Showing papers on "Longest path problem published in 1983"

Optimal paths in graphs with stochastic or multidimensional weights

Abstract: This paper explores computationally tractable formulations of stochastic and multidimensional optimal path problems, each as an extension of the shortest path problem. A single formulation encompassing both problems is considered, in which a utility function defines preference among candidate paths. The result is the ability to state explicit conditions for exact solutions using standard methods, and the applicability of well-understood approximation techniques.

A Design for Directed Graphs with Minimum Diameter

Abstract: This paper proposes a simple procedure for the design of small-diameter graphs. It can be used to construct a directed graph whose diameter is less than or equal to that of any previously proposed graph.

An Algorithm to Compact a VLSI Symbolic Layout with Mixed Constraints

Abstract: A popular algorithm to compact VLSI symbolic layout is to use a graph algorithm similar to finding the "longest path" in a network The algorithm assumes that the spacing constraints on the mask elements are of the lower bound type However, to enable the user to have close control over the compaction result, a desired symbolic layout system should allow the user to add either the equality or the upper bound constraints on selected pairs of mask elements as well This paper proposes an algorithm which uses a graph-theoretic approach to solve efficiently the compaction problem with mixed constraints

Bounds for width two branching programs

Abstract: Branching programs for the computation of Boolean functions were first studied in the Master's thesis of Masek.7 In a rather straightforward manner they generalize the concept of a decision tree to a decision graph. Let P be a branching program with edges labelled by the Boolean variables, x1,...,xn and their complements. Given an input a=(a1,...,an) e {0,1}n, program P computes a function value fp(a) in the following way. The nodes of P play the role of states or configurations. In particular, sinks play the role of final states or stopping configurations. The length of program P is the length of the longest path in P. Following Cobham,2capacity of the program is defined to be the logarithm to the base 2 of the number of nodes in P. Length and capacity are lower bounds on time and space requirements for any reasonable model of sequential computation. Clearly, any n-variable Boolean function can be computed by a branching program of length n if the capacity is not constrained. Since space lower bounds in excess of log n remain a fundamental challenge, we consider restricted branching programs in the hope of gaining insight into this problem and the closely related problem of time-space trade-offs.

Improved Compaction by Minimized Length of Wires

Abstract: The compaction of IC or hybrid layouts by means of the "longest path" method yields a slack in the placement of part of the elements, which, in its turn, can be used to reduce the overall wire-length. The result is an improved electrical performance and a smaller layout. The optimization problem was transformed to a graph-theoretical problem in a way similar to the compaction problem itself. Our procedure starts by adding pieces of information out of the connectivity of the layout to the constraint graph. The succeeding heuristic algorithms generate a new tree of longest paths, taking the linear inequalities and the result of the longest path calculation into consideration. A few examples demonstrate the significant reduction of wire-length and sometimes even an additional reduction of layout area achieved with low computational effort.

Shortest path problems in planar graphs

Abstract: Graph decomposition and data structures techniques are presented that make possible faster algorithms for shortest paths in planar graphs. Improved algorithms are presented for the single source problem, the all pairs problem, and the problem of finding a minimum cut in an undirected graph.

Summarizing graphs by regular expressions

Abstract: This paper shows how to rapidly determine the path relationships between k different elements of a graph (of the type primarily resulting from programs) in time proportional to k log k. Given the path relations between elements u,v, and w, it is easy to answer questions like "is there a path from u to w?" and "is there a path from u to w which does not go through v?" (The elements can be either nodes or edges.)This algorithm can be used in a wide variety of contexts. For example, in order to prove that whenever control reaches a point p, the last assignment of a value to a variable, v, has always been of the form v := c, where c is a certain constant, it is only necessary to know the path relations between the point p and all assignments to that variable.Ordinarily one is interested in the possible points, where a variable was assigned its current value (definition points), and at the points at which that value is used. Previously, all flow analysis algorithms would compute the definition points of all variables at all nodes in the graph, despite the fact that any given node may use only one of those variables.This algorithm may also be a generally useful graph algorithm. It can compute transitive closure more rapidly than the standard algorithm using the current best matrix multiplication algorithm on the kinds of graphs resulting from programs. The algorithm proceeds by preprocessing the graph, creating tables that can be used later to rapidly determine flow relationships between any sections of the program. In addition, small changes to the graph need only result in small changes to the tables. The algorithm is therefore suitable for incremental analysis.

Remark on algorithm 562: shortest path lengths

Abstract: There is an error in the subprogram S H P T H L (). If the deque contains only the current node under discussion (node i), the updating process may be incorrect. This is due to the fact that an erroneous value of the pointer (nt) is used and the tail part of the deque will be lost. The problem is fixed by inserting a line after sequence number SPL00073 in subprogram S H P T H L (): NJ(I) = K SPL00073 IF (NT.EQ.I) NT = K {new line} SPL00073.1 5 CONTINUE SPL00074 This correction has no significant impact on the efficiency of this algorithm.

Hamiltonian path graphs

Abstract: The Hamiltonian path graph H(G) of a graph G is that graph having the same vertex set as G and in which two vertices u and v are adjacent if and only if G contains a Hamiltonian u-v path. A characterization of Hamiltonian graphs isomorphic to their Hamiltonian path graphs is presented.

A single source shortest path algorithm for graphs with separators

Abstract: We show how to solve a single source shortest path problem on a planar network in time O(n3/2log n). The algorithm works for arbitrary edge weights (positive and negative) and is based on the planar separator theorem. More generally, the algorithm works in time O(na+blog n + n3a+ nd) on graphs G=(V, E) which have a separator of size na, have at most nb edges and where the separator can be found in time nd.

A Single Shortest Path Algorithm for Graphs with Separators

Abstract: We show how to solve a single source shortest path problem on a planar network in time O(n3/2log n). The algorithm works for arbitrary edge weights (positive and negative) and is based on the planar separator theorem. More generally, the algorithm works in time O(na+blog n + n3a+ nd) on graphs G=(V, E) which have a separator of size na, have at most nb edges and where the separator can be found in time nd.

Proof of harary's c0njecwre on the reconstruci'ion of trees

Abstract: We prove that a tree with at least three cutvertices is reconstructible from its cutvertexdeleted subgraphs. This answers in the affirmative a question raised by Professor Harary at the Seventh British Combinatorial Conference in 1979. All graphs T = (V(T), E(T)) considered will be finite and simple. For u E V(T), a neighbour of u will mean a vertex adjacent to 0 in T. The dency of u, that is the number of neighbours of 2, is denoted by e(u). When it is clear that we are referring to the valency of 2) in T we simply write this as p(u). A k-vertex is a vertex of valency k. A l-vertex is called an erzduetiex. The subgraph of T resulting from the deletion of the vertex v and all the edges incident to it will be denoted by Tu or T- v. If u and w are adjacent vertices in T, then the edge incident to both v and w will be denoted by VW. The subgraph of T obtained by the deletion of the edge uw is denoted by T-VW. A subgraph C of T is said to be a path from v. to u, if V(C) is (vo, vl, . . . , v,} and E(C) is {ViVi+l: i ~0, 1, . . . , t - 1). We say that v, and u, are joined by C. The length of C is t. A graph is connected if every pair of vertices are joined by path. A maximal connected subgraph of T is called a component of T. A cutvertex of T is a vertex whose removal increases the number of components of T. A cutvertex will be called heavy if it is adjacent to at least three other cutvertices. The distance d(u, u) between two vertices u aud v in T is the length of a shortest path joining them, if any. A forest is a graph in which every pair of vertices is joined by at most one path. A tree is a connected forest. If T is a tree, then the diameter d(T) of T is the length of a longest path in T. The centre of T is the set of all central vertices of T (see [4n. We shall use the well known result that a tree is either central or ¢ral. If T is bicentral, the edge adjacent to both central vertices is called the central edge. A dial vertex of a tree is one which is at a maximum distance from the centre. A graph T is said to be reconstructible if it can be determined uniquely (up to