Rearrangeable and Nonblocking $[w,f]$-Distributors

Abstract

We formulate a graph model called [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> ] <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-distributors</i> which is useful in analyzing the structures and comparing the quantitative complexities and qualitative features of optical multicast cross-connects. Using the formulation we show that two strictly nonblocking multicast optical cross-connects under two different request models are equivalent topologically, even though one request model is much less restrictive than the other. We then investigate the tradeoff between the depth and the complexity of an optical multicast cross-connect using the graph model. Upper and lower complexity bounds are proved. In the process, we also give a generic recursive construction that can be used to construct optimal and near-optimal [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> ]-distributors. The recursive construction can also be used to construct cost-effective optical multicast cross-connects. Another important result that follows is the exact asymptotic behavior of the size of optimal [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> ] -connectors, the unicast version of [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> ]-distributors.