New Routing Techniques And Their Applications

In this paper we present two new routing techniques that allow us to obtain the following new routing schemes:A routing scheme for n-nodes, m-edges unweighted graphs that uses (O) over tilde (1/epsilon n(2/3)) space at each vertex and (O) over tilde (1/epsilon)-bit headers, to route a message between any pair of vertices u, v is an element of V on a (2 + epsilon, 1)-stretch path, i.e., a path of length at most (2 + epsilon).d + 1, where d is the distance between u and v. This should be compared to the (2, 1)-stretch and (O) over tilde (n(5/3)) space distance oracle of Patrascu and Roditty [FOCS'10 and SIAM J. Comput. 2014] and to the (2, 1)-stretch routing scheme of Abraham and Gavoille [DISC'11] that uses (O) over tilde (n(3/4)) space at each vertex. It follows from Patrascu, Thorup and Roditty [FOCS'12] that a 2-stretch distance oracle with (O) over tilde (m(2/3)) space at each vertex is optimal, assuming a hardness conjecture on set intersection holds.A routing scheme for n-nodes weighted graphs with normalized diameter D, that uses (O) over tilde (1/epsilon n(1/3) log D) space at each vertex and (O) over tilde (1/epsilon log D)-bit headers, to route a message between any pair of vertices on a (5 + epsilon)stretch path. This should be compared to the 5-stretch and (O) over tilde (n(4/3)) space distance oracle of Thorup and Zwick [STOC'01 and J. ACM. 2005] and to the 7-stretch routing scheme of Thorup and Zwick [SPAA'01] that uses (O) over tilde (n(1/3)) space at each vertex. Since a 5-stretch routing scheme must use tables of Omega(n(1/3)) space our result is almost tight.For an integer l > 1, a routing scheme for n-nodes unweighted graphs that uses (O) over tilde (l1/epsilon n(l/(2l+/-1))) space at each vertex and (O) over tilde (1/epsilon)-bit headers, to route a message between any pair of vertices on a (3 +/- 2/l + epsilon; 2)-stretch path. This should be compared to the distance oracles of Patrascu, Thorup and Roditty [FOCS'12] for weighted graphs with a stretch of (3 +/- 2l) and (O) over tilde (lm(1+l/(2l+/-1))) total space.A routing scheme for n-nodes weighted graphs, that for any integer k > 2, uses (O) over tilde (1/epsilon n(1/k) log D) space at each vertex and (O) over tilde (1/epsilon log D)-bit headers, to route a message between any pair of vertices on a (4k - 7 + epsilon)stretch path. This improves the (4k 5)-stretch routing scheme of Thorup and Zwick [SPAA'01] and can also be used in the recent ((4)k)-stretch routing scheme of Chechik [PODC'13] to obtain slightly better values for alpha and beta.