The complexity of shortest path and dilation bounded interval routing

Abstract

Interval 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(n 1/2−ε) , any ε>0, for SE, Ω( n/ log n ) for CCC and BF, and Ω(n( log log n/ log n) 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( n log n ) 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).