Interval Routing on Layered Cross Product of Trees and Cycles

Interval routing is an attractive space-efficient routing method for point-to-point networks (introduced in [13] and [15]) which has found industrial applications in the INMOS T9000 transputer design. Surveys of the principal theoretical results as well as recent trends in the area of interval routing can be found in [16, 5, 11]. Interval routing is based on compact routing tables, in which the set of nodes reachable via outgoing links is compactly represented in the form of intervals. The space efficiency can be measured by compactness, that is the maximum number of intervals per link. Previous research mostly concentrated on shortest path interval routing schemes (IRS for short). Shortest path IRS of compactness 1 have been designed for a number of well-known interconnection networks including trees, rings, complete bipartite graphs, meshes, tori, and hypercubes. However, there are interconnection networks having provably large [8] compactness for shortest path IRS, for example shuffle-exchange, De Bruijn, cube-connected cycles, butterfly, pancake, and star graphs. Several generalizations of IRS were therefore proposed. Multidimensional interval routing schemes (MIRS for short) were introduced in [3] and were used to represent the information on all shortest paths. MIRS with low memory requirements were proposed in [3] for hypercubes, grids, tori and certain types of chordal rings. Other efficient MIRS were designed in [12]. Certain graph operators have been found interesting in the design of communication networks. The impact of some graph operators on the compactness of interval routing has been previously studied in [6, 4, 10]. These results characterize the effect of the cartesian product, the composition, and the join of graphs on the minimum number of linear intervals needed for the optimal deterministic routing. We present the study of another graph-theoretic operation, namely the layered cross product of graphs. The Layered Cross Product (LCP in short) was introduced in [2] as a technique for constructing some complex interconnection