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24
Visibility with a moving point of view
 Algorithmica
, 1994
"... We investigate 3d visibility problems in which the viewing position moves along a straight flightpath. Specifically we focus on two problems: determining the points along the flightpath at which the topology of the viewed scene changes, and answering rayshooting queries for rays with origin on the ..."
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Cited by 28 (1 self)
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We investigate 3d visibility problems in which the viewing position moves along a straight flightpath. Specifically we focus on two problems: determining the points along the flightpath at which the topology of the viewed scene changes, and answering rayshooting queries for rays with origin on the flightpath. Three progressively more specialized problems are considered: general scenes, terrains, and terrains with vertical flightpaths. 1.
Taking a Walk in a Planar Arrangement
 SIAM J. Comput
, 1999
"... We present a new randomized algorithm for computing portions of an arrangement of n arcs in the plane, each pair of which intersect in at most t points. We use this algorithm to perform online walks inside such an arrangement (i.e., compute all the faces that a curve crosses, where the curve is g ..."
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Cited by 26 (6 self)
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We present a new randomized algorithm for computing portions of an arrangement of n arcs in the plane, each pair of which intersect in at most t points. We use this algorithm to perform online walks inside such an arrangement (i.e., compute all the faces that a curve crosses, where the curve is given in an online manner), and to compute a level in an arrangement, both in an outputsensitive manner. The expected running time of the algorithm is O( t+2 (m+n) log n), where m is the number of intersections between the walk and the given arcs. No algorithm with similar performance is known for the general case of arcs. For the case of lines and segments, our algorithm improves the best known algorithm of [OvL81] by almost a logarithmic factor. 1 Introduction Let S be a set of n xmonotone arcs in the plane. Computing the whole (or parts of the) arrangement A( S), induced by the arcs of S, is one of the fundamental problems in computational geometry, and has received a lot o...
Generalized DavenportSchinzel Sequences
, 1993
"... The extremal function Ex(u; n) (introduced in the theory of DavenportSchinzel sequences in other notation) denotes for a fixed finite alternating sequence u = ababa : : : the maximum length of a finite sequence v over n symbols with no immediate repetition which does not contain u. Here (following ..."
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Cited by 19 (3 self)
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The extremal function Ex(u; n) (introduced in the theory of DavenportSchinzel sequences in other notation) denotes for a fixed finite alternating sequence u = ababa : : : the maximum length of a finite sequence v over n symbols with no immediate repetition which does not contain u. Here (following the idea of J. Nesetril) we generalize this concept for arbitrary sequence u. We summarize the already known properties of Ex(u; n) and we present also two new theorems which give good upper bounds on Ex(u; n) for u consisting of (two) smaller subsequences u i provided we have good upper bounds on Ex(u i ; n). We use these theorems to describe a wide class of sequences u ("linear sequences") for which Ex(u; n) = O(n). Both theorems are used for obtaining new superlinear upper bounds as well. We partially characterize linear sequences over three symbols. We also present several problems about Ex(u; n).
Finding the Best Viewpoints for ThreeDimensional Graph Drawings
 Proc. 5th International Symp. on Graph Drawing (GD ’97
, 1997
"... In this paper we address the problem of finding the best viewpoints for threedimensional straightline graph drawings. We define goodness in terms of preserving the relational structure of the graph, and develop two continuous measures of goodness under orthographic parallel projection. We develop ..."
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Cited by 19 (0 self)
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In this paper we address the problem of finding the best viewpoints for threedimensional straightline graph drawings. We define goodness in terms of preserving the relational structure of the graph, and develop two continuous measures of goodness under orthographic parallel projection. We develop Voronoi variants to find the best viewpoints under these measures, and present results on the complexity of these diagrams.
On Spanning Trees with Low Crossing Numbers
, 1992
"... Every set S of n points in the plane has a spanning tree such that no line disjoint from S has more than O( p n) intersections with the tree (where the edges are embedded as straight line segments). We review the proof of this result (originally proved by Bernard Chazelle and the author in a more ..."
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Cited by 15 (0 self)
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Every set S of n points in the plane has a spanning tree such that no line disjoint from S has more than O( p n) intersections with the tree (where the edges are embedded as straight line segments). We review the proof of this result (originally proved by Bernard Chazelle and the author in a more general setting), point at some methods for constructing such a tree, and describe some algorithmic and combinatorial applications. 1 Introduction. Over the recent years there has been considerable progress in the simplex range searching problem. In the planar version of this problem we are required to store a set S of n points such that the number of points in any query triangle can be determined efficiently. One of the combinatorial tools developed for this problem are spanning trees with low crossing numbers. Let S be set of n points in the plane. For a spanning tree on S and a line h, the crossing number of h in the tree is defined as c h = a + b 2 , where a is the number of edges ...
Optimal Partition Trees
, 2010
"... We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Back in SoCG’92, Matouˇsek gave a partition tree method for ddimensional simplex range searching achieving O(n) space and O(n 1−1/d) query time. Although this method is generally be ..."
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Cited by 15 (2 self)
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We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Back in SoCG’92, Matouˇsek gave a partition tree method for ddimensional simplex range searching achieving O(n) space and O(n 1−1/d) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n 1+ε) preprocessing time for any fixed ε> 0. An earlier method by Matouˇsek (SoCG’91) requires O(n log n) preprocessing time but O(n1−1/d log O(1) n) query time. We give a new method that achieves simultaneously O(n log n) preprocessing time, O(n) space, and O(n1−1/d) query time with high probability. Our method has several advantages: • It is conceptually simpler than Matouˇsek’s SoCG’92 method. Our partition trees satisfy many ideal properties (e.g., constant degree, optimal crossing number at almost all layers, and disjointness of the children’s cells at each node). • It leads to more efficient multilevel partition trees, which are important in many data structural applications (each level adds at most one logarithmic factor to the space and query bounds, better than in all previous methods). • A similar improvement applies to a shallow version of partition trees, yielding O(n log n) time, O(n) space, and O(n 1−1/⌊d/2 ⌋ ) query time for halfspace range emptiness in even dimensions d ≥ 4. Numerous consequences follow (e.g., improved results for computing spanning trees with low crossing number, ray shooting among line segments, intersection searching, exact nearest neighbor search, linear programming queries, finding extreme points,...). 1
When Crossings Count  Approximating the Minimum Spanning Tree
 In Proc. 16th Annu. ACM Sympos. Comput. Geom
, 2002
"... We present an (1+")approximation algorithm for computing the minimumspanning tree of points in a planar arrangement of lines, where the metric is the number of crossings between the spanning tree and the lines. The expected running time of the algorithm is near linear. We also show how to embed ..."
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Cited by 13 (5 self)
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We present an (1+")approximation algorithm for computing the minimumspanning tree of points in a planar arrangement of lines, where the metric is the number of crossings between the spanning tree and the lines. The expected running time of the algorithm is near linear. We also show how to embed such a crossing metric of hyperplanes in ddimensions, in subquadratic time, into highdimensions so that the distances are preserved. As a result, we can deploy a large collection of subquadratic approximations algorithms [IM98, GIV01] for problems involving points with the crossing metric as a distance function. Applications include MST, matching, clustering, nearestneighbor, and furthestneighbor.
Computational Geometry
 in Directions in Computational Geometry
, 1994
"... Computational geometry, the study of algorithms involving relatively simple geometric objects, is an active, exciting field. This chapter samples current research in computational geometry. Three topics are discussed at some length: the theory of arrangements, a randomincremental convex hull alg ..."
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Cited by 10 (0 self)
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Computational geometry, the study of algorithms involving relatively simple geometric objects, is an active, exciting field. This chapter samples current research in computational geometry. Three topics are discussed at some length: the theory of arrangements, a randomincremental convex hull algorithm, and robustness of geometric algorithms. 1 Introduction As pointed out by Robin Forrest and others[39, 79], the term "computational geometry" could be used for a variety of fields, including geometric modeling using curves and surfaces, computer proofs of geometric theorems, geometric design software, and the theory of perceptrons[71]. The purpose of this chapter is to give an impressionistic view of recent research in computational geometry, defined as the study of algorithms involving relatively simple geometric objects such as points and lines. A typical topic in this field is the analysis of an algorithm involving n objects in d dimensions; examples are the computation of the c...
Online Point Location in Planar Arrangements and Its Applications
 GEOM
, 2002
"... Recently, HarPeled [HP99b] presented a new randomized technique for online construction of the zone of a curve in a planar arrangement of arcs. In this paper, we present several applications of this technique, which yield improved solutions to a variety of problems. These applications include: ( ..."
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Cited by 10 (6 self)
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Recently, HarPeled [HP99b] presented a new randomized technique for online construction of the zone of a curve in a planar arrangement of arcs. In this paper, we present several applications of this technique, which yield improved solutions to a variety of problems. These applications include: (i) an efficient mechanism for performing online point location queries in an arrangement of arcs; (ii) an efficient algorithm for computing an approximation to the minimumweight Steinertree of a set of points, where the weight is the number of intersections between the tree edges and a given collection of arcs; (iii) a subquadratic algorithm for cutting a set of pseudoparabolas into pseudosegments; (iv) an algorithm for cutting a set of line segments (`rods') in 3space so as to eliminate all cycles in the vertical depth order; and (v) a nearoptimal algorithm for reporting all bichromatic intersections between a set R of red arcs and a set B of blue arcs, where the unions of the arcs in each set are both connected.
Queries with Segments in Voronoi Diagrams
, 1999
"... In this paper we consider proximity problems in which the queries are line segments in the plane. We build a query structure that for a set of n points P can determine the closest point in P to a query segment outside the convex hull of P in O(log n) time. With this we solve the problem of computing ..."
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Cited by 10 (1 self)
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In this paper we consider proximity problems in which the queries are line segments in the plane. We build a query structure that for a set of n points P can determine the closest point in P to a query segment outside the convex hull of P in O(log n) time. With this we solve the problem of computing the closest point to each of n disjoint line segments in O(n log 3 n) time. Nearest foreign neighbors or Hausdorff distance for disjoint, colored segments can be computed in the same time. We explore some connections to Hopcroft's problem. 1 Introduction Since Knuth [13] posed the post office problem preprocess a set of points, or sites, in the plane to quickly report the nearest to a query pointand Shamos and Hoey [17] suggested Voronoi diagrams as a solution, there have been a number of proximity problems in the plane whose solution is to build some type of Voronoi diagram and query with a point. Note: A Voronoi diagram of a set of sites is the partition of the plane into maxim...