30 research outputs found
Realizing degree sequences in parallel
A sequence of integers is a degree sequence if there exists a (simple) graph such that the components of are equal to the degrees of the vertices of . The graph is said to be a realization of . We provide an efficient parallel algorithm to realize . Before our result, it was not known if the problem of realizing is in
All-pairs min-cut in sparse networks
Algorithms are presented for the all-pairs min-cut problem in bounded tree-width, planar and sparse networks. The approach used is to preprocess the input -vertex network so that, afterwards, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute min-cuts subsequently. In particular, after an preprocessing of a bounded tree-width network, it is possible to find the value of a min-cut between any two vertices in constant time. This implies that for such networks the all-pairs min-cut problem can be solved in time . This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, , of the input network. The parameter varies between 1 and ; the algorithms perform well when . The value of a min-cut can be found in time and all-pairs min-cut can be solved in time for sparse networks. The corresponding running times4 for planar networks are and , respectively. The latter bounds depend on a result of independent interest: outerplanar networks have small ``mimicking'' networks which are also outerplanar
On the parallel complexity of degree sequence problems
We describe a robust and efficient implementation of the Bentley-Ottmann sweep line algorithm based on the LEDA library of efficient data types and algorithms. The program computes the planar graph induced by a set of straight line segments in the plane. The nodes of are all endpoints and all proper intersection points of segments in . The edges of are the maximal relatively open subsegments of segments in that contain no node of . All edges are directed from left to right or upwards. The algorithm runs in time where is the number of segments and is the number of vertices of the graph . The implementation uses exact arithmetic for the reliable realization of the geometric primitives and it uses floating point filters to reduce the overhead of exact arithmetic
All-Pairs Min-Cut in Sparse Networks
Algorithms are presented for the all-pairs min-cut problem in bounded tree-width, planar and sparse networks. The approach used is to preprocess the input -vertex network so that, afterwards, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute min-cuts subsequently. In particular, after an preprocessing of a bounded tree-width network, it is possible to find the value of a min-cut between any two vertices in constant time. This implies that for such networks the all-pairs min-cut problem can be solved in time . This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, , of the input network. The parameter varies between 1 and ; the algorithms perform well when . The value of a min-cut can be found in time and all-pairs min-cut can be solved in time for sparse networks. The corresponding running times for planar networks are and , respectively. The latter bounds depend on a result of independent interest: outerplanar networks have small ``mimicking'' networks which are also outerplanar
All-Pairs Min-Cut in Sparse Networks
Algorithms are presented for the all-pairs min-cut problem in bounded tree-width, planar and sparse networks. The approach used is to preprocess the input -vertex network so that, afterwards, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute min-cuts subsequently. In particular, after an preprocessing of a bounded tree-width network, it is possible to find the value of a min-cut between any two vertices in constant time. This implies that for such networks the all-pairs min-cut problem can be solved in time . This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, , of the input network. The parameter varies between 1 and ; the algorithms perform well when . The value of a min-cut can be found in time and all-pairs min-cut can be solved in time for sparse networks. The corresponding running times4 for planar networks are and , respectively. The latter bounds depend on a result of independent interest: outerplanar networks have small ``mimicking'' networks which are also outerplanar
Approximation algorithms for maximum two-dimensional pattern matching
We introduce the following optimization version of the classical pattern matching problem (referred to as the maximum pattern matching problem). Given a two-dimensional rectangular text and a 2- dimensional rectangular pattern find the maximum number of non- overlapping occurrences of the pattern in the text. Unlike the classical 2-dimensional pattern matching problem, the maximum pattern matching problem is NP - complete. We devise polynomial time approximation algorithms and approximation schemes for this problem. We also briefly discuss how the approximation algorithms can be extended to include a number of other variants of the problem
Engineering molecular recognition in alkane oxidation catalysed by cytochrome P450(cam)
Let <em>G</em> be a geometric graph whose vertex set <em>S</em> is a set of <em>n</em> points in ℝ<sup><em>d</em></sup>. The stretch factor of two distinct points <em>p</em> and <em>q</em> in <em>S</em> is the ratio of their shortest-path distance in <em>G</em> and their Euclidean distance. We consider the problem of approximating the average of the <em>n</em> choose 2 stretch factors determined by all pairs of points in <em>S</em>. We show that for paths, cycles, and trees, this average can be approximated, within a factor of 1+ε, in <em>O</em>(<em>n</em> polylog(<em>n</em>)) time. For plane graphs in ℝ<sup>2</sup>, we present a (2+ε)-approximation algorithm with running time <em>O</em>(<em>n</em><sup>5/3</sup>polylog(<em>n</em>)), and a (4+ε)-approximation algorithm with running time <em>O</em>(<em>n</em><sup>3/2</sup>polylog(<em>n</em>)). Finally, we show that, for any tree in ℝ<sup>2</sup>, the exact average of the squares of the <em>n</em> choose 2 stretch factors can be computed in <em>O</em>(<em>n</em><sup>11/6</sup>) time
Approximate Distance Oracles Revisited
Let G be a geometric t-spanner in E with n points and m edges, where t is a constant. We show that G can be preprocessed in O(m log n) time, such that (1+")-approximate shortest-path queries in G can be answered in O(1) time. The data structure uses O(n log n) space