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177
Efficient Visibility Queries in Simple Polygons
"... We present a method of decomposing a simple polygon that allows the preprocessing of the polygon to efficiently answer visibility queries of various forms in an output sensitive manner. Using O(n3 log n) preprocessing time and O(n3) space, we can, given a query point q inside or outside an n verte ..."
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Cited by 31 (2 self)
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We present a method of decomposing a simple polygon that allows the preprocessing of the polygon to efficiently answer visibility queries of various forms in an output sensitive manner. Using O(n3 log n) preprocessing time and O(n3) space, we can, given a query point q inside or outside an n vertex polygon, recover the visibility polygon of q in O(log n + k) time, where k is the size of the visibility polygon, and recover the number of vertices visible from q in O(log n) time. The key notion
Efficient TriangulationBased Pathfinding
, 2007
"... Pathfinding for commercial games is a challenging problem, and many existing methods use abstractions which lose details of the environment and compromise path quality. Conversely, humans can ignore irrelevant details of an environment that modern search techniques still consider, while maintaining ..."
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Cited by 27 (2 self)
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Pathfinding for commercial games is a challenging problem, and many existing methods use abstractions which lose details of the environment and compromise path quality. Conversely, humans can ignore irrelevant details of an environment that modern search techniques still consider, while maintaining its topography. This thesis describes a technique for extracting features such as dead ends, corridors, and decision points from an environment represented using a constrained Delaunay triangulation. The result is that the pathfinding task is simplified to the point where the search algorithm need only decide to which side of each obstacle to go, while all features of the environment are retained. We present algorithms which search the triangles of the environment (Triangulation
A*) and the decision points identified (Triangulation Reduction A*). We also explore a number of techniques which deal with finding paths for circular objects of nonzero radius and enhancements to various aspects of the search.
A New Approach to Subdivision Simplification
, 1995
"... The line simplification problem is an old and wellstudied problem in cartography. Although there are several algorithms to compute a simplification, there seem to be no algorithms that perform line simplification in the context of other geographical objects. This paper presents a nearly quadratic t ..."
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Cited by 27 (0 self)
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The line simplification problem is an old and wellstudied problem in cartography. Although there are several algorithms to compute a simplification, there seem to be no algorithms that perform line simplification in the context of other geographical objects. This paper presents a nearly quadratic time algorithm for the following line simplification problem: Given a polygonal line, a set of extra points, and a real ffl ? 0, compute a simplification that guarantees (i) a maximum error ffl, (ii) that the extra points remain on the same side of the simplified chain as of the original chain, and (iii) that the simplified chain has no selfintersections. The algorithm is applied as the main subroutine for subdivision simplification. 1 Introduction The line simplification problem is a wellstudied problem in various disciplines including geographic information systems [Buttenfield '85, Cromley '88, Douglas & Peucker '73, Hershberger & Snoeyink '92, Li & Openshaw '92, McMaster '87], digital...
Range mode and range median queries on lists and trees
 In Proceedings of the 14th Annual International Symposium on Algorithms and Computation (ISAAC
, 2003
"... ABSTRACT. We consider algorithms for preprocessing labelled lists and trees so that, for any two nodes u and v we can answer queries of the form: What is the mode or median label in the sequence of labels on the path from u to v. 1 ..."
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Cited by 26 (2 self)
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ABSTRACT. We consider algorithms for preprocessing labelled lists and trees so that, for any two nodes u and v we can answer queries of the form: What is the mode or median label in the sequence of labels on the path from u to v. 1
A compact piecewiselinear Voronoi diagram for convex sites in the plane
, 1993
"... In the plane, the postoffice problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a onedimensional retract of the free space of a robot, are both classically solved by computing a Voronoi diagram. When the sites are k disjoint convex sets, we giv ..."
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Cited by 25 (4 self)
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In the plane, the postoffice problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a onedimensional retract of the free space of a robot, are both classically solved by computing a Voronoi diagram. When the sites are k disjoint convex sets, we give a compact representation of the Voronoi diagram, using O(k) line segments, that is sufficient for logarithmic time postoffice location queries and motion planning. If these sets are polygons with n total vertices, we compute this diagram optimally in O ( k log n) deterministic time for the Euclidean metric and in O(k logn logm) deterministic time for the convex distance function defined by a convex mgon.
More Planar TwoCenter Algorithms
 Comput. Geom. Theory Appl
, 1997
"... This paper considers the planar Euclidean twocenter problem: given a planar npoint set S, find two congruent circular disks of the smallest radius covering S. The main result is a deterministic algorithm with running time O(n log 2 n log 2 log n), improving the previous O(n log 9 n) bound ..."
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Cited by 24 (1 self)
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This paper considers the planar Euclidean twocenter problem: given a planar npoint set S, find two congruent circular disks of the smallest radius covering S. The main result is a deterministic algorithm with running time O(n log 2 n log 2 log n), improving the previous O(n log 9 n) bound of Sharir and almost matching the randomized O(n log 2 n) bound of Eppstein. If a point in the intersection of the two disks is given, then we can solve the problem in O(n log n) time with high probability. Keywords: twocenter, randomization, parametric search 1 Introduction Consider the following "facility location" problem: given a set S of n "demand" points in IR d and a number p, find a set T of p "supply"points in IR d minimizing max s2S min t2T d(s; t), where d(s; t) denotes the Euclidean distance between s and t. Geometrically, the problem is equivalent to finding p congruent disks of the smallest radius covering S and is referred to as the (Euclidean) pcenter problem. Th...
Geodesic Fréchet Distance Inside a Simple Polygon
, 2007
"... We present the first algorithm for the geodesic Fréchet distance between two polygonal curves A and B inside a simple polygon P. If A and B have total complexity N and P has complexity k, then the algorithm runs in O(k+N 2 log kN log N) expected time and O(k+N 2) space. This runtime is quite good a ..."
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Cited by 23 (4 self)
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We present the first algorithm for the geodesic Fréchet distance between two polygonal curves A and B inside a simple polygon P. If A and B have total complexity N and P has complexity k, then the algorithm runs in O(k+N 2 log kN log N) expected time and O(k+N 2) space. This runtime is quite good as it is only a logarithmic factor larger than the nongeodesic Fréchet algorithm [2]. We also unveil an alluring alternative to parametric search that applies to both the nongeodesic and geodesic Fréchet distance algorithms. This randomized approach is based on a variant of redblue intersections and is appealing due to its elegance and practical efficiency when compared to parametric search.
A fast semiLagrangian contouring method for moving interfaces
, 2000
"... General moving interface problems are solved by a new approach: evaluate an explicit semiLagrangian advection formula with ecient geometric algorithms, and extract the moving interface with a fast new contouring technique. The new approach decouples spatial and temporal resolutions, and gridfr ..."
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Cited by 22 (5 self)
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General moving interface problems are solved by a new approach: evaluate an explicit semiLagrangian advection formula with ecient geometric algorithms, and extract the moving interface with a fast new contouring technique. The new approach decouples spatial and temporal resolutions, and gridfree adaptive renement of the interface increases accuracy dramatically. A modular implementation, with a fast new intrinsic geometry module, computes highly accurate solutions to geometric moving interface problems involving merging, anisotropy and faceting; with a high{order embedded geometry module, it solves secondorder problems involving curvature, dynamic topology and nonlocal interactions.
Methods for Achieving Fast Query Times in Point Location Data Structures
, 1997
"... Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linearspace data struc ..."
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Cited by 22 (1 self)
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Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linearspace data structures can be constructed so as to achieve O(log n) query times. But applications, such as those common in geographic information systems, motivate a reexamination of this problem with the goal of improving query times further while also simplifying the methods needed to achieve such query times. In this paper we perform such a reexamination, focusing on the issues that arise in three different classes of pointlocation query sequences: ffl sequences that are reasonably uniform spatially and temporally (in which case the constant factors in the query times become critical), ffl sequences that are nonuniform spatially or temporally (in which case one desires data structures that adapt to s...