Simpler Sequential and Parallel Biconnectivity Augmentation

TL;DR

This paper introduces a simplified sequential algorithm and an optimal parallel algorithm for biconnectivity augmentation in trees, improving efficiency and implementation simplicity over previous methods.

Contribution

It presents a new, simplified sequential algorithm using edge contraction and an optimal parallel algorithm for connectivity augmentation in trees.

Findings

The sequential algorithm is simpler and preserves connectivity information.

The parallel algorithm is optimal with a processor-time product of O(n).

Implementation on CREW PRAM model with O(Δ) processors.

Abstract

For a connected graph, a vertex separator is a set of vertices whose removal creates at least two components and a minimum vertex separator is a vertex separator of least cardinality. The vertex connectivity refers to the size of a minimum vertex separator. For a connected graph $G$ with vertex connectivity $k (k \geq 1)$, the connectivity augmentation refers to a set $S$ of edges whose augmentation to $G$ increases its vertex connectivity by one. A minimum connectivity augmentation of $G$ is the one in which $S$ is minimum. In this paper, we focus our attention on connectivity augmentation of trees. Towards this end, we present a new sequential algorithm for biconnectivity augmentation in trees by simplifying the algorithm reported in \cite{nsn}. The simplicity is achieved with the help of edge contraction tool. This tool helps us in getting a recursive subproblem preserving all connectivity information. Subsequently, we present a parallel algorithm to obtain a minimum connectivity augmentation set in trees. Our parallel algorithm essentially follows the overall structure of sequential algorithm. Our implementation is based on CREW PRAM model with $O(\Delta)$ processors, where $\Delta$ refers to the maximum degree of a tree. We also show that our parallel algorithm is optimal whose processor-time product is O(n) where $n$ is the number of vertices of a tree, which is an improvement over the parallel algorithm reported in \cite{hsu}.