On finding an ear decomposition of an undirected graph distributively

An ear decomposition of a connected undirected graph G= (V ,E) is a partition of the edge (link) set E into a collection of edge-disjoint paths P0,P1, . . . ,P|E|−|V |+1, called ears, such that P0 is a link, P0 ∪P1 is a closed path (i.e., a cycle), and for each i , 2 i |E| − |V | + 1, Pi is a path of which each terminating node belongs to some Pj , j < i , and no internal node belongs to any Pj , j < i . It has been used as a means in solving graph problems such as st-numbering, planarity testing, and graph connectivity in the sequential and parallel settings. Recently, Kazmierczak and Radhakrishnan [2] presented a distributed algorithm for constructing an ear decomposition of a 2-edge-connected (i.e., connected and bridgeless) undirected graph. They claimed that their algorithm has an O(m) time and message complexity where m is the number of links in the graph (Theorem 4 of [2]2). We shall show that the message complexity of their algorithm is actually (mn), where n is the number of nodes in the graph. Unfortunately, there is no easy fix for the error without completely changing the algorithm. This is owing to the fact that the algorithm uses a top-down approach to determine the ears which requires corrections to be made repeatedly over some part of the graphs resulting in substantial overhead, and a depth-first search method that would not shift from the current node to another node until the current node has received an acknowledgment for every message it sent out which results in long delay. We then outline a distributed algorithm that requires 2n− 2 time and 4m− n+ 1 messages. As these time and message complexity are the same as those of the best known distributed depth-first search algorithms [3,5], this is the best result one could ever achieve with a depth-first search based distributed algorithm [3]. Besides being more time and message efficient, our algorithm also uses a distributed model weaker then that of Kazmierczak et al. in that it does not require the messages sent along the