Cycle Vector Space Algorithms for Enumerating all Cycles of a Planar Graph

We present a new and elegant cycle vector space algorithm that runs in O(n 2 \Delta ff) steps and needs O(n) space for enumerating all the cycles of a planar graph with n vertices, where ff is the total number of simple cycles in the graph. Unlike backtrack algorithms, cycle vector space algorithms for this problem are suitable for parallelization. A parallel version of this algorithm along with a parallel version of Syslo's O(n \Delta ff) step algorithm for the same problem are given on an exclusiveread, exclusive-write parallel RAM model with p processors. The results of an implementation of our parallel algorithm on a mesh-connected SIMD computer are also presented. Keywords: enumerating all cycles, cycle basis, planar graphs, EREW PRAM algorithm, meshconnected SIMD computer. 1 Introduction A graph G = (V; E) is a collection of objects, represented by a set of vertices V and relations among them, represented by a set of edges E. A graph G is said to be planar if there exists so..

Hamiltonian Cycle Problem for Triangle Graphs

We show that the Hamiltonian cycle problem is NP-complete for a class of planar graphs named triangle graphs that are closely related to inner-triangulated graphs. We present a lineartime heuristic algorithm that finds a solution at most one-third longer than the optimum solution and use it to obtain a fast rendering algorithm on triangular meshes in computer graphics. Keywords: Planar graph, inner-triangulated graph, triangle graph, NP-complete, heuristic algorithm, rendering triangular meshes. 1 Introduction. The Hamiltonian cycle problem (HC) is that of determining whether or not a given graph contains a cycle that passes through every vertex exactly once and has numerous applications in different areas [3, 9, 10, 11]. HC is NP-complete for various classes of graphs including perfect graphs, planar bipartite graphs, grid graphs and 3-connected planar graphs but a polynomial time algorithm was presented for 4-connected planar graphs by Chiba and Nishizeki [5]. Garey, Johnson, and T..

Parallel Algorithms for Finding Cycles and Cutsets of Graphs

We first present VLSI algorithms that run in O(n 2 ) steps for finding (and reporting) a fundamental set of cycles and a fundamental set of cutsets of an undirected graph on an nxn mesh of processors (SIMD). Both algorithms are decomposable and run in O(n 4 =k 2 ) steps for a graph with n vertices when the size of the mesh-connected computer is kxk and k ! n. An idea similar to the one used in finding a fundamental set of cycles of an undirected graph yields an O(n 2 ) algorithm for generating shortest paths between all pairs of vertices of a graph on the same model. Then, we present a new and elegant cycle vector space algorithm that runs in O(n 2 \Delta ff) steps and needs O(n) space for enumerating all the cycles of a planar graph with n vertices, where ff is the total number of simple cycles in the graph. Unlike backtrack algorithms, cycle vector space algorithms for this problem are suitable for parallelization. A parallel version of this algorithm along with a parall..

Efficient algorithms for finding submasses in weighted strings

We study the Submass Finding Problem: Given a string s over a weighted alphabet, i.e., an alphabet S with a weight function µ : S ¿ N, decide for an input mass M whether s has a substring whose weights sum up to M. If M is indeed a submass, then we want to find one or all occurrences of such substrings. We present efficient algorithms for both the decision and the search problem. Furthermore, our approach allows us to compute efficiently the number of different submasses of s. The main idea of our algorithms is to define appropriate polynomials such that we can determine the solution for the Submass Finding Problem from the coefficients of the product of these polynomials. We obtain very efficient running times by using Fast Fourier Transform to compute this product. Our main algorithm for the decision problem runs in time O(µs log µs), where µs is the total mass of string s. Employing standard methods for compressing sparse polynomials, this runtime can be viewed as O(s(s) log2 s(s)), where s(s) denotes the number of different submasses of s. In this case, the runtime is independent of the size of the individual masses of characters

On Open Problems in Biological Network Visualization

Much of the data generated and analyzed in the life sciences can be interpreted and represented by networks or graphs. Network analysis and visualization methods help in investigating them, and many universal as well as specialpurpose tools and libraries are available for this task. However, the two fields of graph drawing and network biology are still largely disconnected. Hence, visualization of biological networks does typically not apply state-of-the-art graph drawing techniques, and graph drawing tools do not respect the drawing conventions of the life science community. In this paper, we analyze some of the major problems arising in biological network visualization. We characterize these problems and formulate a series of open graph drawing problems. These use cases illustrate the need for efficient algorithms to present, explore, evaluate, and compare biological network data. For each use case, problems are discussed and possible solutions suggested