On the complexity of multi-dimensional interval routing schemes

Multi-dimensional interval routing schemes (MIRS) introduced in [4] are an extension of interval routing schemes (IRS). We give an evidence that multi-dimensional interval routing schemes really help for certain well-known interconnection networks, e.g. for n-dimensional butterfly there exists a full-information shortest path 〈2, 3〉-DIS-MIRS, while in [12] it was shown that a shortest path IRS needs Ω(2) intervals. Further, we compare the DIS-MIRS model and the CONMIRS model introduced in [4]. The main result is that the DIS-MIRS model is asymptotically stronger than the CON-MIRS model. We prove this by showing that for n-dimensional cubeconnected-cycles there exists a full-information shortest path 〈2n, n〉-DIS-MIRS while for any full-information shortest path 〈k, d〉-CON-MIRS it holds kd = Ω(2 √ ). We also show that for n-dimensional star any full-information shortest path 〈k, d〉-CON-MIRS requires kd = Ω(2). The congestion is a common phenomenon in networks which can completely degrade their performance. Therefore it is reasonable to study routing schemes which are not necessarily shortest path, but allow high network throughput. We show that there exists a 〈2, n + 2〉-DIS-MIRS of n-dimensional cube-connected-cycles with asymptotically optimal congestion. Moreover, we give an upper bound on the tradeoff between congestion and space complexity of multipath MIRS for general graphs. For any graph G and given 1 ≤ s ≤ |V | there exists a multipath 〈2 + d|V |/2se, 1〉MIRS with congestion F + |V | · ∆ · s, where F is the forwarding index of the graph G and ∆ is the maximum degree of vertices in the graph G. As a consequence, for planar graphs of constant bounded degree there exists a multipath 〈O( √ |V |), 1〉-MIRS with asymptotically optimal congestion. We also show that for planar graphs of bounded degree there exists (deterministic) O( √ |V | log |V |)-IRS with asymptotically optimal congestion.