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28
DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
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Cited by 416 (115 self)
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An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between DavenportSchinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Data structures for mobile data
 JOURNAL OF ALGORITHMS
, 1997
"... A kinetic data structure (KDS) maintains an attribute of interest in a system of geometric objects undergoing continuous motion. In this paper we develop a conceptual framework for kinetic data structures, propose a number of criteria for the quality of such structures, and describe a number of fund ..."
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Cited by 231 (50 self)
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A kinetic data structure (KDS) maintains an attribute of interest in a system of geometric objects undergoing continuous motion. In this paper we develop a conceptual framework for kinetic data structures, propose a number of criteria for the quality of such structures, and describe a number of fundamental techniques for their design. We illustrate these general concepts by presenting kinetic data structures for maintaining the convex hull and the closest pair of moving points in the plane; these structures behavewell according to the proposed quality criteria for KDSs.
Kinetic Data Structures  A State of the Art Report
, 1998
"... ... In this paper we present a general framework for addressing such problems and the tools for designing and analyzing relevant algorithms, which we call kinetic data structures. We discuss kinetic data structures for a variety of fundamental geometric problems, such as the maintenance of convex hu ..."
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Cited by 90 (27 self)
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... In this paper we present a general framework for addressing such problems and the tools for designing and analyzing relevant algorithms, which we call kinetic data structures. We discuss kinetic data structures for a variety of fundamental geometric problems, such as the maintenance of convex hulls, Voronoi and Delaunay diagrams, closest pairs, and intersection and visibility problems. We also briefly address the issues that arise in implementing such structures robustly and efficiently. The resulting techniques satisfy three desirable properties: (1) they exploit the continuity of the motion of the objects to gain efficiency, (2) the number of events processed by the algorithms is close to the minimum necessary in the worst case, and (3) any object may change its `flight plan' at any moment with a low cost update to the simulation data structures. For computer applications dealing with motion in the physical world, kinetic data structures lead to simulation performance unattainable by other means. In addition, they raise fundamentally new combinatorial and algorithmic questions whose study may prove fruitful for other disciplines as well.
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 75 (20 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Voronoi Diagrams of Moving Points
, 1995
"... Consider a set of n points in ddimensional Euclidean space, d 2, each of which is continuously moving along a given individual trajectory. At each instant in time, the points define a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in ..."
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Cited by 46 (6 self)
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Consider a set of n points in ddimensional Euclidean space, d 2, each of which is continuously moving along a given individual trajectory. At each instant in time, the points define a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Voronoi diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, while showing that the number of topological events has an upper bound of O(n d s (n)), where s (n) is the maximum length of a (n; s)DavenportSchinzel sequence [AgShSh 89, DaSc 65] and s is a constant depending on the motions of the point sites. Our results are a linearfactor improvement over the naive O(n d+2 ) upper bound on the number of topological events. In addition, we show that if only k points are moving (while leaving the other n \Gamma k points fixed), there is an upper bound of O(kn d\Gamma1 s (n) + (n \Gamma k)...
On Dynamic Voronoi Diagrams and the Minimum Hausdorff Distance for Point Sets Under Euclidean Motion in the Plane
, 1992
"... We show that the dynamic Voronoi diagram of k sets of points in the plane, where each set consists of n points moving rigidly, has complexity O(n 2 k 2 s (k)) for some fixed s, where s (n) is the maximum length of a (n; s) DavenportSchinzel sequence. This improves the result of Aonuma et. al ..."
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Cited by 45 (3 self)
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We show that the dynamic Voronoi diagram of k sets of points in the plane, where each set consists of n points moving rigidly, has complexity O(n 2 k 2 s (k)) for some fixed s, where s (n) is the maximum length of a (n; s) DavenportSchinzel sequence. This improves the result of Aonuma et. al., who show an upper bound of O(n 3 k 4 log k) for the complexity of such Voronoi diagrams. We then apply this result to the problem of finding the minimum Hausdorff distance between two point sets in the plane under Euclidean motion. We show that this distance can be computed in time O((m + n) 6 log(mn)), where the two sets contain m and n points respectively. This work was supported in part by NSF grant IRI9057928 and matching funds from General Electric and Kodak, and in part by AFOSR under contract AFOSR910328. The second author was also supported by the Eshkol grant 0460190 from The Israeli Ministry of Science and Technology. 1. Introduction Determining the degree to ...
A Practical Evaluation of Kinetic Data Structures
 In Proc. 13th Annu. ACM Sympos. Comput. Geom
, 1997
"... this paper is to study and validate the use of the kinetic data structures proposed in [BGH97] in practice. The implementation and experimental evaluation of such structures brings to light several important issues which were not addressed in the original paper. For instance, KDSs implement the cont ..."
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Cited by 21 (6 self)
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this paper is to study and validate the use of the kinetic data structures proposed in [BGH97] in practice. The implementation and experimental evaluation of such structures brings to light several important issues which were not addressed in the original paper. For instance, KDSs implement the continuous maintenance of the function of interest through a discreteevent simulation. The calculation of the event times poses numerical difficulties which raise questions about the integrity of the structure if nearby events should happen out of order, or if the time of an event changes when recalculated. Also, the algorithmic measures for evaluating the quality of KDSs in [BGH97] are all worstcase measures, which may not reflect behavior under ordinary motions which occur in practice. It may well be that simple data structures do better in such situations. In order to study these issues, we have compared the KDSs of [BGH97] with simpler variants under various common data/motion distributions, each meant to generate a different amount of
Geometric algorithms for conflict detection/resolution in air traffic management
 In 36th IEEE Conference on Decision and Control
, 1997
"... We consider the problems of conflict detection and resolution in air traffic management (ATM) from the perspective of computational geometry and give algorithms for solving these problems efficiently. For conflict resolution, we propose a simple method that can route multiple aircraft, conflictfree ..."
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Cited by 15 (0 self)
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We consider the problems of conflict detection and resolution in air traffic management (ATM) from the perspective of computational geometry and give algorithms for solving these problems efficiently. For conflict resolution, we propose a simple method that can route multiple aircraft, conflictfree, through a cluttered airspace, using a prioritized routing scheme in spacetime. Our algorithms have been implemented into a simulation system that tracks a large set of flights, having multiple conflicts, and proposes modified routes to resolve them. We report on the preliminary results from an extensive set of experiments that are under way to determine the effectiveness of our methods. 1
A tight bound for the complexity of Voronoi diagrams under polyhedral convex distance functions in 3D
 PROC. 33RD ANNU. ACM SYMPOS. THEORY OF COMPUT
, 2001
"... We consider the Voronoi diagram of a set of n points in three dimensions under a convex distance function induced by an arbitrary, fixed polytope. The combinatorial complexity, i. e. the number of vertices, edges, and facets, of this diagram is shown to be in Θ(n²), which constitutes a considerable ..."
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Cited by 7 (0 self)
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We consider the Voronoi diagram of a set of n points in three dimensions under a convex distance function induced by an arbitrary, fixed polytope. The combinatorial complexity, i. e. the number of vertices, edges, and facets, of this diagram is shown to be in Θ(n²), which constitutes a considerable improvement to the results known so far. Unlike previous work, we do not need probabilistic arguments or recurrence techniques, but we exploit properties of the involved geometric structures. Key observations are that the lower envelope of n polygonal chains in the plane with a total of O(n) line segments and with only a fixed number of slopes is linear in n, and that the number of slopes of Voronoi edges is bounded by a constant that only depends on the complexity of the polytope.
Compact Voronoi Diagrams for Moving Convex Polygons
 In Proc. Scand. Workshop Alg. and Data Structures (SWAT
, 2000
"... We describe a kinetic data structure for maintaining a compact Voronoilike diagram of convex polygons moving around in the plane. We use a compact diagram for the polygons, dual to the Voronoi, first presented in [MKS96]. A key feature of this diagram is that its size is only a function of the n ..."
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Cited by 6 (4 self)
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We describe a kinetic data structure for maintaining a compact Voronoilike diagram of convex polygons moving around in the plane. We use a compact diagram for the polygons, dual to the Voronoi, first presented in [MKS96]. A key feature of this diagram is that its size is only a function of the number of polygons and not of their complexity. We demonstrate a local certifying property of that diagram, akin to that of Delaunay triangulations of points. We then obtain a method for maintaining this diagram that is outputsensitive and costs O(log n) per update. Furthermore, we show that for a set of k polygons with a total of n vertices moving along bounded degree algebraic motions, this dual diagram, and thus their compact Voronoi diagram, changes combinatorially## ) and O(kn #(k)#(n)) times, where #() is an extremely slowly growing function. This compact Voronoi diagram can be used for collision detection or retraction motion planning among the moving polygons.