Results 1  10
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18
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.
On Levels in Arrangements of Curves
 Proc. 41st IEEE
, 2002
"... Analyzing the worstcase complexity of the klevel in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk 9 2 s 3 )) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously ..."
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Cited by 23 (3 self)
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Analyzing the worstcase complexity of the klevel in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk 9 2 s 3 )) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously, nontrivial results were known only for the case s = 1 and s = 2. We also improve the earlier bound for pseudoparabolas (curves that pairwise intersect at most twice) to O(nk k). The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudoparabolas into pseudosegments, as well as a new observation for cutting pseudosegments into pieces that can be extended to pseudolines. We mention applications to parametric and kinetic minimum spanning trees.
Approximate Shape Fitting via Linearization
 In Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci
, 2001
"... Shape fitting is a fundamental optimization problem in computer science. In this paper, we present a general and unified technique for solving a certain family of such problems. Given a point set P in R d, this technique can be used to εapproximate: (i) the minwidth annulus and shell that contains ..."
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Cited by 14 (7 self)
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Shape fitting is a fundamental optimization problem in computer science. In this paper, we present a general and unified technique for solving a certain family of such problems. Given a point set P in R d, this technique can be used to εapproximate: (i) the minwidth annulus and shell that contains P, (ii) minimum width cylindrical shell containing P, (iii) diameter, width, minimum volume bounding box of P, and (iv) all the previous measures for the case the points are moving. The running time of the resulting algorithms is O(n + 1/ε c), where c is a constant that depends on the problem at hand. Our new general technique enable us to solve those problems without resorting to a careful and painful case by case analysis, as was previously done for those problems. Furthermore, for several of those problems our results are considerably simpler and faster than what was previously known. In particular, for the minimum width cylindrical shell problem, our solution is the first algorithm whose running time is subquadratic in n. (In fact we get running time linear in n.) 1
On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences
 In Proc. 44th IEEE Sympos. Found. Comput. Sci
, 2003
"... We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the klevel has subquadratic (O(n 2s )) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which provided a ..."
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Cited by 9 (2 self)
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We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the klevel has subquadratic (O(n 2s )) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which provided a weaker bound for a restricted class of curves (graphs of degrees polynomials) only. When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved klevel results for most of the curve families studied earlier, including a nearO(n ) bound for parabolas.
Soft kinetic data structures
 In SODA ’01: Proceedings of the twelfth annual ACMSIAM symposium on Discrete algorithms
"... We introduce the framework of soft kinetic data structures (SKDS). A soft kinetic data structure is an approximate data structure that can be used to answer queries on a set of moving objects with unpredictable motion. We analyze the quality of a soft kinetic data structure by giving a competitive a ..."
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Cited by 8 (0 self)
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We introduce the framework of soft kinetic data structures (SKDS). A soft kinetic data structure is an approximate data structure that can be used to answer queries on a set of moving objects with unpredictable motion. We analyze the quality of a soft kinetic data structure by giving a competitive analysis with respect to the dynamics of the system. We illustrate our approach by presenting soft kinetic data structures for maintaining classical data structures: sorted arrays, balanced search trees, heaps, and range trees. We also describe soft kinetic data structures for maintaining the Euclidean minimum spanning trees. 1 Introduction. The need of storing and processing continuously moving data arises in a broad variety of applications, including
Multiplesource shortest paths in embedded graphs
, 2012
"... Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of ..."
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Cited by 7 (5 self)
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Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)time algorithm of Klein [Multiplesource shortest paths in planar graphs. In Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms, 2005] for multiplesource shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortestpath tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in O(g² n log n) time.
ALGORITHMS FOR SMOOTH AND DEFORMABLE SURFACES IN 3D
, 2002
"... In this thesis, we study the skin surface as a new paradigm for the deformable surfaces. The skin surface handles deformation and topology changes robustly, supported by the underlying structure of Delaunay triangulations and alpha shapes. The surface serves as a deformable manifold in various disc ..."
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Cited by 4 (2 self)
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In this thesis, we study the skin surface as a new paradigm for the deformable surfaces. The skin surface handles deformation and topology changes robustly, supported by the underlying structure of Delaunay triangulations and alpha shapes. The surface serves as a deformable manifold in various disciplines, such as computer graphics, molecular modeling, and mechanical engineering. We develop an algorithm and software for the construction and visualization of the skin surface in 3D in various ways, namely, a parametric representation, static and dynamic triangulations. The triangulation algorithm is guaranteed to terminate with a high quality triangle mesh. In our investigation, geometric properties of the skin serve as the foundation of our proofs and insights for the algorithms. The proofs can be extended to the meshing of other low degree surfaces, such as NURBS. The surfaces created by the software bring stability in finite element methods and visualization of molecular structures to scientists.
Maintaining Approximate Minimum Steiner Tree and kcenter for Mobile Agents in a Sensor Network
"... Abstract—We study the problem of maintaining group communication between m mobile agents, tracked and helped by n static networked sensors. We develop algorithms to maintain a O(lg n)approximation to the minimum Steiner tree of the mobile agents such that the maintenance message cost is on average ..."
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Cited by 1 (1 self)
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Abstract—We study the problem of maintaining group communication between m mobile agents, tracked and helped by n static networked sensors. We develop algorithms to maintain a O(lg n)approximation to the minimum Steiner tree of the mobile agents such that the maintenance message cost is on average O(lg n) per each hop an agent moves. The key idea is to extract a ‘hierarchical wellseparated tree (HST) ’ on the sensor nodes such that the tree distance approximates the sensor network hop distance by a factor of O(lg n). We then prove that maintaining the subtree of the mobile agents on the HST uses logarithmic messages per hop movement. With the HST we can also maintain O(lg n) approximate kcenter for the mobile agents with the same message cost. Both the minimum Steiner tree and the kcenter problems are NPhard and our algorithms are the first efficient algorithms for maintaining approximate solutions in a distributed setting. I.
Competitive Maintenance of Minimum Spanning Trees in Dynamic Graphs ⋆ Extended abstract
"... Abstract. We consider the problem of maintaining a minimum spanning tree within a graph with dynamically changing edge weights. An online algorithm is confronted with an input sequence of edge weight changes and has to choose a minimum spanning tree after each such change in the graph. The task of t ..."
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Cited by 1 (0 self)
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Abstract. We consider the problem of maintaining a minimum spanning tree within a graph with dynamically changing edge weights. An online algorithm is confronted with an input sequence of edge weight changes and has to choose a minimum spanning tree after each such change in the graph. The task of the algorithm is to perform as few changes in its minimum spanning tree as possible. We compare the number of changes in the minimum spanning tree produced by an online algorithm and that produced by an optimal offline algorithm. The number of changes is counted in the number of edges changed between spanning trees in consecutive rounds. For any graph with n vertices we provide a deterministic algorithm achieving a competitive ratio of O(n 2). We show that this result is optimal up to a constant. Furthermore we give a lower bound for randomized algorithms of Ω(log n). We show a randomized algorithm achieving a competitive ratio of O(n log n) for general graphs and O(log n) for planar graphs. 1