Parallel algorithms for k-tree recognition and its applications

The class of k-trees generalize the notion of trees, maximal outer-planar graphs, and caterpillars. We present new parallel algorithms to recognize k-trees and to find a collection of k vertex disjoint paths between a specified vertex pair (u,v), called a uv-cable, in a k-tree. A parallel algorithm to compute the k paths in a k-tree is also presented. The algorithms are based on parallel construction of a directed graph as a representation for R-trees, referred to as underlying trees in this paper. The model of parallel computation used is the CRCW PRAM (concurrent read, concurrent write, parallel RAM), where more than one processor can concurrently read from or write into the same memory location during the same memory cycle. Writing conflicts are resolved in a nondeterministic fashion. The recognition algorithm runs in O(log/sup 2/ n) time using O(m+n) processors. Constructing a spatial graph and the underlying tree representations of the k-tree for fixed k takes only O(log n) time using O(n) processors. The parallel algorithms for a uv-cable and k paths take O(log n) time using O(n) processors if the underlying tree structure of the k-tree is available.<>