On the space requirement of interval routing

Interval routing is a space-efficient method for point-to-point networks. It is based on labeling the edges of a network with intervals of vertex numbers (called interval labels). An M-label scheme allows up to M labels to be attached on an edge. For arbitrary graphs of size n, n the number of vertices, the problem is to determine the minimum M necessary for achieving optimality in the length of the longest routing path. The longest routing path resulted from a labeling is an important indicator of the performance of any algorithm that runs on the network. We prove that there exists a graph with D = Ω(n1/3) such that if M ≤ n/18D - O(√n/D), the longest path is no shorter than D + Θ(D/√M). As a result, for any M-label IRS, if the longest path is to be shorter than D + Θ(D/√M), at least M = Ω(n/D) labels per edge would be necessary.published_or_final_versio

The complexity of shortest path and dilation bounded interval routing

AbstractInterval routing is a popular compact routing method for point-to-point networks which found industrial applications in novel transputer routing technology (May and Thompson, Transputers and Routers: Components for Concurrent Machines, Inmos, 1991).Recently much effort is devoted to relate the efficiency (measured by the dilation or the stretch factor) to space requirements (measured by the compactness or the total number of memory bits) in a variety of compact routing methods (Eilam, Moran and Zaks, 10th International Workshop on Distributed Algorithms (WDAG), Lecture Notes in Computer Science, vol. 1151, Springer, Berlin, 1996, pp. 191–205; Fraigniaud and Gavoille, 8th Annual ACM Symp. on Parallel Algorithms and Architectures (SPAA), ACM Press, New York, 1996; Gavoille and Pérennes, Proc. SIRCCO’96, Carleton Press, 1996, pp. 88–103; Kranakis, Krizanc, 13th Annual Symp. on Theoretical Aspects of Computer Science (STACS), Lecture Notes in Computer Science, vol. 1046, Springer, Berlin, 1996, pp. 529–540; Meyer auf der Heide and Scheideler, Proc. 37th Symp. on Foundations of Computer Science (FOCS), November 1996; Peleg and Upfal, J. ACM 36 (1989) 510–530; Tse and Lau, Proc. SIROCCO'95, Carleton Press, 1995, pp. 123–134). We add new results in this direction for interval routing.For the shortest path interval routing we apply a technique from Flammini, van Leeuwen and Marchetti-Spaccamela (MFCS’95, Lecture Notes in Computer Science, vol. 969, Springer, Berlin, 1995, pp. 37–49) to some interconnection networks (shuffie exchange (SE), cube connected cycles (CCC), butterfly (BF) and star (S)) and get improved lower bounds on compactness in the form Ω(n1/2−ε), any ε>0, for SE, Ω(n/logn) for CCC and BF, and Ω(n(loglogn/logn)5) for S, where n is the number of nodes in the corresponding network. Previous lower bounds for these networks were only constant (Fraigniaud and Gavoille, CONPAR’94, Lecture Notes in Computer Science, vol. 854, Springer, Berlin, 1994, pp. 785–796).For the dilation bounded interval routing we give a routing algorithm with the dilation ⌈1.5D⌉ and the compactness O(nlogn) on n-node networks with the diameter D. It is the first nontrivial upper bound on the dilation bounded interval routing on general networks. Moreover, we construct a network on which each interval routing with the dilation 1.5D−3 needs the compactness at least Ω(n). It is an asymptotical improvement over the previous lower bounds in Tse and Lau (Proc. SIROCCO’95, Carleton Press, 1995, pp. 123–134) and it is also better than independently obtained lower bounds in Tse and Lau (Proc. Computing: The Australasian Theory Symp. (CATS’97), Sydney, Australia, February 1997)

The Complexity of Shortest Path and Dilation Bounded Interval Routing

Interval routing is a popular compact routing method for point-to-point networks which found industrial applications in novel transputer routing technology [13]. Recently much effort is devoted to relate the efficiency (measured by the dilation or the stretch factor) to space requirements (measured by the compactness or the total number of memory bits) in a variety of compact routing methods [3, 8, 9, 12, 14, 15, 19]. We add new results in this direction for interval routing. For the shortest path interval routing we apply a technique from [4] to some interconnection networks (shuffle exchange (SE), cube connected cycles (CCC), butterfly (BF) and star (S)) and get improved lower bounds on compactness in the form\Omega\Gamma n 1=2\Gamma" ), any " ? 0, for SE,\Omega\Gamma p n=logn) for CCC and BF, and\Omega\Gamma n(log log n=logn) 5 ) for S, where n is the number of nodes in the corresponding network. Previous lower bounds for these networks were only constant [7]. For the dila..