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( 1 /ε n2/3) space at each vertex and O (1=")-bit headers, to route a message between any pair of vertices u; v 2 V on a (2 + ε 1)-stretch path, i.e., a path of length at most (2+ε) d+1, where d is the distance between u and v. This should be compared to the (2; 1)-stretch and O(n5/3) 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 (n3=4) space at each vertex. It follows from Patrascu, Thorup and Roditty [FOCS'12] that a 2-stretch distance oracle with O (m2=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( 1 "n1=3 logD) space at each vertex and O( 1 " logD)-bit headers, to route a message between any pair of vertices on a (5 + ") stretch path. This should be compared to the 5-stretch and O(n4=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 (n1=3) space at each vertex. Since a 5-stretch routing scheme must use tables of (n1=3) space our result is almost tight. For an integer > 1, a routing scheme for n-nodes unweighted graphs that uses O(1 "n=(21)) space at each vertex and O( 1 " )-bit headers, to route a message between any pair of vertices on a (3 ± 2= + "; 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 ± 2=) and O (m1+=(21)) total space. A routing scheme for n-nodes weighted graphs, that for any integer k > 2, uses O( 1 "n1=k logD) space at each vertex and O( 1 " logD)-bit headers, to route a message between any pair of vertices on a (4k - 7 + ")- 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-α)k-β)-stretch routing scheme of Chechik [PODC'13] to obtain slightly better values for α and β.