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63
An Optimal Algorithm for Approximate Nearest Neighbor Searching in Fixed Dimensions
 ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 1994
"... Consider a set S of n data points in real ddimensional space, R d , where distances are measured using any Minkowski metric. In nearest neighbor searching we preprocess S into a data structure, so that given any query point q 2 R d , the closest point of S to q can be reported quickly. Given any po ..."
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Cited by 786 (31 self)
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Consider a set S of n data points in real ddimensional space, R d , where distances are measured using any Minkowski metric. In nearest neighbor searching we preprocess S into a data structure, so that given any query point q 2 R d , the closest point of S to q can be reported quickly. Given any positive real ffl, a data point p is a (1 + ffl)approximate nearest neighbor of q if its distance from q is within a factor of (1 + ffl) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set of n points in R d in O(dn log n) time and O(dn) space, so that given a query point q 2 R d , and ffl ? 0, a (1 + ffl)approximate nearest neighbor of q can be computed in O(c d;ffl log n) time, where c d;ffl d d1 + 6d=ffle d is a factor depending only on dimension and ffl. In general, we show that given an integer k 1, (1 + ffl)approximations to the k nearest neighbors of q can be computed in additional O(kd log n) time.
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 be the sum of the weights of t ..."
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Cited by 147 (12 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
A Simple and Fast Incremental Randomized Algorithm for Computing Trapezoidal Decompositions and for Triangulating Polygons
 Comput. Geom. Theory Appl
, 1991
"... This paper presents a very simple incremental randomized algorithm for computing the trapezoidal decomposition induced by a set S of n line segments in the plane. If S is given as a simple polygonal chain the expected running time of the algorithm is O(n log n). This leads to a simple algorithm of t ..."
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Cited by 99 (2 self)
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This paper presents a very simple incremental randomized algorithm for computing the trapezoidal decomposition induced by a set S of n line segments in the plane. If S is given as a simple polygonal chain the expected running time of the algorithm is O(n log n). This leads to a simple algorithm of the same complexity for triangulating polygons. More generally, if S is presented as a plane graph with k connected components, then the expected running time of the algorithm is O(n log n k log n). As a byproduct our algorithm creates a search structure of expected linear size that allows point location queries in the resulting trapezoidation in logarithmic expected time. The analysis of the expected performance is elementary and straightforward. All expectations are with respect to "coinflips" generated by the algorithm and are not based on assumptions about the geometric distribution of the input. Large Portions of the research reported here were conducted while the author visit...
VisibilityBased PursuitEvasion in a Polygonal Environment
 International Journal of Computational Geometry and Applications
, 1997
"... This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. This problem was first introduced by Suzuki and Yamashita. Our stud ..."
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Cited by 86 (26 self)
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This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. This problem was first introduced by Suzuki and Yamashita. Our study of this problem is motivated in part by robotics applications, such as surveillance with a mobile robot equipped with a camera that must find a moving target in a cluttered workspace. A few bounds are introduced, and a complete algorithm is presented for computing a successful motion strategy for a single pursuer. For simplyconnected free spaces, it is shown that the minimum number of pursuers required is \Theta(lg n). For multiplyconnected free spaces, the bound is \Theta( p h + lg n) pursuers for a polygon that has n edges and h holes. A set of problems that are solvable by a single pursuer and require a linear number of recontaminations is shown. The complete algorithm searches a f...
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
 SIAM J. Comput
, 1997
"... We propose an optimaltime algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worstcase time O(n log n) and requires O(n log n) space, where n is the total number of vertice ..."
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Cited by 86 (1 self)
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We propose an optimaltime algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worstcase time O(n log n) and requires O(n log n) space, where n is the total number of vertices in the obstacle polygons. The algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles, and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane; the map can be used to answer singlesource shortest path queries in O(logn) time. The time complexity of our algorithm is a significant improvement over all previously published results on the shortest path problem. Finally, we also discuss extensions to more general shortest path problems, involving nonpoint and multiple sources. 1 Introduction 1.1 The Background and Our Result The Euclidean shortest path problem is one of the o...
Computing Minimum Length Paths of a Given Homotopy Class
 Comput. Geom. Theory Appl
, 1991
"... In this paper, we show that the universal covering space of a surface can be used to unify previous results on computing paths in a simple polygon. We optimize a given path among obstacles in the plane under the Euclidean and link metrics and under polygonal convex distance functions. Besides reveal ..."
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Cited by 74 (7 self)
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In this paper, we show that the universal covering space of a surface can be used to unify previous results on computing paths in a simple polygon. We optimize a given path among obstacles in the plane under the Euclidean and link metrics and under polygonal convex distance functions. Besides revealing connections between the minimum paths under these three distance functions, the framework provided by the universal cover leads to simplified lineartime algorithms for shortest path trees, for minimumlink paths in simple polygons, and for paths restricted to c given orientations. 1 Introduction If a wire, a pipe, or a robot must traverse a path among obstacles in the plane, then one might ask what is the best route to take. For the wire, perhaps the shortest distance is best; for the pipe, perhaps the fewest straightline segments. For the robot, either might be best depending on the relative costs of turning and moving. In this paper, we find shortest paths and shortest closed curve...
A VisibilityBased PursuitEvasion Problem
 SUBMITTED TO THE INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY AND APPLICATIONS
"... This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. A visibility region is associated witheach pursuer, and the goal is ..."
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Cited by 65 (1 self)
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This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. A visibility region is associated witheach pursuer, and the goal is to guarantee that the evader will ultimately lie in at least one visibility region. The study of this problem is motivated inpart by robotics applications, such as surveillance with a mobile robot equipped withacamera that must nd a moving target in a cluttered workspace. A few bounds are introduced, and a complete algorithm is presented for computing a successful motion strategy. For a simplyconnected free space, a logarithmic bound is established on the minimum of pursuers needed. Loose bounds for multiplyconnected free spaces are also given. A set of problems that are solvable by a single pursuer and require a linear number of recontaminations is shown. The complete algorithm searches a nite cell complex that is constructed onthebasis of critical information changes. This concept can be applied in principle to multiplepursuer problems, and the case of a single pursuer has been implemented. Several solution strategies are shown, most of which were computed in a few seconds on a standard workstation.
Planar Separators and Parallel Polygon Triangulation
, 1992
"... We show how to construct an O( p n)separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree where each node corresponds to a subgraph of G and stores an O( p n)separator of that subgraph. We also show how to construct an O(n ffl )way decomposition tree ..."
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Cited by 51 (7 self)
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We show how to construct an O( p n)separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree where each node corresponds to a subgraph of G and stores an O( p n)separator of that subgraph. We also show how to construct an O(n ffl )way decomposition tree in parallel in O(log n) time so that each node corresponds to a subgraph of G and stores an O(n 1=2+ffl )separator of that subgraph. We demonstrate the utility of such a separator decomposition by showing how it can be used in the design of a parallel algorithm for triangulating a simple polygon deterministically in O(log n) time using O(n= log n) processors on a CRCW PRAM. Keywords: Computational geometry, algorithmic graph theory, planar graphs, planar separators, polygon triangulation, parallel algorithms, PRAM model. 1 Introduction Let G = (V; E) be an nnode graph. An f(n)separator is an f(n)sized subset of V whose removal disconnects G into two subgraphs G 1 and G 2 each...
Dynamic Trees and Dynamic Point Location
 In Proc. 23rd Annu. ACM Sympos. Theory Comput
, 1991
"... This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using ..."
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Cited by 46 (11 self)
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This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the linkcut trees structure of Sleator and Tarjan, leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of kedge monotone chains in O(log n + k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial pointlocation in a 3dimensional convex subdivision. In addition, the interlacedtree approach is applied to online pointlo...
Triangulation and shapecomplexity
 ACM Trans. Graph
, 1984
"... This paper describes a new method for triangulating a simple nsided polygon. The algorithm runs in time O(n log s), with s _< n. The quantity s measures the sinuosity of the polygon, that is, the number of times the boundary alternates between complete spirals of opposite orientation. The value of ..."
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Cited by 41 (1 self)
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This paper describes a new method for triangulating a simple nsided polygon. The algorithm runs in time O(n log s), with s _< n. The quantity s measures the sinuosity of the polygon, that is, the number of times the boundary alternates between complete spirals of opposite orientation. The value of s is in practice a very small constant, even for extremely winding polygons. Our algorithm is the first method whose performance is linear in the number of vertices, up to within a factor that depends only on the shapecomplexity of the polygon. Informally, this notion of shapecomplexity measures how entangled a polygon is, and is thus highly independent of the number of vertices. A practical advantage of the algorithm is that it does not require sorting or the use of any balanced tree structure. Aside from the notion of sinuosity, we are also able to characterize a large class of polygons for which the algorithm can be proven to run in O(n log log n) time. The algorithm has been implemented, tested, and empirical evidence has confirmed its theoretical claim to efficiency.