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89
Dynamic planar convex hull
 Proc. 43rd IEEE Sympos. Found. Comput. Sci
, 2002
"... In this paper we determine the amortized computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage o ..."
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Cited by 53 (1 self)
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In this paper we determine the amortized computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage of the data structure is O(n). The data structure supports extreme point queries in a given direction, tangent queries through a given point, and queries for the neighboring points on the convex hull in O(log n) time. The extreme point queries can be used to decide whether or not a given line intersects the convex hull, and the tangent queries to determine whether a given point is inside the convex hull. We give a lower bound on the amortized asymptotic time complexity that matches the performance of this data structure.
Proximity Problems on Moving Points
 In Proc. 13th Annu. ACM Sympos. Comput. Geom
, 1997
"... A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dimensions: the first structure maintains the closest pair o ..."
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Cited by 50 (15 self)
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A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dimensions: the first structure maintains the closest pair of a set of continuously moving points, and is provably e#cient. The second structure maintains a spanning tree of the moving points whose cost remains within some prescribed factor of the minimum spanning tree. The method for maintaining the closest pair of points can be extended to the maintenance of closest pair of other distance functions which allows us to maintain the closest pair of a set of moving objects with similar sizes and of a set of points on a smooth manifold.
Raising Roofs, Crashing Cycles, and Playing Pool: Applications of a Data Structure for Finding Pairwise Interactions
 In Proc. 14th Annu. ACM Sympos. Comput. Geom
, 1998
"... The straight skeleton of a polygon is a variant of the medial axis, introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an ngon with r reflex vertices in time O(n 1+" +n 8=11+" r ..."
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Cited by 46 (0 self)
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The straight skeleton of a polygon is a variant of the medial axis, introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an ngon with r reflex vertices in time O(n 1+" +n 8=11+" r 9=11+" ), for any fixed " ? 0, improving the previous best upper bound of O(nr log n). Our algorithm simulates the sequence of collisions between edges and vertices during the shrinking process, using a technique of Eppstein for maintaining extrema of binary functions to reduce the problem of finding successive interactions to two dynamic range query problems: (1) maintain a changing set of triangles in IR 3 and answer queries asking which triangle would be first hit by a query ray, and (2) maintain a changing set of rays in IR 3 and answer queries asking for the lowest intersection of any ray with a query triangle. We also exploit a novel characterization of the straight skeleton as a ...
Geometric Range Searching
, 1994
"... In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in c ..."
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Cited by 46 (2 self)
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In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.
Coresets for kMeans and kMedian Clustering and their Applications
 In Proc. 36th Annu. ACM Sympos. Theory Comput
, 2003
"... In this paper, we show the existence of small coresets for the problems of computing kmedian and kmeans clustering for points in low dimension. In other words, we show that given a point set P in IR , one can compute a weighted set S P , of size log n), such that one can compute the kmed ..."
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Cited by 46 (13 self)
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In this paper, we show the existence of small coresets for the problems of computing kmedian and kmeans clustering for points in low dimension. In other words, we show that given a point set P in IR , one can compute a weighted set S P , of size log n), such that one can compute the kmedian/means clustering on S instead of on P , and get an (1 + ")approximation.
Dynamizing static algorithms with applications to dynamic trees and history independence
 In ACMSIAM Symposium on Discrete Algorithms (SODA
, 2004
"... We describe a machine model for automatically dynamizing static algorithms and apply it to historyindependent data structures. Static programs expressed in this model are dynamized automatically by keeping track of dependences between code and data in the form of a dynamic dependence graph. To study ..."
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Cited by 43 (26 self)
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We describe a machine model for automatically dynamizing static algorithms and apply it to historyindependent data structures. Static programs expressed in this model are dynamized automatically by keeping track of dependences between code and data in the form of a dynamic dependence graph. To study the performance of such automatically dynamized algorithms we present an analysis technique based on trace stability. As an example of the use of the model, we dynamize the Parallel Tree Contraction Algorithm of Miller and Reif to obtain a historyindependent data structure for the dynamic trees problem of Sleator and Tarjan. 1
Dynamic ThreeDimensional Linear Programming
, 1992
"... We perform linear programming optimizations on the intersection of k polyhedra in R³, represented by their outer recursive decompositions, in expected time O(k log k log n + √k log k log³ n). We use this result to derive efficient algorithms for dynamic linear programming problems in ..."
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Cited by 39 (5 self)
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We perform linear programming optimizations on the intersection of k polyhedra in R³, represented by their outer recursive decompositions, in expected time O(k log k log n + √k log k log³ n). We use this result to derive efficient algorithms for dynamic linear programming problems in which constraints are inserted and deleted, and queries must optimize specified objective functions. As an application, we describe an improved solution to the planar 2center and 2median problems.
Dynamic Euclidean Minimum Spanning Trees and Extrema of Binary Functions
, 1995
"... We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in amortized time O(n 1/2 log 2 n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in time O(n # ) per update. Our algorithm uses a novel ..."
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Cited by 38 (4 self)
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We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in amortized time O(n 1/2 log 2 n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in time O(n # ) per update. Our algorithm uses a novel construction, the ordered nearest neighbor path of a set of points. Our results generalize to higher dimensions, and to fully dynamic algorithms for maintaining minima of binary functions, including the diameter of a point set and the bichromatic farthest pair. 1 Introduction A dynamic geometric data structure is one that maintains the solution to some problem, defined on a geometric input such as a point set, as the input undergoes update operations such as insertions or deletions of single points. Dynamic algorithms have been studied for many geometric optimization problems, including closest pairs [7, 23, 25, 26], diameter [7, 26], width [4], convex hulls [15, 22], linear ...
An experimental analysis of selfadjusting computation
 In Proceedings of the ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI
, 2006
"... Selfadjusting computation uses a combination of dynamic dependence graphs and memoization to efficiently update the output of a program as the input changes incrementally or dynamically over time. Related work showed various theoretical results, indicating that the approach can be effective for a r ..."
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Cited by 35 (19 self)
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Selfadjusting computation uses a combination of dynamic dependence graphs and memoization to efficiently update the output of a program as the input changes incrementally or dynamically over time. Related work showed various theoretical results, indicating that the approach can be effective for a reasonably broad range of applications. In this article, we describe algorithms and implementation techniques to realize selfadjusting computation and present an experimental evaluation of the proposed approach on a variety of applications, ranging from simple list primitives to more sophisticated computational geometry algorithms. The results of the experiments show that the approach is effective in practice, often offering orders of magnitude speedup from recomputing the output from scratch. We believe this is the first experimental evidence that incremental computation of any type is effective in practice for a reasonably broad set of applications.
Dynamic Planar Convex Hull Operations in NearLogarithmic Amortized Time
 JOURNAL OF THE ACM
, 1999
"... We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P , such as membership and tangentfinding. Updates take O(log 1+" n) amortized time and queries take O(log n) time each, where n is the maximum size of ..."
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Cited by 35 (6 self)
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We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P , such as membership and tangentfinding. Updates take O(log 1+" n) amortized time and queries take O(log n) time each, where n is the maximum size of P and " is any fixed positive constant. For some advanced queries such as bridgefinding, both our bounds increase to O(log 3=2 n). The only previous fully dynamic solution was by Overmars and van Leeuwen from 1981 and required O(log 2 n) time per update. 1 Introduction Although the algorithmic study of convex hulls is as old as computational geometry itself, the basic problem of optimally maintaining the planar convex hull under insertions and deletions of points [30, 34] remains unsolved and has been regarded by some as one of the foremost open problems in the area [14, 26]. Besides its natural appeal, such a dynamic data structure has a wide range of applications, as it is often us...