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Growing Fat Graphs
 In 18th Annual Symposium on Computational Geometry
, 2002
"... INTRODUCTION We present an algorithm for growing fat graphs. Traditionally, graph drawing algorithms represent vertices as circles and edges as closed curves connecting the vertices. The thickness of an edge is often used as a visualization cue, to indicate importance, or to convey some additional ..."
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Cited by 8 (4 self)
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INTRODUCTION We present an algorithm for growing fat graphs. Traditionally, graph drawing algorithms represent vertices as circles and edges as closed curves connecting the vertices. The thickness of an edge is often used as a visualization cue, to indicate importance, or to convey some additional information. We show how to grow fat graphs with edges of variable thickness. For the purpose of the demonstration we focus on a restricted class of graphs that occur in VLSI wire routing. This class corresponds to planar, maxdegree1 graphs. The underlying algorithm also extends to general planar graphs as shown in [2]. In VLSI wire routing it is often desirable to maximize the distance between di#erent wires. Maximizing the distance between wires is equivalent to finding the drawing in which the edges are drawn as thick as possible, i.e., allowing the graph to grow as fat as possible. The continuous homotopic routing problem [1, 3, 5] is a classic VLSI problem. The input is an initial sk
Thick NonCrossing Paths and MinimumCost Flows in Polygonal Domains
"... We study the problem of finding shortest noncrossing thick paths in a polygonal domain, where a thick path is the Minkowski sum of a usual (zerothickness, or thin) path and a disk. Given K pairs of terminals on the boundary of a simple ngon, we compute in O(n + K) time a representation of the set ..."
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Cited by 5 (3 self)
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We study the problem of finding shortest noncrossing thick paths in a polygonal domain, where a thick path is the Minkowski sum of a usual (zerothickness, or thin) path and a disk. Given K pairs of terminals on the boundary of a simple ngon, we compute in O(n + K) time a representation of the set of K shortest noncrossing thick paths joining the terminal pairs; using the representation, any particular path can be output in time proportional to its complexity. We compute K shortest thick noncrossing paths in a polygon with h holes in O ` (K + 1) h h! poly(n, K) ´ time, using an efficient method to compute any one of the K thick paths if the “threadings ” of all paths amidst the holes are specified. We show that if h is not constant, the problem is NPhard; we also show the hardness of approximation. We give a pseudopolynomialtime algorithm for some rectilinear versions of the problem. We apply our thick paths algorithms to obtain the first algorithmic results for the minimumcost continuous flow problem — an extension of the standard discrete minimumcost network flow problem to continuous domains. The results are based on showing a continuous analog of the Network
NodeDisjoint Paths on the Mesh and a New TradeOff in VLSI Layout
, 1996
"... A number of basic models for VLSI layout are based on the construction of nodedisjoint paths between terminals on a multilayer grid. In this setting, one is interested in minimizing both the number of layers required and the area of the underlying grid. Building on work of Cutler and Shiloach, and ..."
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Cited by 5 (1 self)
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A number of basic models for VLSI layout are based on the construction of nodedisjoint paths between terminals on a multilayer grid. In this setting, one is interested in minimizing both the number of layers required and the area of the underlying grid. Building on work of Cutler and Shiloach, and Aggarwal, Klawe, et al., we prove an upperbound tradeoff between these two quantities in a general multilayer grid model. As a special case of our main result, we obtain significantly improved bounds for the problem of routing a full permutation on the mesh using nodedisjoint paths; our new bound here is within polylogarithmic factors of the bisection bound. Our algorithms involve some new techniques for analyzing the structure of nodedisjoint paths in planar graphs, and indicate some respects in which this problem, at least in the planar case, is fundamentally different from its edgedisjoint counterpart. 1 Introduction The basic nodedisjoint paths problem is as follows. We are give...
Feasible Offset and Optimal Offset for General SingleLayer Channel Routing
, 1995
"... . This paper provides an efficient method to find all feasible offsets for a given separation in a VLSI channel routing problem in one layer. The prior literature considers this task only for problems with no singlesided nets. When singlesided nets are included, the worstcase solution time increa ..."
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Cited by 4 (2 self)
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. This paper provides an efficient method to find all feasible offsets for a given separation in a VLSI channel routing problem in one layer. The prior literature considers this task only for problems with no singlesided nets. When singlesided nets are included, the worstcase solution time increases from \Theta(n) to \Omega\Gamma n 2 ), where n is the number of nets. But if the number of columns c is O(n), the problem can be solved in time O(n 1:5 lg n), which improves upon a "naive" O(cn) approach. As a corollary of this result, the same time bound suffices to find the optimal offset (the one that minimizes separation). Better running times result when there are no twosided nets or all singlesided nets are on one side of the channel. This paper also gives improvements upon the naive approach for c<F NaN> 6= O(n), including an algorithm with running time independent of c. An interesting algorithmic aspect of the paper is a connection to discrete convolution. Key words. VLSI...
Geometric Transformations for a Rubberband Sketch
, 1992
"... vi Acknowledgements vii 1. Introduction 1 1.1 Overview of the Surf System : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1.1 Topological Routing : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1.2 Spoke Creation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.1.3 ..."
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Cited by 4 (1 self)
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vi Acknowledgements vii 1. Introduction 1 1.1 Overview of the Surf System : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1.1 Topological Routing : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1.2 Spoke Creation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.1.3 Geometric Wiring : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.2 Organization : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2. Overview of Approach 7 2.1 Basic Method : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 2.2 Why does it work? : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3. Segment Transformation Order 14 3.1 Producing the graph G : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19 3.2 Time complexity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 3.3 Implementation Details : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 4. Producing an Initial Transf...
RubberBand Based Topological Router
, 1997
"... A multilayer, topological detailedrouter is described. This is the first router ever reported that uses a rubberband sketch (RBS) to represent the interconnect. The detailedrouter is part of SURF, a routing system for multichip modules and VLSI that was designed to handle efficiently large m ..."
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Cited by 1 (0 self)
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A multilayer, topological detailedrouter is described. This is the first router ever reported that uses a rubberband sketch (RBS) to represent the interconnect. The detailedrouter is part of SURF, a routing system for multichip modules and VLSI that was designed to handle efficiently large multilayer problems. The detailedrouter supports various routing goals and can generate layouts for rectilinear, octilinear and anyangle wiring rules. It uses a novel approach of unconstrained layerassignment that makes a better usage of the routing resources by considering a continuous metric of conflict between nets as opposed to the binary go/nogo approach. The layerassignment is formulated as an optimization problem and various routing goals such as wire and via minimization or constrainedlayers can be achieved by simple modifications to the cost function. The layerassignment partitions the multilayer problem into a set of singlelayer subproblems that are routed independently by a topological planar router.
Costdriven Layout for Thinfilm MCMs
"... The performance advantages of MCM packaging are well known and have received much attention. It is becoming increasingly clear however that if MCMs are to succeed, they will have to be attractive in terms of cost as well as performance for consumer oriented designs. The SURF rubberband router is de ..."
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The performance advantages of MCM packaging are well known and have received much attention. It is becoming increasingly clear however that if MCMs are to succeed, they will have to be attractive in terms of cost as well as performance for consumer oriented designs. The SURF rubberband router is designed to meet many of the needs of thinfilm MCM layout that are not met by existing PCB tools. This paper describes a few of the ways that SURF can exploit the flexibility of the rubberband wiring model to generate layouts that help reduce substrate cost. Specifically, approaches for reducing the number of metal layers and increasing substrate yield are presented. The results of applying these techniques to several MCM designs are shown. Finally, some future directions for research are described.
APPROXIMATE SHORTEST HOMOTOPIC PATHS IN WEIGHTED REGIONS
"... Let P be a path between two points s and t in a polygonal subdivision T with obstacles and weighted regions. Given a relative error tolerance ε ∈ (0,1), we present the first algorithm to compute a path between s and t that can be deformed to P without passing over any obstacle and the path cost is w ..."
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Let P be a path between two points s and t in a polygonal subdivision T with obstacles and weighted regions. Given a relative error tolerance ε ∈ (0,1), we present the first algorithm to compute a path between s and t that can be deformed to P without passing over any obstacle and the path cost is within a factor 1+ε of the optimum. The running time is O ( h3 ε2 knpolylog(k,n, 1)), where k is the number of segments in P and ε h and n are the numbers of obstacles and vertices in T, respectively. The constant in the running time of our algorithm depends on some geometric parameters and the ratio of the maximum region weight to the minimum region weight.
Combinatorial Optimization of Cycles and Bases
 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
"... We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimumweight cycles in a given homotopy or homology class, and computing minimumweight cycle bases for the fundamental group or various homology groups. ..."
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We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimumweight cycles in a given homotopy or homology class, and computing minimumweight cycle bases for the fundamental group or various homology groups.
Abstract
, 2008
"... This paper addresses the problem of finding shortest paths homotopic to a given disjoint set of paths that wind amongst point obstacles in the plane. We present a faster algorithm than previously known. 0 ..."
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This paper addresses the problem of finding shortest paths homotopic to a given disjoint set of paths that wind amongst point obstacles in the plane. We present a faster algorithm than previously known. 0