On the complexity of multi-dimensional interval routing schemes

Abstract

Multi-dimensional interval routing schemes (MIRS) have been introduced in Flammini et al. [5] as an extension of interval routing schemes (IRS) defined by Santoro and Khatib [21] and van Leeuwen and Tan [16]. In this paper we study the efficiency of multi-dimensional interval routing schemes with respect to the space complexity and the congestion. We bring certain comparative complexity results for general graphs as well as for specific networks. In the first part of the paper, we present three main contributions to the complexity of shortest path MIRS. • For certain hypercube-like graphs there are efficient multi–dimensional interval routing schemes despite of provable nonexistence of efficient deterministic interval routing schemes. • We compare the DIS-MIRS and the CON-MIRS models introduced in [5] and prove that the DIS-MIRS model is asymptotically stronger than the CON-MIRS model when considering space requirements of the full-information shortest path routing schemes. • We introduce a powerful lower bound technique on the CON-MIRS model and prove that not for every Cayley graph there exists space-efficient shortest path MIRS. (On the contrary, there does not exist a powerful space lower bound technique for DIS-MIRS model.) 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. In the second part of the paper we present congestion results for MIRS. • We show that there exists a multipath 〈2,n+2〉-DIS-MIRS of n-dimensional cube-connected cycles with asymptotically optimal congestion. • We give a tradeoff between the congestion and the space complexity of multipath MIRS on general graphs. For any graph G and given 1⩽s⩽|V| there exists a multipath 〈2+⌈|V|/(2s)⌉,1〉-MIRS with congestion π+|V|·Δ·s, where π is the forwarding index of the graph G and Δ is the maximum degree of vertices in the graph G. • For planar graphs of constant bounded degree there exist a multipath 〈 O( |V| ),1〉 -MIRS and a (deterministic) O( |V| log|V|) -IRS, both with asymptotically optimal congestion.