New routing techniques and their applications

Let $G=(V,E)$ be an undirected graph with $n$ vertices and $m$ edges. We obtain the following new routing schemes: - A routing scheme for unweighted graphs that uses $\tilde O(\frac{1}{\epsilon} n^{2/3})$ space at each vertex and $\tilde O(1/\epsilon)$-bit headers, to route a message between any pair of vertices $u,v\in V$ on a $(2 + \epsilon,1)$-stretch path, i.e., a path of length at most $(2+\epsilon)\cdot d(u,v)+1$. This should be compared to the $(2,1)$-stretch and $\tilde O(n^{5/3})$ 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 $\tilde O( n^{3/4})$ space at each vertex. - A routing scheme for weighted graphs with normalized diameter $D$, that uses $\tilde O(\frac{1}{\epsilon} n^{1/3}\log D)$ space at each vertex and $\tilde O(\frac{1}{\epsilon}\log D)$-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 $\tilde O(n^{4/3})$ 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 $\tilde O( n^{1/3})$ 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 $\ell>1$, a routing scheme for unweighted graphs that uses $\tilde O(\ell\frac{1}{\epsilon} n^{\ell/(2\ell \pm 1)})$ space at each vertex and $\tilde O(\frac{1}{\epsilon})$-bit headers, to route a message between any pair of vertices on a $(3\pm2/\ell+\epsilon,2)$-stretch path. - A routing scheme for weighted graphs, that uses $\tilde O(\frac{1}{\epsilon}n^{1/k}\log D)$ space at each vertex and $\tilde O(\frac{1}{\epsilon}\log D)$-bit headers, to route a message between any pair of vertices on a $(4k-7+\epsilon)$-stretch path