Strictly nonblocking conference networks using high-dimensional meshes

This paper introduces a conferencing server design based on an innovative configurable computing architecture to support the information transmission and distributed processing associated with creating and maintaining simultaneous disjoint conferences among sets of N conferees, where rdimensional meshes can be used as the conferencing components. An r-dimensional conferencing mesh network is strictly nonblocking if, regardless of the existing conferences implemented, a new conference among any subset of the idle conferees can be implemented using a connected set of idle processing elements without any disturbance to the existing conferences. Using arguments employing the isoperimetric ratios of the sizes of edge and node sets in a graph, we give necessary and sufficient conditions such that an r-dimensional conferencing mesh of M nodes provides strictly nonblocking conferencing to N conferees. We show that a necessary and sufficient condition for r-dimensional meshes to be strictly nonblocking, when dimension r is fixed, is that M Å O (N ( r/1)/ r ) . For general r-dimensional meshes, M Å O ( r ( r01)/ N ( r/1)/ r ) nodes are sufficient to support strictly nonblocking capabilities. A fundamental relationship is established between the requirements on M for strictly nonblocking conferencing among N conferees using certain graph structures and the isoperimetric ratios for those structures. q 1999 John Wiley & Sons, Inc. Networks 33: 293–308, 1999