Results 1  10
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23
Dynamic Generators of Topologically Embedded Graphs
, 2003
"... We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(logn + logg(loglogg) 3) per update on a surface of genus g ..."
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Cited by 36 (1 self)
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We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(logn + logg(loglogg) 3) per update on a surface of genus g; we can also test orientability of the surface in the same time, and maintain the minimum and maximum spanning tree of the graph in time O(log n + log 4 g) per update. Our data structure allows edge insertion and deletion as well as the dual operations; these operations may implicitly change the genus of the embedding surface. We apply similar ideas to improve the constant factor in a separator theorem for lowgenus graphs, and to find in linear time a treedecomposition of lowgenus lowdiameter graphs.
Homology flows, cohomology cuts
 ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2009
"... We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time fo ..."
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Cited by 16 (6 self)
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We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time for integer capacities that sum to C, or in (g log n) O(g) n time for real capacities. Except for the special case of planar graphs, for which an O(n log n)time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surfaceembedded graphs follow from algorithms for general sparse graphs. Our key insight is to optimize the relative homology class of the flow, rather than directly optimizing the flow itself. A dual formulation of our algorithm computes the minimumcost cycle or circulation in a given (real or integer) homology class.
Improved Algorithms for Dynamic Shortest Paths
, 2000
"... We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge ca ..."
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Cited by 15 (3 self)
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We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. In the case of outerplanar digraphs, our data structures can be updated after any such change in only logarithmic time. A distance query is also answered in logarithmic time. In the case of planar digraphs, we give an interesting tradeoff between preprocessing, query, and update times depending on the value of a certain topological parameter of the graph. Our results can be extended to nvertex digraphs of genus O(n1−ε) for any ε>0.
Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time
, 2008
"... Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bou ..."
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Cited by 10 (3 self)
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Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bound for infinite precision models. As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a threedimensional point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higherdimensional extensions and applications are also discussed. Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this longstanding limitation, answering, for example, a question of Willard (SODA’92).
Persistent Predecessor Search and Orthogonal Point Location on the Word RAM
"... We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of int ..."
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Cited by 6 (3 self)
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We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of integers in {1,..., U} under an arbitrary sequence of n insertions and deletions, with O(log log U) expected query time and expected amortized update time, and O(n) space. The query bound is optimal in U for linearspace structures and improves previous nearO((log log U) 2) methods. The same method solves a fundamental problem from computational geometry: point location in orthogonal planar subdivisions (where edges are vertical or horizontal). We obtain the first static data structure achieving O(log log U) worstcase query time and linear space. This result is again optimal in U for linearspace structures and improves the previous O((log log U) 2) method by de Berg, Snoeyink, and van Kreveld (1992). The same result also holds for higherdimensional subdivisions that are orthogonal binary space partitions, and for certain nonorthogonal planar subdivisions such as triangulations without small angles. Many geometric applications follow, including improved query times for orthogonal range reporting for dimensions ≥ 3 on the RAM. Our key technique is an interesting new vanEmdeBoas–style recursion that alternates between two strategies, both quite simple.
Counting Models for 2SAT and 3SAT Formulae
"... We here present algorithms for counting models and maxweight models for 2sat and 3sat formulae. They use polynomial space and run in O(1.2561^n) and O(1.6737^n) time, respectively, where n is the number of variables. This is faster than the previously best algorithms for counting nonweighted model ..."
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Cited by 5 (0 self)
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We here present algorithms for counting models and maxweight models for 2sat and 3sat formulae. They use polynomial space and run in O(1.2561^n) and O(1.6737^n) time, respectively, where n is the number of variables. This is faster than the previously best algorithms for counting nonweighted models for 2sat and 3sat, which run in O(1.3247^n) and O(1.6894^n) time, respectively. In order to prove these time bounds, we develop new measures of formula complexity, allowing us to conveniently analyze the eoeects of certain factors with a large impact on the total running time. We also provide an algorithm for the restricted case of separable 2sat formulae, with fast running times for wellstudied input classes. For all three algorithms we present interesting applications, such as computing the permanent of sparse 0/1 matrices.
Collective Tree Spanners in Graphs with Bounded Parameters
 ALGORITHMICA
, 2006
"... In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded treewidth, graphs with bounded cliquewidth, and graphs with bounded chordality. We say that a graph G = (V, E) admits a system of μ colle ..."
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Cited by 4 (4 self)
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In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded treewidth, graphs with bounded cliquewidth, and graphs with bounded chordality. We say that a graph G = (V, E) admits a system of μ collective additive tree rspanners if there is a system T (G) of at most μ spanning trees of G such that for any two vertices x,y of G a spanning tree T ∈ T (G) exists such that dT (x, y) ≤ dG(x, y) + r. We describe a general method for constructing a “small” system of collective additive tree rspanners with small values of r for “well ” decomposable graphs, and as a byproduct show (among other results) that any weighted planar graph admits a system of O ( √ n) collective additive tree 0spanners, any weighted graph with treewidth at most k − 1 admits a system of k log 2 n collective additive tree 0spanners, any weighted graph with cliquewidth at most k admits a system of k log 3/2 n collective additive tree (2w)spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log 2 n collective additive tree (2⌊c/2⌋w)spanners and a system of 4 log 2 n collective additive tree (2(⌊c/3⌋+1)w)spanners (here, w is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4 log 2 n collective additive tree (2w)spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs. Results of this paper were partially presented at the ISAAC’05 conference [14].
Approximate Shortest Path Queries on Weighted Polyhedral Surfaces
"... We consider the classical geometric problem of determining shortest paths between pairs of points lying on a weighted polyhedral surface P consisting of n triangular faces. We present query algorithms that compute approximate distances and/or approximate (weighted) shortest paths. Our algorithm take ..."
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Cited by 4 (1 self)
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We consider the classical geometric problem of determining shortest paths between pairs of points lying on a weighted polyhedral surface P consisting of n triangular faces. We present query algorithms that compute approximate distances and/or approximate (weighted) shortest paths. Our algorithm takes as input an approximation parameter ε ∈ (0, 1) and a query time parameter q and builds a data structure which is then used for answering ǫapproximate distance queries in O(q) time. This algorithm is source point independent and improves significantly on the best previous solution. For the case where one of the query points is fixed we build a data structure that can answer ǫapproximate distance queries to any query point in P in O(log 1) time. This is an improveε ment upon the previously known solution for the Euclidean fixed source query problem. Our algorithm also generalizes the setting from previously studied unweighted polyhedral to weighted polyhedral surfaces of arbitrary genus. Our solutions are based on a novel graph separator algorithm introduced here which extends and generalizes previously known separator algorithms.
Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces
, 2008
"... We consider the well known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P, where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the subpaths w ..."
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Cited by 3 (0 self)
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We consider the well known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P, where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the subpaths within each face of P. We present query algorithms that compute approximate distances and/or approximate shortest paths on P. Our allpairs query algorithms take as input an approximation parameter ε ∈ (0,1) and a query time parameter q, in a certain range, and builds a data structure APQ(P,ε;q), which is then used for answering εapproximate distance queries in O(q) time. As a building block of the APQ(P,ε;q) data structure, we develop a single source query data structure SSQ(a;P,ε) that can answer εapproximate distance queries from a fixed point a to any query point on P in logarithmic time. Our algorithms answer shortest path queries in weighted surfaces, which is an important extension, both theoretically and practically, to the extensively studied Euclidean distance case. In addition, our algorithms improve upon previously known query algorithms for shortest paths on surfaces. The algorithms are based on a novel graph separator algorithm introduced and analyzed here, which extends and generalizes previously known separator algorithms.