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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 194 (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
Rectilinear Paths among Rectilinear Obstacles
 Discrete Applied Mathematics
, 1996
"... Given a set of obstacles and two distinguished points in the plane the problem of finding a collision free path subject to a certain optimization function is a fundamental problem that arises in many fields, such as motion planning in robotics, wire routing in VLSI and logistics in operations resear ..."
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Cited by 31 (3 self)
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Given a set of obstacles and two distinguished points in the plane the problem of finding a collision free path subject to a certain optimization function is a fundamental problem that arises in many fields, such as motion planning in robotics, wire routing in VLSI and logistics in operations research. In this survey we emphasize its applications to VLSI design and limit ourselves to the rectilinear domain in which the goal path to be computed and the underlying obstacles are all rectilinearly oriented, i.e., the segments are either horizontal or vertical. We consider different routing environments, and various optimization criteria pertaining to VLSI design, and provide a survey of results that have been developed in the past, present current results and give open problems for future research. 1 Introduction Given a set of obstacles and two distinguished points in the plane, the problem of finding a collision free path subject to a certain optimization function is a fundamental probl...
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.
A Basis for SelfRepair Robots Using SelfReconfiguring Crystal Modules
 In Proc. of Intelligent Autonomous Systems 6
, 2000
"... Selfrepair robots are modular robots that have the capability of detecting and recovering from failures. Typically, such robots are unitmodular and carry a number of redundant modules on their bodies. Selfrepair consists of detecting the failure of a module, ejecting the bad module and replacing ..."
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Cited by 6 (3 self)
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Selfrepair robots are modular robots that have the capability of detecting and recovering from failures. Typically, such robots are unitmodular and carry a number of redundant modules on their bodies. Selfrepair consists of detecting the failure of a module, ejecting the bad module and replacing it with one of the extra modules. In this paper we show how selfrepair can be accomplished by selfreconfiguring Crystalline robots. We describe the Crystalline robots, which consist of modules that can aggregate together to form distributed robot systems and are actuated by expanding and contracting each unit. This actuation mechanism permits automated shape metamorphosis. We also describe an algorithm that uses this actuation mechanism for selfrepair.
Approximation algorithms for the minimum bends traveling salesman problem
 In Proceedings of the 8th Conference on Integer Programming and Combinatorial Optimization
, 2000
"... The problem of traversing a set of points in the order that minimizes the total distance traveled (traveling salesman problem) is one of the most famous and wellstudied problems in combinatorial optimization. It has many applications, and has been a testbed for many of the must useful ideas in algo ..."
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Cited by 5 (1 self)
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The problem of traversing a set of points in the order that minimizes the total distance traveled (traveling salesman problem) is one of the most famous and wellstudied problems in combinatorial optimization. It has many applications, and has been a testbed for many of the must useful ideas in algorithm design and analysis. The usual metric, minimizing the total distance traveled, is an important one, but many other metrics are of interest. In this paper, we introduce the metric of minimizing the number of turns in the tour, given that the input points are in the Euclidean plane. To our knowledge this metric has not been studied previously. It is motivated by applications in robotics and in the movement of other heavy machinery: for many such devices turning is an expensive operation. We give approximation algorithms for several variants of the traveling salesman problem for which the metric is to minimize the number of turns. We call this the minimum bend traveling salesman problem. For the case of an arbitrary set of n points in the Euclidean plane, we give an O(lg z)approximation algorithm, where z is the maximum number of collinear points. In the worst case z can be as big as n, but z will often be much smaller. For the case when the lines are restricted to being either horizontal or vertical, we give a 2approximation algorithm. If we have the further restriction that no two points are allowed to have the same x or ycoordinate, we give an algorithm that finds a tour which makes at most two turns more than the optimal tour. Thus we have an approximation algorithm with an additive, rather than a multiplicative error bound. Beyond the additive error bound, our algorithm for this problem introduces several interesting algorithmic techniques for decomposing sets of points in the Euclidean plane that we believe to be of independent interest.
On routing in VLSI design and communication networks
 Discrete Applied Mathematics
"... In this paper, we study the global routing problem in VLSI design and the multicast routing problem in communication networks. We first propose new and realistic models for both problems. In the global routing problem in VLSI design, we are given a lattice graph and subsets of the vertex set. The g ..."
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Cited by 5 (1 self)
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In this paper, we study the global routing problem in VLSI design and the multicast routing problem in communication networks. We first propose new and realistic models for both problems. In the global routing problem in VLSI design, we are given a lattice graph and subsets of the vertex set. The goal is to generate trees spanning these vertices in the subsets to minimize a linear combination of overall wirelength (edge length) and the number of bends of trees with respect to edge capacity constraints. In the multicast routing problem in communication networks, a graph is given to represent the network, together with subsets of the vertex set. We are required to find trees to span the given subsets and the overall edge length is minimized with respect to capacity constraints. Both problems are APXhard. We present the integer linear programming formulation of both problems and solve the linear programming (LP) relaxations by the fast approx
Finding Rectilinear Paths Among Obstacles In A TwoLayer Interconnection Model
, 1996
"... Finding the best rectilinear path with respect to the bends and the lengths of paths connecting two given points in the presence of rectilinear obstacles in a twolayer model is studied in this paper. In this model, rectilinear obstacles on each layer are specified separately, and the orientation of ..."
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Cited by 2 (0 self)
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Finding the best rectilinear path with respect to the bends and the lengths of paths connecting two given points in the presence of rectilinear obstacles in a twolayer model is studied in this paper. In this model, rectilinear obstacles on each layer are specified separately, and the orientation of routing in each layer is fixed. An algorithm is presented to transform any problem instance in the twolayer model to one in a onelayer model, so that almost all algorithms for finding rectilinear paths among obstacles in the plane can be applied. The transformation algorithm runs in O(e log e) time where e is the number of edges on obstacles lying on both layers. A direct graphtheoretic approach to finding a shortest path in the twolayer model, which is easier to implement is also presented. The algorithm runs in O(e log 2 e) time. Keywords: Rectilinear shortest path, grid graph, minimum link path, VLSI routing. 1. Introduction When we do routing, i.e., connecting two points in the...
Path Planning Algorithms under the LinkDistance Metric
, 2006
"... The Traveling Salesman Problem and the Shortest Path Problem are famous problems in computer science which have been well studied when the objective is measured using the Euclidean distance. Here we examine these geometric problems under a different set of optimization criteria. Rather than consider ..."
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Cited by 1 (0 self)
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The Traveling Salesman Problem and the Shortest Path Problem are famous problems in computer science which have been well studied when the objective is measured using the Euclidean distance. Here we examine these geometric problems under a different set of optimization criteria. Rather than considering the total distance traversed by a path, this thesis looks at reducing the number of times a turn is made along that path, or equivalently, at reducing the number of straight lines in the path. Minimizing this objective value, known as the linkdistance, is useful in situations where continuing in a given direction is cheap, while turning is a relatively expensive operation. Applications exist in VLSI, robotics, wireless communications, space travel, and other fields where it is desirable to reduce the number of turns. This thesis examines rectilinear and nonrectilinear variants of the Traveling Salesman Problem under this metric. The objective of these problems is to find a path visiting a set of points which has the smallest number of bends. A 2approximation algorithm is given for the rectilinear problem, while for the nonrectilinear problem, an O(log n)approximation algorithm is given. The latter problem is also shown to be NPComplete.
An Efficient Direct Approach for Computing Shortest Rectilinear Paths among Obstacles in a TwoLayer Interconnection Model
, 1998
"... In this paper, we present a direct approach for routing a shortest rectilinear path between two points among a set of rectilinear obstacles in a twolayer interconnection model. The previously best known direct approach for this problem takes O(n log n) time and O(n log n) space. By using integer ..."
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In this paper, we present a direct approach for routing a shortest rectilinear path between two points among a set of rectilinear obstacles in a twolayer interconnection model. The previously best known direct approach for this problem takes O(n log n) time and O(n log n) space. By using integer data structures and an implicit graph representation scheme, we reduce the time bound to O(n log n) while maintaining the space bound to O(n log n). Comparing with the indirect approach for the problem, our algorithm is simpler to implement and faster when the input size is moderate.
We consider the
, 2002
"... This is a presentation of a ”workinprogress”, on which which we are still working out many details, but which we feel may still be of interest to the community. ..."
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This is a presentation of a ”workinprogress”, on which which we are still working out many details, but which we feel may still be of interest to the community.