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195
Geometric Range Searching and Its Relatives
 CONTEMPORARY MATHEMATICS
"... ... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems. ..."
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Cited by 256 (40 self)
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... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems.
Approximation Algorithms for Disjoint Paths Problems
, 1996
"... The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for w ..."
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Cited by 140 (0 self)
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The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in highspeed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.
Approximating Maximum Independent Sets by Excluding Subgraphs
 BIT
, 1992
"... An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known to O(n/(log n)²). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strong simultaneous performance guarantee ..."
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Cited by 137 (10 self)
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An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known to O(n/(log n)²). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strong simultaneous performance guarantee for the clique and coloring problems. The framework
Art gallery and illumination problems
 In Handbook on Computational Geometry, Elsevier Science Publishers, J.R. Sack and
, 2000
"... How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O’Rourke pub ..."
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Cited by 86 (3 self)
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How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O’Rourke published his book Art Gallery Theorems and Algorithms which has further fueled this area of research. The present book is being written almost 10 years since the publication of O’Rourke’s book, and the need for an uptodate manuscript on Art Gallery or Illumination Problems is evident. Some important open problems stated in O’Rourke’s book, such as... have been solved. New directions of research have since been investigated, including: watchman routes, floodlight illumination problems, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this book, we study these results and try to give a complete
Range Searching
, 1996
"... Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , an ..."
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Cited by 70 (1 self)
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Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.
Iterated Nearest Neighbors and Finding Minimal Polytopes
, 1994
"... Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based o ..."
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Cited by 56 (6 self)
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Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convex kvertex polygons and polytopes, and to improvemany previous results. Weachievemany of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n+kn). We also demonstrate a related technique for finding minimum area kpoint sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.
Approximations of Weighted Independent Set and Hereditary Subset Problems
 JOURNAL OF GRAPH ALGORITHMS AND APPLICATIONS
, 2000
"... The focus of this study is to clarify the approximability of weighted versions of the maximum independent set problem. In particular, we report improved performance ratios in boundeddegree graphs, inductive graphs, and general graphs, as well as for the unweighted problem in sparse graphs. Wher ..."
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Cited by 53 (6 self)
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The focus of this study is to clarify the approximability of weighted versions of the maximum independent set problem. In particular, we report improved performance ratios in boundeddegree graphs, inductive graphs, and general graphs, as well as for the unweighted problem in sparse graphs. Where possible, the techniques are applied to related hereditary subgraph and subset problem, obtaining ratios better than previously reported for e.g. Weighted Set Packing, Longest Common Subsequence, and Independent Set in hypergraphs.
Approximating the independence number via the ϑfunction
, 1994
"... We study the relationship between the independence number of a graph and its semidefinite relaxation, the Lov'asz `function. We deduce an improved approximation algorithm for the independence number. If a graph on n vertices has an independence number n=k + m, for some fixed integer k 3 and some ..."
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Cited by 31 (5 self)
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We study the relationship between the independence number of a graph and its semidefinite relaxation, the Lov'asz `function. We deduce an improved approximation algorithm for the independence number. If a graph on n vertices has an independence number n=k + m, for some fixed integer k 3 and some m ? 0, the algorithm finds, in random polynomial time, an independent set of size ~ \Omega\Gamma m 3=(k+1) ). This is the first improvement upon the Ramsey Theory based algorithm of Boppana and Halldorsson that finds an independent set of size\Omega\Gamma m 1=(k\Gamma1) ) in such a graph. The algorithm is based on semidefinite programming, some properties of the `function, and the recent algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a graph. If the `function of an n vertex graph is at least Mn 1\Gamma2=h , for some absolute constant M , we describe another, related algorithm that finds an independent set of size h. Finally, while it is e...
On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 31 (19 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
On Disjoint Cycles
 International Journal of Foundations of Computer Science
, 1990
"... It is shown, that for each constant k _ 1, the following problems can be solved in O(n) time: given a graph G, determine whether G has k vertex disjoint cycles, determine whether G has k edge disjoint cycles, determine whether G has a feedback vertex set of size _ k. Also, every class G, that is ..."
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Cited by 29 (4 self)
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It is shown, that for each constant k _ 1, the following problems can be solved in O(n) time: given a graph G, determine whether G has k vertex disjoint cycles, determine whether G has k edge disjoint cycles, determine whether G has a feedback vertex set of size _ k. Also, every class G, that is closed under minor taking, or that is closed under immersion taking, and that does not contain the graph formed by taking the disjoint union of k copies of Ks, has an O(n) membership test algorithm.