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    Minimal contention-free matrices with application to multicasting

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    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×qp \times q matrix MM with 0-1 entries, the {\em shadow} of MM is defined as a boolean vector xx of qq entries such that xi=0x_i=0 if and only if there is no 1-entry in the iith column of MM, and xi=1x_i=1 otherwise. (The shadow xx can also be seen as the binary expression of the integer x=∑i=1qxi2q−ix=\sum_{i=1}^{q}x_i 2^{q-i}. Similarly, every row of MM 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×qp \times q matrix MM with 0-1 entries, finding a matrix M∗M^* such that: 1- M∗M^* has at most one 1-entry per column; 2- every row rr of M∗M^* (viewed as the binary expression of an integer) is larger than the corresponding row rr of MM, 1≤r≤p1 \leq r \leq p; and 3- the shadow of M∗M^* (viewed as an integer) is minimum. We show that there is an O(q(p+q))O(q(p+q)) algorithm that returns M∗M^* for any p×qp \times q boolean matrix MM. The application of this result is the following: Given a {\em directed} tree TT whose arcs are oriented from the root toward the leaves, and a subset of nodes DD, there exists a polynomial-time algorithm that computes an optimal multicast protocol from the root to all nodes of DD in the all-port line model.Peer Reviewe
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