Decomposition of binary images—A survey and comparison

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.

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.

For a bipartite graph G = (V, E), (1) perfect, (2) maximum and (3) maximal matchings are matchings (1) such that all vertices are incident to some matching edges, (2) whose cardinalities are maximum among all matchings, (3) which are contained in no other matching. In this paper, we present three algorithms for enumerating these three types of matchings. Their time complexities are O(|V |) per a matching.

Vertex-independent sets and vertex coverings as also edge-independent sets and edge coverings of graphs occur very naturally in many practical situations and hence have several potential applications. In this chapter, we study the properties of these sets. In addition, we discuss matchings in graphs and, in particular, in bipartite graphs. Matchings in bipartite graphs have varied applications in operations research. We also present two celebrated theorems of graph theory, namely, Tutte’s 1-factor theorem and Hall’s matching theorem. All graphs considered in this chapter are loopless.

Finding all the perfect matchings in bipartite graphs

It is difficult to obtain high performance when computing matchings on parallel processors because matching algorithms explicitly or implicitly search for paths in the graph, and when these paths become long, there is little concurrency. In spite of this limitation, we present a new algorithm and its shared-memory parallelization that achieves good performance and scalability in computing maximum cardinality matchings in bipartite graphs. This algorithm searches for augmenting paths via specialized breadth-first searches (BFS) from multiple source vertices, hence creating more parallelism than single source algorithms. Algorithms that employ multiple-source searches cannot discard a search tree once no augmenting path is discovered from the tree, unlike algorithms that rely on single-source searches. We describe a novel tree-grafting method that eliminates most of the redundant edge traversals resulting from this property of multiple-source searches. We also employ the recent direction-optimizing BFS algorithm as a subroutine to discover augmenting paths faster. Our algorithm compares favorably with the current best algorithms in terms of the number of edges traversed, the average augmenting path length, and the number of iterations. Here, we provide a proof of correctness for our algorithm. Our NUMA-aware implementation is scalable to 80 threads of an Intel multiprocessor and to 240 threads on an Intel Knights Corner coprocessor. On average, our parallel algorithm runs an order of magnitude faster than the fastest algorithms available. The performance improvement is more significant on graphs with small matching number.

Maximum weight bipartite matching in matrix multiplication time

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 article presents the results of an experiment comparing scheduling procedures (for example, sanatorium «Denyshi» (Zhytomyr region).) Methods: genetic algorithm, ant algorithm and branch and bound method, patients resort to different methods (ant, branch and bound, depletion of finding a solution options) and algorithms (for example genetic algorithms) that have been modified by the author for finding maximum matching in bipartite graphs and consider limiting compatibility. The experiment was conducted to compare different methods of computing platforms using software copyright ICS_DENISH. Reviewed by time spent on calculations and algorithms efficiency. The basis of comparative numerical experiment is necessary to select the algorithm with the least computational complexity for application use in health resorts of Ukraine.

Graph search algorithms and maximum bipartite matching algorithm on the hypercube network model

In this paper, using a digraph model we present both exact and suboptimum algorithms for finding the maximum level of diagnosis capability offered by a given network of processors. The exact algorithm is based on the Chwa-Hakimi characterization for t-diagnosability and the suboptimum method uses an algorithm for maximum matching in bipartite graphs. Finally, using the notion of syndrome information set, an O(t3) algorithm is presented for t-diagnosis in a large D(N,t,X) system of interconnected processors. Syndrome processing is assumed to be done in a separate fault-free processor.

The problem of finding an optimal semi-matching is a generalization of the problem of finding classical matching in bipartite graphs. A semi-matching in a bipartite graph G=(U, V, E) with n vertices and m edges is a set of edges M⊆E, such that each vertex in U is incident to at most one edge in M. An optimal semi-matching is a semi-matching with degM(u)=1 for all u∈U and the minimal value of $\sum_{v \in V} \frac{deg_M(v).(deg_M(v)+1)}2$ . We propose a schema that allows a reduction of the studied problem to a variant of the maximum bounded-degree semi-matching problem. The proposed schema yields to two algorithms for computing an optimal semi-matching. The first one runs in time $O(\sqrt{n} \cdot m \cdot \log{n})$ that is the same as the time complexity of the currently best known algorithm. However, our algorithm uses a different approach that enables some improvements in practice (e.g. parallelization, faster algorithms for special graph classes). The second one is randomized and it computes an optimal semi-matching with high probability in O(nω ·log1+o(1)n), where ω is the exponent of the best known matrix multiplication algorithm. Since ω≤2.38, this algorithms breaks through O(n2.5) barrier for dense graphs.

For best performance in image compression, wavelet transforms require filters that combine some important properties, such as orthogonality and symmetry. However, in the designing of wavelets this combination is limited because to have these simultaneous properties is not possible, except in the Haar case. The multiwavelet system allows more design options than wavelet, and therefore, several simultaneous characteristics such as: compact support, smoothness, higher order approach, symmetry and orthogonality, among others. Therefore, establishing parameters which allow comparing of scalar wavelets and multiwavelets performance, is a necesity. This paper provides an overview of filter banks, wavelets, multiwavelets, basic principles, and connections between each one, in addition to some experimental results from the compression of test images, comparing each system performance.

Advances in wavelet transforms and quantization methods have produced algorithms capable of surpassing the existing image compression standards like the Joint Photographic Experts Group (JPEG) algorithm. For best performance in image compression, wavelet transforms require filters that combine a number of desirable properties, such as orthogonality and symmetry. However, the design possibilities for wavelets are limited because they cannot simultaneously possess all of these desirable properties. The relatively new field of Multiwavelets shows promise in removing some of the limitations of wavelets. Multiwavelets offer more design options and hence can combine all desirable transform features. It also shows that the latest Multiwavelet filters can give performance equal to, or in many cases superior to, the current wavelet filters. The performance of Multiwavelet packets is also explored and is shown to be competitive to that of wavelet packets in some cases.

A matrix-algebraic formulation of distributed-memory maximal cardinality matching algorithms in bipartite graphs