Set Of Parallel Edges Research Articles

Larger crossing angles make graphs easier to read

Objective: Aesthetics are important in algorithm design and graph evaluation. This paper presents two user studies that were conducted to investigate the impact of crossing angles on human graph comprehension.Method and results: These two studies together demonstrate our newly proposed two-step approach for testing graph aesthetics. The first study is a controlled experiment with purposely-generated graphs. Twenty-two subjects participated in the study and were asked to determine the length of a path which was crossed by a set of parallel edges at different angles. The result of an analysis of variance showed that larger crossing angles induced better task performance. The second study was a non-controlled experiment with general real world graphs. Thirty-seven subjects participated in the study and were asked to find the shortest path of two pre-selected nodes in a set of graph drawings. The results of simple regression tests confirmed the negative effect of small crossing angles. This study also showed that among our four proposed candidates, the minimum crossing angle on the path was the best measure for the aesthetic when path finding is important.Conclusion: Larger crossing angles make graphs easier to read.Implications: In situations where crossings cannot be completely removed (for example, graphs are non-planar, or a drawing convention is applied), or where effort needed to remove all crossings cannot be justified, the crossing angle should be maximized to reduce the negative impact of crossings to the minimum.

On the complexity of the planar directed edge-disjoint paths problem

The edge-disjoint paths problem and many special cases of it are known to be NP-complete. We present a new NP-completeness result for a special case of the problem, namely the directed edge-disjoint paths problem restricted to planar supply graphs and demand graphs consisting of two sets of parallel edges.

Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

Given a graph and terminal pairs $(s_i,t_i)$, $i\in[k]$, the edge-disjoint paths problem is to determine whether there exist $s_{i}t_{i}$ paths, $i\in[k]$, that do not share any edges. We consider this problem on acyclic digraphs. It is known to be NP-complete and solvable in time $n^{O(k)}$ where n is the number of nodes. It has been a long-standing open question whether it is fixed-parameter tractable in k, i.e., whether it admits an algorithm with running time of the form $f(k)\,n^{O(1)}$. We resolve this question in the negative: we show that the problem is $W[1]$-hard, hence unlikely to be fixed-parameter tractable. In fact it remains $W[1]$-hard even if the demand graph consists of two sets of parallel edges. On a positive side, we give an $O(m+k^{O(1)}\,k!\,n)$ algorithm for the special case when G is acyclic and $G+H$ is Eulerian, where H is the demand graph. We generalize this result (1) to the case when $G+H$ is “nearly” Eulerian, and (2) to an analogous special case of the unsplittable flow problem, a generalized version of disjoint paths that has capacities and demands.

Allocating Bandwidth for Bursty Connections

In this paper, we undertake the first study of statistical multiplexing from the perspective of approximation algorithms. The basic issue underlying statistical multiplexing is the following: in high-speed networks, individual connections (i.e., communication sessions) are very bursty, with transmission rates that vary greatly over time. As such, the problem of packing multiple connections together on a link becomes more subtle than in the case when each connection is assumed to have a fixed demand. We consider one of the most commonly studied models in this domain: that of two communicating nodes connected by a set of parallel edges, where the rate of each connection between them is a random variable. We consider three related problems: (1) stochastic load balancing, (2) stochastic bin-packing, and (3) stochastic knapsack. In the first problem the number of links is given and we want to minimize the expected value of the maximum load. In the other two problems the link capacity and an allowed overflow probability p are given, and the objective is to assign connections to links, so that the probability that the load of a link exceeds the link capacity is at most $p$. In bin-packing we need to assign each connection to a link using as few links as possible. In the knapsack problem each connection has a value, and we have only one link. The problem is to accept as many connections as possible. For the stochastic load balancing problem we give an O(1)-approximation algorithm for arbitrary random variables. For the other two problems we have algorithms restricted to on-off sources (the most common special case studied in the statistical multiplexing literature), with a somewhat weaker range of performance guarantees. A standard approach that has emerged for dealing with probabilistic resource requirements is the notion of effective bandwidth---this is a means of associating a fixed demand with a bursty connection that represents its distribution as closely as possible. Our approximation algorithms make use of the standard definition of effective bandwidth and also a new one that we introduce; the performance guarantees are based on new results showing that a combination of these measures can be used to provide bounds on the optimal solution.

NP-completeness of some edge-disjoint paths problems

We prove: The directed edge-disjoint paths problem is NP-complete, even if (a) the underlying graph G is acyclic, the demand graph H consists just of three sets of parallel edges and G + H is Eulerian, or (b) G + H is planar, or (c) G is planar and acyclic. (d) The undirected edge-disjoint paths problem is NP-complete, even if G + H is Eulerian and H consists just of three sets of parallel edges.

A general and flexible switchbox router: CARIOCA

A switchbox router utilizing layers is presented. Each net is first partitioned into a set of subnets. This decomposition is driven by the minimum Steiner tree. The construction is then carried out on a step-by-step basis, which allows the system to dynamically take new information on the problem into account. The information is examined by a set of expert systems which decide which subnet should be routed next. This part has been implemented with a blackboard architecture. The router can handle pins that are not on a grid on one set of parallel edges. The program has been implemented in the LISP programming language. Many test cases have been run successfully, and some results, including Burstein's difficult switchbox, are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Elasticity Solution of Deep Beam Under Settlement of Supports

The present study investigates the two-dimensional stresses and displacements in a finite rectangle whose one set of parallel edges is given a relative tangential displacement by means of rigidly attached planes. The other set of parallel edges is free from tractions. The problem is formulated in terms of a singular integral equation of the first kind, which yields the correct singular behavior of stresses at the corners. The integral equation is solved numerically by employing Gauss-Jacobi quadrature in conjunction with certain collocation technique. Numerical results of quantities of practical interests are shown graphically and also compared with the classical bending analysis.

On the Phenomenon of Separation During Compression and Frictional Sliding of an Elastic Rectangle

The plane strain compression and sliding of an elastic rectangle is considered whose one set of parallel edges are free from tractions. The remaining set of parallel edges is simultaneously subjected by rigid rough planes to a compressive force P and a sliding force F = μP, where μ is the coefficient of sliding friction. The rigid planes are prevented from rotation and a limiting equilibrium is assumed by taking the contact shear stress to be equal to the product of the normal compressive stress with μ. It is found that the presence of free edges significantly affects the deformation of the rectangle. In particular, it is shown that during sliding the leading edge of the rectangle is compressed most severely against the rigid plane, whereas separation takes place at the trailing edge. The separation occurs at the slightest trace of the application of load and although the magnitude depends upon the level of loading, the extent of the zone depends only on Poisson’s ratio, frictional coefficient, and the aspect ratio. Numerical results of the quantities of practical interest are reported.

Indentation of an elastic rectangle by a pair of rough punches with finite friction

The present paper considers an elastic rectangle indented on a set of parallel edges by flat punches with finite coefficient of sliding friction. The contact area is divided into an inner adhesive region in which the surface displacements are known, surrounded by regions in which the friction is limiting and the displacement parallel to the interface is not known. The remaining extreme portions of the edges as well as the other set of parallel edges are free from tractions. The problem is formulated in terms of a set of three singular integral equations which are nonhomogenious. Solutions of the integral equations which satisfy the finiteness of stresses at the point which separates the adhesive from the slip zone, determine the extent of adhesion. This is found to be independent of the magnitude of the load, but depends on the values of the frictional coefficient, Poisson's ratio and the geometrical parameters. Numerical results of the quantities of practical interest are reported.

Effect of Sliding Friction on Contact Stresses in a Rectangle Compressed by Rigid Planes

The plane-strain compression of an elastic rectangle by rigid, rough planes with finite coefficient of friction between the surfaces is considered. The contact area is divided into an inner adhesive region in which the surface displacements are known, surrounded by regions in which the friction is limiting and the displacement parallel to the interface is not known. The remaining set of parallel edges of the rectangle is free from tractions. The problem is formulated in terms of Papkovich-Fadle eigenfunctions which lead to the solution of a set of two integral equations of the second kind. Solution of the integral equations which satisfies the finiteness of stresses at the point which separates the adhesive from the slip zone, determines the extent of adhesion. This is found to be independent of the magnitude of load, but depends on the values of frictional coefficients, Poisson’s ratio and the aspect ratio. Numerical results of the quantities of practical interest are reported.

An Enumeration of the Five Parallelohedra

A parallelohedron is a convex polyhedron, in real affine three-dimensional space, which can be repeated by translation to fill the whole space without interstices. It has centrally symmetrical faces [4, p. 120] and hence is centrally symmetrical.Let Fi denote the number of faces each having exactly i edges, Vi denote the number of vertices each incident with exactly i edges, E denote the number of edges, n denote the number of sets of parallel edges, F denote the total number of faces, V denote the total number of vertices.