Results 1  10
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11
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...
Localizing a Robot with Minimum Travel
, 1995
"... We consider the problem of localizing a robot in a known environment modeled by a simple polygon P . We assume that the robot has a map of P but is placed at an unknown location inside P . From its initial location, the robot sees a set of points called the visibility polygon V of its location. I ..."
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Cited by 44 (4 self)
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We consider the problem of localizing a robot in a known environment modeled by a simple polygon P . We assume that the robot has a map of P but is placed at an unknown location inside P . From its initial location, the robot sees a set of points called the visibility polygon V of its location. In general, sensing at a single point will not suffice to uniquely localize the robot, since the set H of points in P with visibility polygon V may have more than one element. Hence, the robot must move around and use range sensing and a compass to determine its position (i.e.
Approximate convex decomposition of polygons
 In Proc. 20th Annual ACM Symp. Computat. Geom. (SoCG
, 2004
"... We propose a strategy to decompose a polygon, containing zero or more holes, into “approximately convex” pieces. For many applications, the approximately convex components of this decomposition provide similar benefits as convex components, while the resulting decomposition is significantly smaller ..."
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Cited by 22 (3 self)
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We propose a strategy to decompose a polygon, containing zero or more holes, into “approximately convex” pieces. For many applications, the approximately convex components of this decomposition provide similar benefits as convex components, while the resulting decomposition is significantly smaller and can be computed more efficiently. Moreover, our approximate convex decomposition (ACD) provides a mechanism to focus on key structural features and ignore less significant artifacts such as wrinkles and surface texture. We propose a simple algorithm that computes an ACD of a polygon by iteratively removing (resolving) the most significant nonconvex feature (notch). As a by product, it produces an elegant hierarchical representation that provides a series of ‘increasingly convex ’ decompositions. A user specified tolerance determines the degree of concavity that will be allowed in the lowest level of the hierarchy. Our algorithm computes an ACD of a simple polygon with n vertices and r notches in O(nr) time. In contrast, exact convex decomposition is NPhard or, if the polygon has no holes, takes O(nr 2) time. Models and movies can be found on our webpages at:
Generalized Hidden Surface Removal
 Comput. Geom. Theory Appl
, 1993
"... In this paper we study the following generalization of the classical hidden surface removal problem: given a set S of objects, a view point and a point light source, compute which parts of the objects in S are visible, subdivided into parts that are lit and parts that are not lit. ..."
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Cited by 4 (1 self)
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In this paper we study the following generalization of the classical hidden surface removal problem: given a set S of objects, a view point and a point light source, compute which parts of the objects in S are visible, subdivided into parts that are lit and parts that are not lit.
Approximate convex decomposition and its applications
, 2006
"... Geometric computations are essential in many realworld problems. One important issue in geometric computations is that the geometric models in these problems can be so large that computations on them have infeasible storage or computation time requirements. Decomposition is a technique commonly us ..."
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Cited by 4 (1 self)
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Geometric computations are essential in many realworld problems. One important issue in geometric computations is that the geometric models in these problems can be so large that computations on them have infeasible storage or computation time requirements. Decomposition is a technique commonly used to partition complex models into simpler components. Whereas decomposition into convex components results in pieces that are easy to process, such decompositions can be costly to construct and can result in representations with an unmanageable number of components. In this work, we have developed an approximate technique, called Approximate Convex Decomposition (ACD), which decomposes a given polygon or polyhedron into “approximately convex ” pieces that may provide similar benefits as convex components, while the resulting decomposition is both significantly smaller (typically by orders of magnitude) and can be computed more efficiently. Indeed, for many applications, an ACD can represent the important structural features of the model more accurately by providing a mechanism for ignoring less significant features, such as wrinkles and surface texture. Our study of a wide range of applications shows that in addition to providing computational efficiency, ACD also provides natural multiresolution or hierarchical representations. In this dissertation, we provide some examples of ACD’s many potential applications, such as particle simulation, mesh generation, motion planning, and skeleton extraction.
Shooting permanent rays among disjoint polygons
 in the plane, in: Proc. 25th SCG, 2009, ACM
"... We present a data structure for ray shootingandinsertion in the free space among disjoint polygonal obstacles with a total of n vertices in the plane, where each ray starts at the boundary of some obstacle. The portion of each query ray between the starting point and the first obstacle hit is inse ..."
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Cited by 3 (2 self)
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We present a data structure for ray shootingandinsertion in the free space among disjoint polygonal obstacles with a total of n vertices in the plane, where each ray starts at the boundary of some obstacle. The portion of each query ray between the starting point and the first obstacle hit is inserted permanently as a new obstacle. Our data structure uses O(n log n) space and preprocessing time, and it supports m successive ray shootingandinsertion queries in O(n log 2 n + m log m) total time. We present two applications for our data structure: (1) Our data structure supports efficient implementation of autopartitions in the plane i.e. binary space partitions where each partition is done along the supporting line of an input segment. If n input line segments are fragmented into m pieces by an autopartition, then it can now be implemented in O(n log 2 n + m log m) time. This improves the expected runtime of Patersen and Yao’s classical randomized autopartition algorithm for n disjoint line segments to O(n log 2 n). (2) If we are given disjoint polygonal obstacles with a total of n vertices in the plane, a permutation of the reflex vertices, and a halfline at each reflex vertex that partitions the reflex angle into two convex angles, then the folklore convex partitioning algorithm draws a ray emanating from each reflex vertex in the prescribed order in the given direction until it hits another obstacle, a previous ray, or infinity. The previously best implementation (with a semidynamic ray shooting data structure) requires O(n 3/2−ε/2) time using O(n 1+ε) space. Our data structure improves the runtime to O(n log 2 n). 1
Pareto envelopes in simple polygons ∗
"... For a set T of n points in a metric space (X, d), a point y ∈ X is dominated by a point x ∈ X if d(x, t) ≤ d(y, t) for all t ∈ T and there exists t ′ ∈ T such that d(x, t ′ ) < d(y, t ′). The set of nondominated points of X is called the Pareto envelope of T. H. Kuhn (1973) established that in Euc ..."
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For a set T of n points in a metric space (X, d), a point y ∈ X is dominated by a point x ∈ X if d(x, t) ≤ d(y, t) for all t ∈ T and there exists t ′ ∈ T such that d(x, t ′ ) < d(y, t ′). The set of nondominated points of X is called the Pareto envelope of T. H. Kuhn (1973) established that in Euclidean spaces, the Pareto envelopes and the convex hulls coincide. Chalmet et al. (1981) characterized the Pareto envelopes in the rectilinear plane (R 2, d1) and constructed them in O(n log n) time. In this note, we investigate the Pareto envelopes of pointsets in simple polygons P endowed with geodesic d2 or d1metrics (i.e., Euclidean and Manhattan metrics). We show that Kuhn’s characterization extends to Pareto envelopes in simple polygons with d2metric, while that of Chalmet et al. extends to simple rectilinear polygons with d1metric. These characterizations provide efficient algorithms for construction of these Pareto envelopes. 1
Graph Drawing Heuristics for Path Finding in Large Dimensionless Graphs
"... Abstract — This paper presents a heuristic for guiding A*search for approximating the shortest path between two vertices in arbitrarilysized dimensionless graphs. First we discuss methods by which these dimensionless graphs are laid out into Euclidean drawings. Next, two heuristics are computed bas ..."
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Abstract — This paper presents a heuristic for guiding A*search for approximating the shortest path between two vertices in arbitrarilysized dimensionless graphs. First we discuss methods by which these dimensionless graphs are laid out into Euclidean drawings. Next, two heuristics are computed based on drawings of the graphs. We compare the performance of an A*search using these heuristics with breadthfirst search on graphs with various topological properties. The results show a large savings in the number of vertices expanded for large graphs.
www.elsevier.com/locate/tcs Querypoint visibility constrained shortest paths in simple polygons ✩
"... In this paper, we study the problem of finding the shortest path between two points inside a simple polygon such that there is at least one point on the path from which a query point is visible. We provide an algorithm which preprocesses the input in O(n 2 +nK) time and space and provides logarithmi ..."
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In this paper, we study the problem of finding the shortest path between two points inside a simple polygon such that there is at least one point on the path from which a query point is visible. We provide an algorithm which preprocesses the input in O(n 2 +nK) time and space and provides logarithmic query time. The input polygon has n vertices and K is a parameter dependent on the input polygon which is O(n 2) in the worst case but is much smaller for most polygons. The preprocessing algorithm sweeps an angular interval around every reflex vertex of the polygon to store the optimal contact points between the shortest paths and the windows separating the visibility polygons of the query points from the source and the destination. c ○ 2007 Elsevier B.V. All rights reserved.
www.elsevier.com/locate/ipl Shortest paths in simple polygons with polygonmeet constraints ✩
, 2004
"... We study a constrained version of the shortest path problem in simple polygons, in which the path must visit a given target polygon. We provide a worstcase optimal algorithm for this problem and also present a method to construct a subdivision of the simple polygon to efficiently answer queries to ..."
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We study a constrained version of the shortest path problem in simple polygons, in which the path must visit a given target polygon. We provide a worstcase optimal algorithm for this problem and also present a method to construct a subdivision of the simple polygon to efficiently answer queries to retrieve the shortest polygonmeeting paths from a singlesource to the query point. The algorithms are linear, both in time and space, in terms of the complexity of the two polygons. © 2004 Elsevier B.V. All rights reserved.