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40
Optimal Simplification of Polygonal Chain for Rendering
, 2007
"... For a given polygonal chain, we study the min # problem, which consists in finding an approximate and ordered subchain with a minimum number of vertices. Previous approaches simplify the input chain relative to an approximation criterion which minimizes the gap between the original chain and the si ..."
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For a given polygonal chain, we study the min # problem, which consists in finding an approximate and ordered subchain with a minimum number of vertices. Previous approaches simplify the input chain relative to an approximation criterion which minimizes the gap between the original chain and the simplified subchain. Nevertheless, no criterion allows us to directly control the visual quality of the final rendered result. Moreover, efficient methods produce peculiar simplifications or entail a useless increase in the number of vertices. A quadratic complexity is then required to bypass these misbehaviors and to obtain a good perceptual quality. We define a new criterion which retains the shape of the original chain and which guarantees that the distance between the rendered simplification is at most half a pixel away from the original chain. Thus, our criterion does not produce an incorrect simplification. Based on a flexible tolerance region, it does not involve any side effect that would increase the number of vertices of the simplification. Moreover, our method reaches a nearlinear time complexity and its implementation is based on classical functions. To our knowledge, this is the first algorithm providing all these advantages.
Distributed Ranking Methods for Geographic Information Retrieval
 In 20th European Workshop on Computational Geometry
, 2004
"... Geographic Information Retrieval is concerned with retrieving documents in response to a spatially related query. This paper addresses the ranking of documents by both textual and spatial relevance. To this end, we introduce distributed ranking, where similar documents are ranked spread in the li ..."
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Geographic Information Retrieval is concerned with retrieving documents in response to a spatially related query. This paper addresses the ranking of documents by both textual and spatial relevance. To this end, we introduce distributed ranking, where similar documents are ranked spread in the list instead of consecutively. The e#ect of this is that documents close together in the ranked list have less redundant information. We present various ranking methods, e#cient algorithms to implement them, and experiments to show the outcome of the methods.
Relative εApproximations in Geometry
, 2007
"... We reexamine relative εapproximations, previously studied in [Pol86, Hau92, LLS01, CKMS06], and their relation to certain geometric problems. We give a simple constructive proof of their existence in general range spaces with finite VCdimension, and of a sharp bound on their size, close to the be ..."
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Cited by 5 (2 self)
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We reexamine relative εapproximations, previously studied in [Pol86, Hau92, LLS01, CKMS06], and their relation to certain geometric problems. We give a simple constructive proof of their existence in general range spaces with finite VCdimension, and of a sharp bound on their size, close to the best known one. We then give a construction of smallersize relative εapproximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure—spanning trees with small relative crossing number, which we believe to be of independent interest. We also consider applications of the new structures for approximate range counting and related problems.
Polygonal chain approximation: a query based approach
, 2005
"... In this paper we present a new, query based approach for approximating polygonal chains in the plane. We give a few results based on this approach, some of more general interest, and propose a greedy heuristic to speed up the computation. Our algorithms are simple, based on standard geometric operat ..."
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Cited by 4 (0 self)
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In this paper we present a new, query based approach for approximating polygonal chains in the plane. We give a few results based on this approach, some of more general interest, and propose a greedy heuristic to speed up the computation. Our algorithms are simple, based on standard geometric operations, and thus suitable for efficient implementation. We also show that the query based approach can be used to obtain a subquadratic time exact algorithm with infinite beam criterion and Euclidean distance metric if some condition on the input path holds. Although in a special case, this is the first subquadratic result for path approximation with Euclidean distance metric.
Improved upper bounds on the reflexivity of point sets
 COMPUTATIONAL GEOMETRY, VOLUME 42, ISSUE
, 2009
"... Given a set S of n points in the plane, the reflexivity of S, ρ(S), is the minimum number of reflex vertices in a simple polygonalization of S. Arkin et al. [4] proved that ρ(S) ≤ ⌈n/2 ⌉ for any set S, and conjectured that the tight upper bound is ⌊n/4⌋. We show that the reflexivity of any set of n ..."
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Cited by 3 (1 self)
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Given a set S of n points in the plane, the reflexivity of S, ρ(S), is the minimum number of reflex vertices in a simple polygonalization of S. Arkin et al. [4] proved that ρ(S) ≤ ⌈n/2 ⌉ for any set S, and conjectured that the tight upper bound is ⌊n/4⌋. We show that the reflexivity of any set of n points is at most 3 7n + O(1) ≈ 0.4286n. Using computeraided abstract order type extension the upper bound can be further improved to 5 12n + O(1) ≈ 0.4167n. We also present an algorithm to compute polygonalizations with at most this number of reflex vertices in O(nlog n) time.
Relative convex hulls in semidynamic subdivisions
 in: Proc. 16th ESA
"... Abstract. We present data structures for maintaining the relative convex hull of a set of points (sites) in the presence of pairwise noncrossing line segments (barriers) that subdivide a bounding box into simply connected faces. Our data structures have O((n + m) log n) size for n sites and m barri ..."
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Abstract. We present data structures for maintaining the relative convex hull of a set of points (sites) in the presence of pairwise noncrossing line segments (barriers) that subdivide a bounding box into simply connected faces. Our data structures have O((n + m) log n) size for n sites and m barriers. They support O(m) barrier insertions and O(n) site deletions in O((m + n) polylog (mn)) total time, and can answer analogues of standard convex hull queries in O(polylog (mn)) time. Our data structures support a generalization of the sweep line technique, in which the sweep wavefront may have arbitrary polygonal shape, possibly bending around obstacles. We reduce the total time of m online updates of a polygonal sweep wavefront from O(m √ n polylog n) to O((m + n) polylog (mn)). 1
Tight Bounds for Dynamic Convex Hull Queries (Again) ABSTRACT
"... The dynamic convex hull problem was recently solved in O(lg n) time per operation, and this result is best possible in models of computation with bounded branching (e.g., algebraic computation trees). From a data structures point of view, however, such models are considered unrealistic because they ..."
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The dynamic convex hull problem was recently solved in O(lg n) time per operation, and this result is best possible in models of computation with bounded branching (e.g., algebraic computation trees). From a data structures point of view, however, such models are considered unrealistic because they hide intrinsic notions of information in the input. In the standard wordRAM and cellprobe models of computation, we prove that the optimal query time for dynamic convex hulls is, in fact, Θ ` ´ lg n, for polylogarithmic uplg lg n date time (and word size). Our lower bound is based on a reduction from the markedancestor problem, and is one of the first data structural lower bounds for a nonorthogonal geometric problem. Our upper bounds follow a recent trend of attacking nonorthogonal geometric problems from an informationtheoretic perspective that has proved central to advanced data structures. Interestingly, our upper bounds are the first to successfully apply this perspective to dynamic geometric data structures, and require substantially different ideas from previous work.
Outputsensitive algorithms for Tukey depth and related problems
, 2006
"... The Tukey depth (Tukey 1975) of a point p with respect to a finite set S of points is the minimum number of elements of S contained in any closed halfspace that contains p. Algorithms for computing the Tukey depth of a point in various dimensions are considered. The running times of these algorithms ..."
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The Tukey depth (Tukey 1975) of a point p with respect to a finite set S of points is the minimum number of elements of S contained in any closed halfspace that contains p. Algorithms for computing the Tukey depth of a point in various dimensions are considered. The running times of these algorithms depend on the value of the output, making them suited to situations, such as outlier removal, where the value of the output is typically small.
Uniquely Represented Data Structures for Computational Geometry
, 2008
"... We present new techniques for the construction of uniquely represented data structures in a RAM, and use them to construct efficient uniquely represented data structures for orthogonal range queries, line intersection tests, point location, and 2D dynamic convex hull. Uniquely represented data stru ..."
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Cited by 2 (2 self)
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We present new techniques for the construction of uniquely represented data structures in a RAM, and use them to construct efficient uniquely represented data structures for orthogonal range queries, line intersection tests, point location, and 2D dynamic convex hull. Uniquely represented data structures represent each logical state with a unique machine state. Such data structures are strongly historyindependent. This eliminates the possibility of privacy violations caused by the leakage of information about the historical use of the data structure. Uniquely represented data structures may also simplify the debugging of complex parallel computations, by ensuring that two runs of a program that reach the same logical state reach the same physical state, even if various parallel processes executed in different orders during the two runs. 1
Covering point sets with two disjoint disks or squares
, 2007
"... We study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks CR and CB with disjoint interiors such that the number of red points covered by CR plus the number of blue points covered by CB is maximized. We give an algorithm to solve this proble ..."
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Cited by 2 (1 self)
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We study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks CR and CB with disjoint interiors such that the number of red points covered by CR plus the number of blue points covered by CB is maximized. We give an algorithm to solve this problem in O(n 8/3 log² n) time, where n denotes the total number of points. We also show that the analogous problem of finding two axisaligned unit squares SR and SB instead of unit disks can be solved in O(n log n) time, which is optimal. If we do not restrict ourselves to axisaligned squares, but require that both squares have a common orientation, we give a solution using O(n³ log n) time.