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Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to ..."
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Cited by 196 (15 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
DUNE: A MultiLayer Gridless Routing System with Wire Planning
, 2000
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Cited by 17 (5 self)
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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 13 (7 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
Geometric Interconnection and Placement Algorithms
, 1995
"... This dissertation examines a number of geometric interconnection, partitioning, and placement problems arising in the field of VLSI physical design automation. In particular, many of the results concern the geometric Steiner tree problem: given a set of terminals in the plane, find a minimumlength ..."
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Cited by 12 (3 self)
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This dissertation examines a number of geometric interconnection, partitioning, and placement problems arising in the field of VLSI physical design automation. In particular, many of the results concern the geometric Steiner tree problem: given a set of terminals in the plane, find a minimumlength interconnection of those terminals according to some geometric distance metric. Two new algorithms are introduced that compute optimal rectilinear Steiner trees. Both are provably faster than any previous algorithm for instances small enough to solve in practice, and both are also fast in practice. The first algorithm is a dynamic programming algorithm based on decomposing a rectilinear Steiner tree into full trees. A full tree is a Steiner tree in which every terminal is a leaf. Its time complexity is O(n3^n), where n is the number of terminals. The second algorithm modifies the first by the use of fullset screening, which is a process by which some candidate full trees are eliminated f...
Computational Geometry
 in optimization 2.5D and 3D NC surface machining. Computers in Industry
, 1996
"... Introduction Computational geometry evolves from the classical discipline of design and analysis of algorithms, and has received a great deal of attention in the last two decades since its inception in 1975 by M. Shamos[108]. It is concerned with the computational complexity of geometric problems t ..."
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Cited by 11 (0 self)
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Introduction Computational geometry evolves from the classical discipline of design and analysis of algorithms, and has received a great deal of attention in the last two decades since its inception in 1975 by M. Shamos[108]. It is concerned with the computational complexity of geometric problems that arise in various disciplines such as pattern recognition, computer graphics, computer vision, robotics, VLSI layout, operations research, statistics, etc. In contrast with the classical approach to proving mathematical theorems about geometryrelated problems, this discipline emphasizes the computational aspect of these problems and attempts to exploit the underlying geometric properties possible, e.g., the metric space, to derive efficient algorithmic solutions. The classical theorem, for instance, that a set S is convex if and only if for any 0 ff 1 the convex combination ffp + (1 \Gamma<F
Heterogeneous selfreconfiguring robotics
, 2002
"... Selfreconfiguring robots are modular systems that can change shape, or reconfigure, to match structure to task. They comprise many small, discrete, often identical modules that connect together and that are minimally actuated. Global shape transformation is achieved by composing local motions. Syst ..."
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Cited by 11 (5 self)
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Selfreconfiguring robots are modular systems that can change shape, or reconfigure, to match structure to task. They comprise many small, discrete, often identical modules that connect together and that are minimally actuated. Global shape transformation is achieved by composing local motions. Systems with a single module type, known as homogeneous systems, gain fault tolerance, robustness and low production cost from module interchangeability. However, we are interested in heterogeneous systems, which include multiple types of modules such as those with sensors, batteries or wheels. We believe that heterogeneous systems offer the same benefits as homogeneous systems with the added ability to match not only structure to task, but also capability to task. Although significant results have been achieved in understanding homogeneous systems, research in heterogeneous systems is challenging as key algorithmic issues remain unexplored. We propose in this thesis to investigate questions in four main areas: 1) how to classify heterogeneous systems, 2) how to develop efficient heterogeneous reconfiguration algorithms with desired characteristics, 3) how to characterize the complexity of key algorithmic problems, and 4) how to apply these heterogeneous algorithms to perform useful new tasks in simulation and in the physical world. Our goal is to develop
A FaultTolerant Adaptive and Minimal Routing Approach in nD Meshes
"... In this paper a sufficient condition is given for minimal routing in ndimensional (nD) meshes with faulty nodes contained in a set of disjoint fault regions. It is based on an early work of the author on minimal routing in low dimension meshes (such as 2D meshes with faulty blocks). Unlike many t ..."
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Cited by 9 (3 self)
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In this paper a sufficient condition is given for minimal routing in ndimensional (nD) meshes with faulty nodes contained in a set of disjoint fault regions. It is based on an early work of the author on minimal routing in low dimension meshes (such as 2D meshes with faulty blocks). Unlike many traditional models that assume all the nodes know global fault distribution, our approach is based on the concept of limited global fault information. First, a fault model called fault region is used in which all faulty nodes in the system are contained in a set of disjoint regions. Fault information is coded in a 2ntuple called extended safety level associated with each node of an nD mesh to support minimal routing. Specifically, we study the existence of minimal paths at a given source node, limited distribution of fault information, minimal routing, and deadlockfree routing. Our results show that any minimal routing that is partially adaptive can still be applied as long as the destinat...
On Geometric Path Query Problems
, 1997
"... In this paper, we study several geometric path query problems. Given a scene of disjoint polygonal obstacles with totally n vertices in the plane, we construct efficient data structures that enable fast reporting of an "optimal" obstacleavoiding path (or its length, cost, directions, etc) ..."
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Cited by 9 (0 self)
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In this paper, we study several geometric path query problems. Given a scene of disjoint polygonal obstacles with totally n vertices in the plane, we construct efficient data structures that enable fast reporting of an "optimal" obstacleavoiding path (or its length, cost, directions, etc) between two arbitrary query points s and t that are given in an online fashion. We consider geometric paths under several optimality criteria: Lm length, number of edges (called links), monotonicity with respect to a certain direction, and some combinations of length and links. Our methods are centered around the notion of gateways, a small number of easily identified points in the plane that control the paths we seek. We give efficient solutions for several special cases based upon new geometric observations. We also present solutions for the general cases based upon the computation of the minimum size visibility polygon for query points.
An Efficient Algorithm for Shortest Paths in Vertical and Horizontal Segments
 In Proc. 5th Worksh. Algorithms and Data Structures
, 1997
"... . Suppose one has a line segment arrangement consisting entirely ..."
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Cited by 8 (2 self)
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. Suppose one has a line segment arrangement consisting entirely
Orthogonal Connector Routing
"... Abstract. Orthogonal connectors are used in a variety of common network diagrams. Most interactive diagram editors provide orthogonal connectors with some form of automatic connector routing. However, these tools use adhoc heuristics that can lead to strange routes and even routes that pass through ..."
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Cited by 8 (7 self)
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Abstract. Orthogonal connectors are used in a variety of common network diagrams. Most interactive diagram editors provide orthogonal connectors with some form of automatic connector routing. However, these tools use adhoc heuristics that can lead to strange routes and even routes that pass through other objects. We present an algorithm for computing optimal objectavoiding orthogonal connector routings where the route minimizes a monotonic function of the connector length and number of bends. The algorithm is efficient and can calculate connector routings fast enough to reroute connectors during interaction. 1