Minimal contention-free matrices with application to multicasting

In this paper we show that the multicast problem in trees can be expressed in term of arranging rows and columns of boolean matrices Given a p q matrix M with entries the shadow of M is de ned as a boolean vector x of q entries such that xi if and only if there is no entry in the ith column of M and xi otherwise The shadow x can also be seen as the binary expression of the integer x Pq i xi q i Similarly every row of M can be seen as the binary expression of an integer According to this formalism the key for solving a multicast problem in trees is shown to be the following Given a p q matrixM with entries nding a matrixM such that M has at most one entry per column every row r of M viewed as the binary expression of an integer is larger than the corresponding row r of M r p and the shadow of M viewed as an integer is minimum We show that there is an O q p q algorithm that returnsM for any p q boolean matrixM The application of this result is the following Given a directed tree T whose arcs are oriented from the root toward the leaves and a subset of nodes D there exists a polynomial time algorithm that computes an optimal multi cast protocol from the root to all nodes of D in the all port line model