Multi-Dimensional Interval Routing Schemes

Interval Routing Scheme (k-IRS) is a compact routing scheme on general networks. It has been studied extensively and recently been implemented on the latest generation INMOS Transputer Router chip. In this paper we introduce an extension of the Interval Routing Scheme k-IRS to the multi-dimensional case hk; di-MIRS, where k is the number of intervals and d is the number of dimensions. Whereas k-IRS only represents compactly a single shortest path between any two nodes, with this new extension we are able to represent all shortest paths compactly. This is useful for fault-tolerance and traac distribution in a network. We study eecient representations of all shortest paths between any pair of nodes for general network topologies, for product graphs and for speciic interconnection networks such as rings, grids, tori, hypercubes and chordal rings. For these interconnection networks we show that for about the same space complexity as k-IRS we can represent all shortest paths in hk; di-MIRS (as compared to only a single shortest path in k-IRS). Moreover, trade-oos are derived between the dimension d and the number of intervals k in multi-dimensional interval routing schemes on hypercubes, grids and tori.