Showing papers on "Longest path problem published in 1984"

Euclidean shortest paths in the presence of rectilinear barriers

Abstract: In this paper we address the problem of constructing a Euclidean shortest path between two specified points (source, destination) in the plane, which avoids a given set of barriers. This problem had been solved earlier for polygonal obstacles with the aid of the visibility graph. This approach however, has an Ω(n2) time lower bound, if n is the total number of vertices of the obstacles. Our goal is to find interesting cases for which the solution can be obtained without the explicit construction of the entire visibility graph. The two cases are (i) the path must lie within an n-vertex simple polygon; (ii) the obstacles are n disjoint and parallel line segments. In both instances greedy O(n log n) time algorithms can be developed which solve the problems by constructing the shortest-path tree from the source to all the vertices of the obstacles and to the destination.

Computational study of an improved shortest path algorithm

Abstract: Shortest and/or longest path analysis is a major analytical component of quantitative models used by transportation planners. The importance of shortest path analysis in all phases of transportation planning has given rise to intensive research and software development over the last three decades for solving shortest path problems. This paper presents a new hybrid solution algorithm called THRESH, which integrates the features of label setting and label correcting algorithms, yet appears to have performance characteristics that transcend both. Preliminary computational results indicate that it is substantially more efficient than the methods determined to be best by previous studies.

A polynomial algorithm for the max-cut problem on graphs without long odd cycles

Abstract: Given a graphG=[V, E] with positive edge weights, the max-cut problem is to find a cut inG such that the sum of the weights of the edges of this cut is as large as possible. Letg(K) be the class of graphs whose longest odd cycle is not longer than2K+1, whereK is a nonnegative integer independent of the numbern of nodes ofG. We present an O(n 4K) algorithm for the max-cut problem for graphs ing(K). The algorithm is recursive and is based on some properties of longest and longest odd cycles of graphs.

Long paths and large cycles in finite graphs

Abstract: Ore-type sufficient conditions ensuring the existence of a large cycle passing through any given path of length s for (s + 2)-connected graphs are given, and the extremal cases are characterized.

Intersection graph algorithms

Abstract: An intersection graph for a set of sets C is a graph G together with a bijection from the vertices of G to C such that distinct vertices in G are adjacent if and only if their images under this bijection intersect. Of particular interest have been the classes of chordal graphs, the intersection graphs of sets of subtrees of a tree, and interval graphs, the intersection graphs of sets of intervals of the real line. I examine another class of intersection graphs, the class of directed path graphs: intersection graphs of sets of paths in a directed tree. This class properly contains the class of interval graphs, and is properly contained by the class of chordal graphs. I give a linear time algorithm for recognizing directed path graphs and for constructing intersection representations, and a polynomial time algorithm for deciding directed path graph isomorphism. Both algorithms use a data structure called a partial path tree, which is a kind of skeletal representation of the clique hypergraph of a directed path graph. I present linear time algorithms for finding partial path trees with specific roots and for finding partial path trees with arbitrary roots. I prove that partial path trees with identical roots are identical. Using this fact I develop a polynomial time algorithm for directed path graph isomorphism.

Characteristic polynomials of large graphs. On alternate form of characteristic polynomial [1]

Abstract: A recursion exists between the absolute magnitudes of the coefficients of the characteristic polynomials of certain families of cyclic and acyclic graphs which makes their computation quite easy for very large graphs using a pencil-and-a-paper approach. Structural requirements are given for such families of graphs which are of interest to the problem of recognition defined in [1].

Ic layout generation and compaction using mathematical optimization

Abstract: This Ph.D. research is in the area of automatic IC mask generation and compaction. It develops a new method of mask compaction. It formulates a mixed integer linear programming problem from a user defined stick diagram, a dimensionless topological representation of IC layout. By solving this mixed integer program, a compacted and design rule violation free layout is obtained. A specialized algorithm was developed for solving this mixed integer program for the purpose of efficiency. This algorithm is based on a branch and bound method and a longest path algorithm on acyclic graphs. In the formulation, a vertical graph and a horizontal graph are generated from the input stick diagram. These two graphs are strongly related to each other by binary decision variables. For every fixed set of decision variables, we solve two independent longest path problems, one for the horizontal direction and the other for the vertical one. The interaction between the horizontal and vertical compaction is via the binary decision variables. The branch and bound method is used for setting values of decision variables. The formulation can incorporate the following: (1) Simultaneous consideration of two directions of compaction; (2) Goal directed compaction and generation of the layout. The research is aimed at the final end of a spectrum of research for a silicon compiler. The method developed is suitable to be used in the final stage (mask generation stage) of a silicon compiler. For example, having a library of cells in stick diagram form together with a set of procedures to translate them into layout is much more flexible and therefore more useful than a standard cell library. The cells could be scaled automatically according to fan in and fan out considerations. Their shapes could be altered automatically to fit other parts. Moreover changes in design rules will not render the cells obsolete. The algorithm can also be incorporated into an IC design work station as a compaction procedure. An experimental program was developed which implemented the compaction algorithm. The program demonstrated that the algorithm is flexible, powerful, and efficient.

On a problem of a. kotzig concerning factorizations of 4-regular graphs

Abstract: The paper gives a solution of a problem of A. Kotzǐg. This problem concerns the 4-regular graphs G with the property that in every decomposition of G into two edge-disjoint 2-regular factors at least one factor is a Hamiltonian circuit in G.

Problems in sorting and graph algorithms

Abstract: Five disjoint problems are discussed. The first problem concerns the determination of optimal algorithms with respect to a new model for evaluating sorting algorithms. We did an exhaustive search for such algorithms. The second problem concerns a conjecture that every sorting algorithm on some input involves every key in O(log n) comparisons. We give partial results. The third problem concerns finding efficient algorithms for finding cycles of small fixed length in graphs. We give algorithms for general graphs and O(n log n) algorithms for cycles of length 5 or 6 in planar graphs. The fourth problem concerns the NP-completeness of a wire-routing problem. Specifically, the problem asks for vertex-disjoint paths connecting pairs of points in certain planar graphs. The fifth problem concerns automata traversing graphs in a myopic fashion. We study several cases and show when this can be done and when it is impossible.

An algorithm for generation of s-t acyclic graphs

Abstract: The problem of finding all s-t acyclic graphs between two given vertices of a digraph is interesting in relation to the evaluation of the reliability of a communication network. There is already an algorithm which generates all s-t acyclic digraphs without duplication by opening an arc or a set of arcs one by one starting with a given graph. But the efficiency of that algorithm decreases when the number of cycles in the graph becomes large. This paper proposes an algorithm which generates s-t acyclic graphs without duplication by combining s-t paths obtained by a depth-first search. Compared with the existing algorithms (i) the execution time of this algorithm is not affected much by the number of arc disjoint cycles of the graph; (ii) when the generation is stopped at an arbitrary level i, all the s-t acyclic graphs composed of paths of length i or less will have been generated. Thus they can be used to approximate the reliability.

Path Problems in Graphs and Matrix Multiplication

Abstract: In this chapter we concentrate on path problems in graphs. Typical examples are the problems of computing shortest or longest paths or computing the k shortest path between all pairs of points in a graph. The best known algorithms for these problems differ only slightly. In fact, they are all special cases of an algorithm for solving general path problems on graphs. General path problems over closed semi-rings and Kleene’s algorithm for solving them are dealt with in section 1, special cases are then treated in section 2. The algebraic point of view allows us to formulate the connection between general path problems and matrix multiplication in an elegant way: Matrix multiplication in a semi-ring and solution of a general path problem have the same order of complexity. In section 4 we consider fast algorithms for multiplication of matrices over a ring. This is then applied to boolean matrices. Section 7 contains a lower bound on the complexity of boolean matrix product.