Abstract not found

The minimum linear arrangement problem is widely used and studied in many practical and theoretical applications. In this paper we present a linear-time algorithm for the problem inspired by the algebraic multigrid approach which is based on weighted edge contraction rather than simple contraction. Our results turned out to be better than every known result in almost all cases, while the short running time of the algorithm enabled experiments with very large graphs.

The bigraph crossing problem, embedding the two vertex sets of a bipartite graph G = (V0;V1;E) along two parallel lines so that edge crossings are minimized, has application to circuit layout and graph drawing. We consider the case where both V0 and V1 can be permuted arbitrarily -- both this and the case where the order of one vertex set is fixed are NP-hard. Two new heuristics that perform well on sparse graphs such as occur in circuit layout problems are presented. The new heuristics outperform existing heuristics on graph classes that range from application-specific to random. Our experimental design methodology ensures that differences in performance are statistically significant and not the result of minor variations in graph structure or input order.

A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2-Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar. This problem is NP-complete, as is the 1-Layer Planarization problem in which the permutation of the vertices in one layer is fixed. We give the following fixed parameter tractability results: an O(k ·6 k +|G|) algorithm for 2-Layer Planarization and an O(3 k ·|G|) algorithm for 1-Layer Planarization, thus achieving linear time for fixed k.

Linear ordering problems are combinatorial optimization problems which deal with the minimization of different functionals in which the graph vertices are mapped onto (1, 2,..., n). These problems are widely used and studied in many practical and theoretical applications. In this paper we present a variety of linear-time algorithms for these problems inspired by the Algebraic Multigrid approach which is based on weighted edge contraction. The experimental result for four such problems turned out to be better than every known result in almost all cases, while the short running time of the algorithms enables testing very large graphs.

Abstract { The bigraph crossing problem, embedding the two node sets of a bipartite graph G =(V0;V1;E) along two parallel lines so that edge crossings are minimized, has application to placement optimization for standard cells and other technologies. Iterative improvement heuristics involve repeated application of some transformation on an existing feasible solution to obtain better feasible solutions. Typically an increase in the number of iterations, and therefore execution time, implies an improvement in solution quality. We investigate tradeo s between execution time and solution quality in order to establish the best heuristic for any given time budget. Our experiments show some clear trends for a scalable class of graphs based on actual circuits. These trends, based on statistically signi cant samples of each of several graph graph sizes, suggest promising directions for development of better heuristics.

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. This report introduces an experimental design that discovers a new relationship between a cell placement for minimized wire crossing in bipartite (two-layer) graphs and a cell placement in linear arrangement, optimized for minimum total wire length as measured after rectilinear routing in a single channel. We introduce hypercrossing, a new crossing model for graphs, and demonstrate that the total wire length in the channel, as measured after routing, has a positive linear correlation with the total hypercrossing, and a coefficient of correlation of near 100%. While heuristics to directly minimize total hypercrossing are yet to be invented, our experiments show that the minimization of total (regular) wire crossing leads to placements with significantly reduced wire length and wiring area. A methodology and new datasets are now available to determine, with statistical significance, which of the two cost function classes will yield a better placement in general: costs based on the mode...

This paper is motivated by the need to generate classes of circuits that are closely related. Ideally, such circuits not only have identical size and distributions of cells and nets but also resemble each other closely in terms of cell clusters and interconnect patterns. Such classes are essential for welldefined performance evaluation of layout algorithms. For example, if the heuristic in the algorithm is being optimized for multiplier structures, then all circuits in the class should have a multiplierlike structure. Such tight control of circuit class generation has not been demonstrated with recent techniques, including the generation of clone and mutant classes. We introduce an approach to generating circuit classes called siblings. Siblings retain all significant characteristics of their parent circuit. We demonstrate that siblings inherit the essential characteristics of the parent only if we control, for each connected component, the diameter of the clusters and the arrangement ...

In this paper, we study a crossing minimization problem in a radial drawing of a graph. Radial drawings have strong application in social network visualization, for example, displaying centrality indices of actors [20]. The main problem of this paper is called the one-sided radial crossing minimization between two concentric circles, named orbits, where the positions of vertices in the outer orbit are fixed. The main task of the problem is to find the vertex ordering of the free orbit and edge routing between two orbits in order to minimize the number of edge crossings. The problem is known to be NP-hard [1], and the first polynomial time 15-approximation algorithm was presented in [9]. In this paper, we present a new 3α-approximation algorithm for the case when the free orbit has no leaf vertex, where α represents the approximation ratio of the one-sided crossing minimization problem in a horizontal drawing. Using the best known result of α =1.4664 [13], our new algorithm can achieve 4.3992-approximation. Submitted: