Abstract:
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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 $p \times q$ matrix $M$ with 0-1 entries, the {\em shadow}
of $M$ is defined as a boolean vector $x$ of $q$ entries such that
$x_i=0$ if and only if there is no 1-entry in the $i$th column of
$M$, and $x_i=1$ otherwise. (The shadow $x$ can also be seen as the
binary expression of the integer $x=\sum_{i=1}^{q}x_i 2^{q-i}$.
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 $p
\times q$ matrix $M$ with 0-1 entries, finding a matrix $M^*$ such
that:
1- $M^*$ has at most one 1-entry per column;
2- every row $r$ of $M^*$ (viewed as the binary expression of
an integer) is larger than the corresponding row $r$ of $M$, $1 \leq
r \leq p$; and
3- the shadow of $M^*$ (viewed as an integer) is minimum.
We show that there is an $O(q(p+q))$ algorithm that
returns $M^*$ for any $p \times q$ boolean matrix $M$.
The application of this result is the following: Given a {\em 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 multicast protocol from the root to all
nodes of $D$ in the all-port line model. |
Materia(s):
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-Information and Communication Applications, Inc. -Operations research -Computer systems -Graph theory -Application to Multicasting -Minimal Contention-free Matrices -Investigació operativa -Arquitectura de computadors -Grafs, Teoria de -Classificació AMS::68 Computer science::68M Computer system organization -Classificació AMS::05 Combinatorics::05C Graph theory -Classificació AMS::90 Operations research, mathematical programming::90B Operations research and management science -Classificació AMS::94 Information And Communication, Circuits::94A Communication, information |