Roundtrip spanners and roundtrip routing in directed graphs

Abstract

We introduce the notion of roundtrip-spanners of weighted directed graphs and describe efficient algorithms for their construction. We show that for every integer k ≥ 1 and any ϵ > 0, any directed graph on n vertices with edge weights in the range [1, W ] has a (2 k + ϵ)-roundtrip-spanner with O (min{( k 2 /ϵ) n 1 + 1/ k (log( nW ), ( k /ϵ) 2 n 1 + 1/ k ,(log n ) 2−1/ k }) edges. We then extend these constructions and obtain compact roundtrip routing schemes. For every integer k ≥ 1 and every ϵ > 0, we describe a roundtrip routing scheme that has stretch 4 k + ϵ, and uses at each vertex a routing table of size Õ (( k 2 /ϵ) n 1/ k log( nW )). We also show that any weighted directed graph with arbitrary/ positive edge weights has a 3-roundtrip-spanner with O ( n 3/2 ) edges. This result is optimal. Finally, we present a stretch 3 roundtrip routing scheme that uses local routing tables of size Õ ( n 1/2 ). This routing scheme is essentially optimal. The roundtrip-spanner constructions and the roundtrip routing schemes for directed graphs that we describe are only slightly worse than the best available spanners and routing schemes for undirected graphs. Our roundtrip routing schemes substantially improve previous results of Cowen and Wagner. Our results are obtained by combining ideas of Cohen, Cowen and Wagner, Thorup and Zwick, with some new ideas.