Fast RNC and NC algorithms for maximal path sets

Fast RNC and NC Algorithms for Maximal Path Sets

Extending NC and RNC Algorithms

A technique is presented by which NC and RNC algorithms for some problems can be extended into NC and RNC algorithms, respectively, that solve more general parametric problems. The technique is demonstrated on explicit bounded degree circuits. Applications include parametric extensions of the shortest-path and spanning-tree problems and, in particular, the minimum-ratio-cycle problem, showing a...

An Incremental RNC Algorithm for Generating All Maximal Independent Sets in Hypergraphs of Bounded Dimension∗

We show that for hypergraphs of bounded edge size, the problem of extending a given list of maximal independent sets is NC-reducible to the computation of an arbitrary maximal independent set for an induced sub-hypergraph. The latter problem is known to be in RNC. In particular, our reduction yields an incremental RNC dualization algorithm for hypergraphs of bounded edge size, a problem previou...

RNC Algorithms for theUniform Generation of Combinatorial

We describe several RNC algorithms for generating graphs and subgraphs uniformly at random. For example, unla-belled undirected graphs are generated in O(lg 3 n) time using O "n 2 lg 3 n processors if their number is known in advance and in O(lg n) time using O "n 2 lg n processors otherwise. In both cases the error probability is the inverse of a polynomial in ". Thus " may be chosen to trade-...

RNC-Approximation Algorithms for the Steiner Problem

In this paper we present an RNC-algorithm for nding a minimum spanning tree in a weighted 3-uniform hypergraph, assuming the edge weights are given in unary, and a fully polynomial time randomized approximation scheme if the edge weights are given in binary. From this result we then derive RNC-approximation algorithms for the Steiner problem in networks with approximation ratio (1 +) 5=3 for al...