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31
Subgraph Isomorphism in Planar Graphs and Related Problems
, 1999
"... We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small treewidth, and applying dynamic programming within each piece. The same methods can be used to ..."
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Cited by 113 (1 self)
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We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small treewidth, and applying dynamic programming within each piece. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.
Aspects of Unstructured Grids and FiniteVolume Solvers for the Euler and NavierStokes Equations (Part 4)
, 1995
"... this report, the model was tested on various subsonic and transonic flow problems: flat plates, airfoils, wakes, etc. The model consists of a single advectiondiffusion equation with source term for a field variable which is the product of turbulence Reynolds number and kinematic viscosity, e RT . ..."
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Cited by 56 (0 self)
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this report, the model was tested on various subsonic and transonic flow problems: flat plates, airfoils, wakes, etc. The model consists of a single advectiondiffusion equation with source term for a field variable which is the product of turbulence Reynolds number and kinematic viscosity, e RT . This variable is proportional to the eddy viscosity except very near a solid wall. The model equation is of the form: D( e RT ) Dt =(c ffl 2 f 2 (y + ) \Gamma c ffl 1 ) q e RT P +( + t oe R )r 2 ( e RT ) \Gamma 1 oe ffl (r t ) \Delta r( e RT ): (6:3:3) In this equation P is the production of turbulent kinetic energy and is related to the mean flow velocity rateofstrain and the kinematic eddy viscosity t . Equation (6.3.3) depends on distance to solid walls in two ways. First, the damping function f 2 appearing in equation (6.3.3) depends directly on distance to the wall (in wall units). Secondly, t depends on e R t and damping functions which require distance to the wall
Arboricity and Bipartite Subgraph Listing Algorithms
, 1994
"... In graphs of bounded arboricity, the total complexity of all maximal complete bipartite subgraphs is O(n). We describe a linear time algorithm to list such subgraphs. The arboricity bound is necessary: for any constant k and any n there exists an nvertex graph with O(n) edges and (n/ log n) k ..."
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Cited by 31 (2 self)
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In graphs of bounded arboricity, the total complexity of all maximal complete bipartite subgraphs is O(n). We describe a linear time algorithm to list such subgraphs. The arboricity bound is necessary: for any constant k and any n there exists an nvertex graph with O(n) edges and (n/ log n) k maximal complete bipartite subgraphs K k,# . # Work supported in part by NSF grant CCR9258355. 1 Introduction A number of graph algorithms depend on finding all subgraphs of a certain type in a larger graph. For instance, in interval or chordal graphs, a decomposition into maximal cliques is key; such a decomposition can be constructed in linear time [4, 17]. Optimal triangulation construction [3] and certain planar graph computations [8] require a listing of all triangles. Related subgraph isomorphism problems also occur in a wide variety of practical applications [2, 5, 12, 9, 13, 14, 19]. For planar graphs, or more generally for graphs of bounded arboricity, the problem of listing c...
Fully Dynamic Output Bounded Single Source Shortest Path Problem (Extended Abstract)
 In ACMSIAM Symposium on Discrete Algorithms
"... ) Abstract We consider the problem of maintaining the distances and the shortest paths from a single source in either a directed or an undirected graph with positive real edge weights, handling insertions, deletions and cost updates of edges. We propose fully dynamic algorithms with optimal space ..."
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Cited by 24 (4 self)
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) Abstract We consider the problem of maintaining the distances and the shortest paths from a single source in either a directed or an undirected graph with positive real edge weights, handling insertions, deletions and cost updates of edges. We propose fully dynamic algorithms with optimal space requirements and query time. The cost of update operations depends on the class of the considered graph and on the number of vertices that, due to an edge modification, either change their distance from the source or change their parent in the shortest path tree. In the case of graphs with bounded genus (including planar graphs), bounded degree graphs, bounded treewidth graphs and finearplanar graphs with bounded fi, the update procedures require O(log n) amortized time per vertex update, while for general graphs with n vertices and m edges they require O( p m log n) amortized time per vertex update. The solution is based on a dynamization of Dijkstra's algorithm [6] and requires simple ...
Fully Dynamic Shortest Paths and Negative Cycle Detection on Digraphs with Arbitrary Arc Weights
 In European Symposium on Algorithms
, 1998
"... We study the problem of maintaining the distances and the shortest paths from a source node in a directed graph with arbitrary arc weights, when weight updates of arcs are performed. We propose algorithms that work for any digraph and have optimal space requirements and query time. If a negative ..."
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Cited by 18 (2 self)
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We study the problem of maintaining the distances and the shortest paths from a source node in a directed graph with arbitrary arc weights, when weight updates of arcs are performed. We propose algorithms that work for any digraph and have optimal space requirements and query time. If a negativelength cycle is introduced during weightdecrease operations it is detected by the algorithms. The proposed algorithms explicitly deal with zerolength cycles. The cost of update operations depends on the class of the considered digraph and on the number of the output updates. We show that, if the digraph has a kbounded accounting function (as in the case of digraphs with genus, arboricity, degree, treewidth or pagenumber bounded by k) the update procedures require O(k \Delta n \Delta log n) worst case time. In the case of digraphs with n nodes and m arcs k = O( p m), and hence we obtain O( p m \Delta n \Delta log n) worst case time per operation, which is better for a factor o...
Small InducedUniversal Graphs and Compact Implicit Graph Representations
 In Proc. 43’rd annual IEEE Symp. on Foundations of Computer Science
, 2002
"... We show that there exists a graph G with n 2 nodes, where any forest with n nodes is a nodeinduced subgraph of G. Furthermore, the result implies existence of a graph with n nodes that contains all nnode graphs of fixed arboricity k as nodeinduced subgraphs. We provide a lower bound ..."
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Cited by 12 (0 self)
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We show that there exists a graph G with n 2 nodes, where any forest with n nodes is a nodeinduced subgraph of G. Furthermore, the result implies existence of a graph with n nodes that contains all nnode graphs of fixed arboricity k as nodeinduced subgraphs. We provide a lower bound of the size of such a graph. The upper bound is obtained through a simple labeling scheme for parent queries in rooted trees.
PTAS for geometric hitting set problems via local search
 In Symposium on Computational Geometry
, 2009
"... We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NPhard even for simple geometric objects like ..."
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Cited by 9 (0 self)
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We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NPhard even for simple geometric objects like unit disks in the plane. Therefore, unless P=NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are halfspaces in R 3 and when they are an radmissible set regions in the plane (this includes pseudodisks as they are 2admissible). Quite surprisingly, our algorithm is a very simple local search algorithm which iterates over local improvements only.
Connectivity, Graph Minors, and Subgraph Multiplicity
, 1993
"... It is well known that any planar graph contains at most O(n) complete subgraphs. We extend this to an exact characterization: G occurs O(n) times as a subgraph of any planar graph, if and only if G is threeconnected. We generalize these results to similarly characterize certain other minorclosed f ..."
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Cited by 6 (2 self)
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It is well known that any planar graph contains at most O(n) complete subgraphs. We extend this to an exact characterization: G occurs O(n) times as a subgraph of any planar graph, if and only if G is threeconnected. We generalize these results to similarly characterize certain other minorclosed families of graphs; in particular, G occurs O(n) times as a subgraph of the Kb,cfree graphs, b ≥ c and c ≤ 4, iff G is cconnected. Our results use a simple Ramseytheoretic lemma that may be of independent interest.
Shortest path queries in planar graphs in constant time
 In STOC’03
, 2003
"... We present a new algorithm for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(V ) time a data structure, which allows to check in O(1) time whether two given vertices are distant by at most k in G and if so ..."
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Cited by 6 (2 self)
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We present a new algorithm for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(V ) time a data structure, which allows to check in O(1) time whether two given vertices are distant by at most k in G and if so a shortest path between them is returned. This significantly improves the previous result of D. Eppstein [5] where after a linear preprocessing the queries are answered in O(log V ) time. Our approach can be applied to compute the girth of a planar graph and a corresponding shortest cycle in O(V ) time provided that the constant bound on the girth is known. Our results can be easily generalized to other wide classes of graphs – for instance we can take graphs embeddable in a surface of bounded genus or graphs of bounded treewidth. Categories and Subject Descriptors G.2.2 [Discrete Mathematics]: Graph Theory—graph algorithms, path and circuit problems