maintaining a maximum matching in a convex bipartite graph G = (V,E) under a set of update operations which includes insertions and deletions of vertices and edges. It is not hard to show that it is impossible to maintain an explicit representation of a maximum matching in sub-linear time per operation, even in the amortized sense. Despite this difficulty, we develop a data structure which maintains the set of vertices that participate in a maximum matching in O(log2 |V|) amortized time per update and reports the status of a vertex (matched or unmatched) in constant worst-case time. Our structure can report the mate of a matched vertex in the maximum matching in worst-case O(min{k log2 |V |+log |V|, |V| log |V|}) time, where k is the number of update operations since the last query for the same pair of vertices was made. In addition, we give an O(√|V| log2 |V|)-time amortized bound for this pair query.

The maximum matching algorithm for bipartite graphs can be used to provide optimal scheduling for crossbar based interconnection networks. Unfortunately, maximum matching requires O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) time for an N times N communication system, which has limited its application to real-time network scheduling. In this paper, we show how maximum matching can be reformulated in terms of Boolean operations rather than the more traditional formulations. By taking advantage of the inherent parallelism available in custom hardware design, we introduce three maximum matching implementations in hardware and show how we can trade design complexity for performance. Specifically, we examine a pure logic scheduler with three dimensions of parallelism, a matrix scheduler with two dimensions of parallelism and a vector scheduler with one dimension of parallelism. These designs reduce the algorithmic time complexity down to O(1), O(K), and O(KN), respectively, where K is the number of optimization steps. While an optimal scheduling algorithm requires K=2N-1 steps, our simulation results show that the scheduler can achieve 99% of the optimal schedule when K=9. We examine hardware and time complexity of these architectures for crossbar sizes of up to N=1024. Using FPGA synthesis results, we show that a greedy schedule for various sized crossbars, ranging from 8 times 8 to 256 times 256, can be optimized in less than 20 ns per optimization step. For crossbars reaching 1024 times 1024 the scheduling can be completed in approximately 10 s with current technology and could reach under 90 ns with future technologies

The maximum matching algorithm for bipartite graphs can be used to provide optimal scheduling for crossbar based interconnection networks. Unfortunately, maximum matching requires O(N3) time for an N x N communication system, which has limited its application to real-time network scheduling. In this paper, we show how maximum matching can be reformulated in terms of Boolean operations rather than the more traditional formulations. By taking advantage of the inherent parallelism available in custom hardware design, we introduce three Maximum Matching implementations in hardware and show how we can trade design complexity for performance. Specifically, we examine a Pure Logic Scheduler with three dimensions of parallelism, a Matrix Scheduler with two dimensions of parallelism and a Vector Scheduler with one dimension of parallelism. These designs reduce the algorithmic time complexity down to O(1), O(K), and O(KN), respectively, where K is the number of optimization steps. While an optimal scheduling algorithm requires K = 2N -- 1 steps, our simulation results show that the scheduler can achieve 99% of the optimal schedule when K = 9. We examine hardware and time complexity of these architectures for crossbar sizes of up to N = 1024. Using FPGA synthesis results, we show that a greedy schedule for various sized crossbars, ranging from 8 x 8 to 256 x 256, can be optimized in less than 20 ns per optimization step. For crossbars reaching 1024 x 1024 the scheduling can be completed in approximately 10 μs with current technology and could reach under 90 ns with future technologies.

The Combinatorial Nullstellensatz can be used to solve certain problems in combinatorics. However, one of the major complications in using the Combinatorial Nullstellensatz is ensuring that there exists a nonzero monomial. This dissertation looks at applying the Combinatorial Nullstellensatz to finding perfect matchings in bipartite graphs. The first two chapters provide background material covering topics such as linear algebra, group theory, graph theory and even the discrete Fourier transform. New results start in the third chapter, showing that the Combinatorial Nullstellensatz can be used to solve the problem of finding perfect matchings in bipartite graphs. Using the Combinatorial Nullstellensatz also allows for a vice use of matroid intersection to find the nonzero monomial. By also applying the uncertainty principle, the number of perfect matchings in a bipartite graph can be bound. The fourth chapter examines properties of the polynomials created in the use of the Combinatorial Nullstellensatz to find perfect matchings in bipartite graphs. Many of the properties of the polynomials have analogous properties for the transforms of the polynomials, which are also examined. These properties often relate back to the structure of the graph which gave rise to the polynomial. The fifth chapter provides an application of the results. Since finding a nonzero monomial can be difficult and the polynomials created in this dissertation give polynomials with such a nonzero monomial the application shows how certain polynomials can be rewritten in terms of the matching polynomials. Such a rewriting may permit an easy method to find a nonzero monomial so that the Combinatorial Nullstellensatz can be applied to the polynomial. Finally, the fifth chapter concludes with some open problems that may be areas of further research.

Software tools for Social Network Analysis (SNA) are being developed which support various types of analysis of social networks extracted from social media websites (e.g., Twitter). Once extracted and stored in a database such social networks are amenable to analysis by SNA software. This data analysis often involves searching for occurrences of various subgraph patterns (i.e., graphical representations of entities and relationships). The authors have developed the Graph Matching Toolkit (GMT) which provides an intuitive Graphical User Interface (GUI) for a heuristic graph matching algorithm called the Truncated Search Tree (TruST) algorithm. GMT is a visual interface for graph matching algorithms processing large social networks. GMT enables an analyst to draw a subgraph pattern by using a mouse to select categories and labels for nodes and links from drop-down menus. GMT then executes the TruST algorithm to find the top five occurrences of the subgraph pattern within the social network stored in the database. GMT was tested using a simulated counter-insurgency dataset consisting of cellular phone communications within a populated area of operations in Iraq. The results indicated GMT (when executing the TruST graph matching algorithm) is a time-efficient approach to searching large social networks. GMT’s visual interface to a graph matching algorithm enables intelligence analysts to quickly analyze and summarize the large amounts of data necessary to produce actionable intelligence.

Efficient parallel algorithms and software for compressed octrees with applications to hierarchical methods

High-performance computing for climate models has always been an interesting research area. It is valuable to nest a regional climate model within a global climate model, but large-scale simulation of the nesting or coupling severely challenges to the development of efficient parallel algorithms that fit well into multi-core clusters. This paper first presents research on the coupling of the Institute of Atmospheric Physics of Chinese Academy of Sciences Atmospheric General Circulation Model version 4.0 and the Weather Research and Forecasting model, then proposes an efficient parallel algorithm of the coupling. The algorithm includes initialization of input data, decomposition of computing grid and processes, parallel computing of component models, and data exchange by a coupler. By calling some subroutines of the Model Coupling Toolkit, the parallelization of the proposed algorithm is implemented. Experiments show that the parallel algorithm is very effective and scalable. The parallel efficiency of the algorithm on 1,024 CPU cores can reach up to 70%. Moreover, its parallel efficiency with respect to weak scalability is 72.56% on a multi-core cluster.

Over the years many interesting and efficient parallel algorithms have been developed to solve a wide variety of problems, but not much attention has been paid to studying the inherent parallelism in sequential algorithms---i.e., understanding the depth of their dependence structure, and how shallow dependence structures might beused to develop efficient parallel implementations. In this talk I will describe recent work on analyzing the dependence depth of iterative sequential algorithms---ones that loop over a collection of elements. Many of these algorithms have deep dependence chains in the worst case, but shallow chains (polylog w.h.p.) if the elements are randomly ordered. Examples include many fundamental algorithms: the Knuth shuffle for random permutations, sorting by insertion into a binary search tree, greedy maximal independent set (MIS), greedy maximal matching, greedy graph-coloring, counting cycles in a permutation, incremental k-dimensional linear programming, and incremental 2d Delaunay triangulation. An advantage of the approach is that it can lead to very simple and efficient parallel algorithms. Our MIS algorithm, for example can be coded in a dozen or so lines, and is significantly faster than Luby's algorithm on modern multicore machines. Also the approach encourages snapping the view that sequential and parallel algorithms are distinct, and instead thinking of algorithms, in general, as collections of instructions with dependences among them.

This paper examines the use of binary trees in the design of efficient parallel algorithms. Using binary trees, we develop efficient algorithms for several scheduling problems. The shared memory model for parallel computation is used. Our success in using binary trees for parallel computations, indicates that the binary tree is an important and useful design tool for parallel algorithms.

Operator Splitting and Domain Decomposition for Multiclusters