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41
Faster ShortestPath Algorithms for Planar Graphs
 STOC 94
, 1994
"... We give a lineartime algorithm for singlesource shortest paths in planar graphs with nonnegative edgelengths. Our algorithm also yields a lineartime algorithm for maximum flow in a planar graph with the source and sink on the same face. The previous best algorithms for these problems required\O ..."
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Cited by 160 (13 self)
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We give a lineartime algorithm for singlesource shortest paths in planar graphs with nonnegative edgelengths. Our algorithm also yields a lineartime algorithm for maximum flow in a planar graph with the source and sink on the same face. The previous best algorithms for these problems required\Omega\Gamma n p log n) time where n is the number of nodes in the input graph. For the case where negative edgelengths are allowed, we give an algorithm requiring O(n 4=3 log nL) time, where L is the absolute value of the most negative length. Previous algorithms for shortest paths with negative edgelengths required \Omega\Gamma n 3=2 ) time. Our shortestpath algorithm yields an O(n 4=3 log n)time algorithm for finding a perfect matching in a planar bipartite graph. A similar improvement is obtained for maximum flow in a directed planar graph.
Dynamic and efficient key management for access hierarchies
 In Proceedings of the ACM Conference on Computer and Communications Security
, 2005
"... Hierarchies arise in the context of access control whenever the user population can be modeled as a set of partially ordered classes (represented as a directed graph). A user with access privileges for a class obtains access to objects stored at that class and all descendant classes in the hierarchy ..."
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Cited by 64 (8 self)
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Hierarchies arise in the context of access control whenever the user population can be modeled as a set of partially ordered classes (represented as a directed graph). A user with access privileges for a class obtains access to objects stored at that class and all descendant classes in the hierarchy. The problem of key management for such hierarchies then consists of assigning a key to each class in the hierarchy so that keys for descendant classes can be obtained via efficient key derivation. We propose a solution to this problem with the following properties: (1) the space complexity of the public information is the same as that of storing the hierarchy; (2) the private information at a class consists of a single key associated with that class; (3) updates (i.e., revocations and additions) are handled locally in the hierarchy; (4) the scheme is provably secure against collusion; and (5) each node can derive the key of any of its descendant with a number of symmetrickey operations bounded by the length of the path between the nodes. Whereas many previous schemes had some of these properties, ours is the first that satisfies all of them. The security of our scheme is based on pseudorandom functions, without reliance on the Random Oracle Model. 18 Portions of this work were supported by Grants IIS0325345 and CNS06274488 from the
A Randomized Parallel 3D Convex Hull Algorithm For Coarse Grained Multicomputers
 In Proc. ACM Symp. on Parallel Algorithms and Architectures
, 1995
"... We present a randomized parallel algorithm for constructing the 3D convex hull on a generic pprocessor coarse grained multicomputer with arbitrary interconection network and n=p local memory per processor, where n=p p 2+ffl (for some arbitrarily small ffl ? 0). For any given set of n points in ..."
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Cited by 49 (11 self)
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We present a randomized parallel algorithm for constructing the 3D convex hull on a generic pprocessor coarse grained multicomputer with arbitrary interconection network and n=p local memory per processor, where n=p p 2+ffl (for some arbitrarily small ffl ? 0). For any given set of n points in 3space, the algorithm computes the 3D convex hull, with high probaility, in O( n log n p ) local computation time and O(1) communication phases with at most O(n=p) data sent/received by each processor. That is, with high probability, the algorithm computes the 3D convex hull of an arbitrary point set in time O( n logn p + \Gamma n;p ), where \Gamma n;p denotes the time complexity of one communication phase. The assumption n p p 2+ffl implies a coarse grained, limited parallelism, model which is applicable to most commercially available multiprocessors. In the terminology of the BSP model, our algorithm requires, with high probability, O(1) supersteps, synchronization period L = \Th...
Planar spanners and approximate shortest path queries among obstacles
 in the plane, Proc. 4th European Sympos. Algorithms
, 1996
"... Abstract. We consider the problem of finding an obstacleavoiding path between two points s and t in the plane, amidst a set of disjoint polygonal obstacles with a total of n vertices. The length of this path should be within a small constant factor c of the length of the shortest possible obstacle ..."
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Cited by 40 (14 self)
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Abstract. We consider the problem of finding an obstacleavoiding path between two points s and t in the plane, amidst a set of disjoint polygonal obstacles with a total of n vertices. The length of this path should be within a small constant factor c of the length of the shortest possible obstacleavoiding st path measured in the Lvmetric. Such an approximate shortest path is called a cshort path, or a short path with stretch]actor c. The goal is to preprocess the obstaclescattered plane by creating an efficient data structure that enables fast reporting of a cshort path (or its length). In this paper, we give a family of algorithms for the above problem that achieve an interesting tradeoff between the stretch factor, the query time and the preprocessing bounds. Our main results are algorithms that achieve logarithmic length query time, after subquadratic time and space preprocessing. 1
On External Memory MST, SSSP and Multiway Planar Graph Separation (Extended Abstract)
, 2000
"... Recently external memory graph algorithms have received considerable attention because massive graphs arise naturally in many applications involving massive data sets. Even though a large number of I/Oefficient graph algorithms have been developed, a number of fundamental problems still remain ..."
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Cited by 33 (11 self)
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Recently external memory graph algorithms have received considerable attention because massive graphs arise naturally in many applications involving massive data sets. Even though a large number of I/Oefficient graph algorithms have been developed, a number of fundamental problems still remain open. In this paper we develop improved algorithms for the problem of computing a minimum spanning tree of a general graph G = (V; E), as well as new algorithms for the single source shortest paths and the multiway graph separation problems on planar graphs.
Parametric and Kinetic Minimum Spanning Trees
"... We consider the parametric minimum spanning treeproblem, in which we are given a graph with edge weights that are linear functions of a parameter * and wish tocompute the sequence of minimum spanning trees generated as * varies. We also consider the kinetic minimumspanning tree problem, in which * r ..."
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Cited by 29 (7 self)
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We consider the parametric minimum spanning treeproblem, in which we are given a graph with edge weights that are linear functions of a parameter * and wish tocompute the sequence of minimum spanning trees generated as * varies. We also consider the kinetic minimumspanning tree problem, in which * represents time and the graph is subject in addition to changes such as edge insertions, deletions, and modifications of the weight functions as time progresses. We solve both problems in time O(n2=3 log4=3 n) per combinatorial change in the tree (or randomized O(n2=3 log n) per change). Our time bounds reduce to O(n1=2 log3=2 n) per change (O(n1=2 log n) randomized) for planar graphs or other minorclosed families of graphs, and O(n1=4 log3=2 n) per change (O(n1=4 log n) randomized) for planar graphs with weight changes but no insertions or deletions.
On externalmemory MST, SSSP and multiway planar graph separation
 In Proc. 8th Scandinavian Workshop on Algorithmic Theory, volume 1851 of LNCS
, 2000
"... Recently external memory graph algorithms have received considerable attention because massive graphs arise naturally in many applications involving massive data sets. Even though a large number of I/Oefficient graph algorithms have been developed, a number of fundamental problems still remain open ..."
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Cited by 24 (2 self)
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Recently external memory graph algorithms have received considerable attention because massive graphs arise naturally in many applications involving massive data sets. Even though a large number of I/Oefficient graph algorithms have been developed, a number of fundamental problems still remain open. In this paper we develop an improved algorithm for the problem of computing a minimum spanning tree of a general graph, as well as new algorithms for the single source shortest paths and the multiway graph separation problems on planar graphs.
Randomized Parallel Algorithms For Trapezoidal Diagrams
, 1992
"... We describe randomized parallel algorithms for building trapezoidal diagrams of line segments in the plane. The algorithms are designed for a CRCW PRAM. For general segments, we give an algorithm requiring optimal O(A + n log n) expected work and optimal O(logn) time, where A is the number of inters ..."
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Cited by 22 (0 self)
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We describe randomized parallel algorithms for building trapezoidal diagrams of line segments in the plane. The algorithms are designed for a CRCW PRAM. For general segments, we give an algorithm requiring optimal O(A + n log n) expected work and optimal O(logn) time, where A is the number of intersecting pairs of segments. If the segments form a simple chain, we give an algorithm requiring optimal O(n) expected work and O(logn log log n log n) expected time a , and a simpler algorithm requiring O(n log n) expected work. The serial algorithm corresponding to the latter is among the simplest known algorithms requiring O(n log n) expected operations. For a set of segments forming K chains, we give an algorithm requiring O(A + n log n + K log n) expected work and O(logn log log n log n) expected time. The parallel time bounds require the assumption that enough processors are available, with processor allocations every log n steps. Keywords: randomized, parallel, trapez...
Linear Algorithms for Partitioning Embedded Graphs of Bounded Genus
 SIAM Journal of Discrete Mathematics
, 1996
"... This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive " we show that any nvertex graph G of genus g can be divided in O(n + g) time into components whose sizes do not exceed "n by removing a set C of O( ..."
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Cited by 21 (4 self)
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This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive " we show that any nvertex graph G of genus g can be divided in O(n + g) time into components whose sizes do not exceed "n by removing a set C of O( p (g + 1=")n) vertices. Our result improves the best previous ones with respect to the size of C and the time complexity of the algorithm. Moreover, we show that one can cut off from G a piece of no more than (1 \Gamma ")n vertices by removing a set of O( p n"(g" + 1) vertices. Both results are optimal up to a constant factor. Keywords: graph separator, graph genus, algorithm, divideandconquer, topological graph theory AMS(MOS) subject classifications: 05C10, 05C85, 68R10 1 Bulgarian Academy of Sci., CICT, G.Bonchev 25A, 1113 Sofia, Bulgaria 2 Department of Comp.Sci.,Rice University, P.O.Box 1892, Houston, Texas 77251, USA 1 Introduction Let S be a class of graphs closed under t...
Two and ThreeDimensional Point Location in Rectangular Subdivisions
 Journal of Algorithms
, 1995
"... We apply van Emde Boastype stratified trees to point location problems in rectangular subdivisions in 2 and 3 dimensions. In a subdivision with n rectangles having integer coordinates from [0; U \Gamma 1], we locate an integer query point in O((log log U ) d ) query time using O(n) space when d ..."
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Cited by 20 (1 self)
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We apply van Emde Boastype stratified trees to point location problems in rectangular subdivisions in 2 and 3 dimensions. In a subdivision with n rectangles having integer coordinates from [0; U \Gamma 1], we locate an integer query point in O((log log U ) d ) query time using O(n) space when d 2 or O(n log log U ) space when d = 3. Applications and extensions of this "fixed universe" approach include spatial point location using logarithmic time and linear space in rectilinear subdivisions having arbitrary coordinates, point location in coriented polygons or fat triangles in the plane, point location in subdivisions of space into "fat prisms," and vertical ray shooting among horizontal "fat objects." Like other results on stratified trees, our algorithms run on a RAM model and make use of perfect hashing. 1 Introduction The point location problemwhich seeks to preprocess a set of disjoint geometric objects to be able to determine quickly which object contains a query point...