All-Pairs Min-Cut in Sparse Networks

Algorithms are presented for the all-pairs min-cut problem in bounded treewidth, planar, and sparse networks. The approach used is to preprocess the input n-vertex network so that afterward, 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 Onlog Ž n. 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 Ž 2 such networks the all-pairs min-cut problem can be solved in time On.. 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 �Ž. n; the algorithms perform well when � � on. Ž. The value Ž 2 of a min-cut can be found in time On� � log �. and all-pairs min-cut can be Ž 2 4 solved in time On � � log �. for sparse networks. The corresponding runnin

Approximation Algorithms for Maximum Two-Dimensional Pattern Matching

Dessmark y Andrzej Lingas z Madhav V. Marathe

A signature based approach to regularity extraction

Regularity extraction is an important step in the design ow of datapath-dominated circuits. This paper outlines a new method that automatically extracts regular structures from the netlist. The method is general enough to handle two types of designs: designs with structured cluster information for a portion of the datapath components that are identi ed at the HDL level; and designs with no such structured cluster information. The method analyzes the circuit connectivity and uses signature based approaches to recognize regularity.

A correctness certificate for the Stoer–Wagner min-cut algorithm

The Stoer–Wagner algorithm computes a minimum cut in a weighted undirected graph. The algorithm works in n − 1 phases, where n is the number of nodes of G. Each phase takes time O(m+n log n), where m is the number of edges of G, and computes a pair of vertices s and t and a minimum cut separating s and t. We show how to extend the algorithm such that each phase also computes a maximum flow from s to t. The flow is computed in O(m) additional time and certifies the cut computed in the phase

Realizing degree sequences in parallel

A sequence d of integers is a degree sequence if there exists a (simple) graph G such that the components of d are equal to the degrees of the vertices of G. The graph G is said to be a realization of d. We provide an efficient parallel algorithm to realize d; the algorithm runs in O(log n) me using O(n + m) CRCW PRAM processors, where n and m are the number of vertices and edges G. Before our result, it was not known if the problem of realizing d is in NC

Efficient computation of implicit representations of sparse graphs

The problem of finding an implicit representation for a graph such that vertex adjacency can be tested quickly is fundamental to all graph algorithms. In particular, it is possible to represent sparse graphs on n vertices using O(n) space such that vertex adjacency is tested in O(1) time. We show here how to construct such a representation efficiently by providing simple and optimal algorithms, both in a sequential and a parallel setting. Our sequential algorithm runs in O(n) time. The parallel algorithm runs in O(log n) time using O(n=log n) CRCW PRAM processors, or in O(log n log n) time using O(n = log n lo

Efficient Computation of Implicit Representations of Sparse Graphs

The problem of finding an implicit representation for a graph such that vertex adjacency can be tested quickly is fundamental to all graph algorithms. In particular, it is possible to represent sparse graphs on n vertices using O(n) space such that vertex adjacency is tested in O(1) time. We show here how to construct such a representation efficiently by providing simple and optimal algorithms, both in a sequential and a parallel setting. Our sequential algorithm runs in O(n) time. The parallel algorithm runs in O(logn) time using O(n=logn) CRCW PRAM processors, or in O(logn log n) time using O(n= log n log n) EREW PRAM processors. Previous results for this problem are based on matroid partitioning and thus have a high complexity. Keywords: Implicit Representation, Sparse Graphs, Arboricity, Graph Algorithms, Parallel Computation. This work was partially supported by the EU ESPRIT Basic Research Action No. 7141 (ALCOM II). y This research was done while the author was with th..

All-Pairs Min-Cut in Sparse Networks

. Algorithms for the all-pairs min-cut problem in bounded tree-width and sparse networks are presented. The approach used is to preprocess the input network so that, afterwards, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff between the preprocessing time and the time taken to compute min-cuts subsequently is shown. In particular, after O(n log n) 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 O(n 2 ). This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse networks. The running times depend upon a topological property fl of the input network. The parameter fl varies between 1 and \Theta(n); the algorithms perform well when fl = o(n). The value of a min-cut can be found in time O(n + fl 2 ..