Exact Distance Labelings Yield Additive-Stretch Compact Routing Schemes

Abstract

Distance labelings and compact routing schemes have both been active areas of recent research. It was already known that graphs with constant-sized recursive separators, such as trees, outerplanar graphs, series-parallel graphs and graphs of bounded treewidth, support both exact distance labelings and optimal (additive stretch 0, multiplicative stretch 1) compact routing schemes, but there are many classes of graphs known to admit exact distance labelings that do not have constant-sized separators. Our main result is to demonstrate that every unweighted, undirected n-vertex graph which supports an exact distance labeling with l(n)-sized labels also supports a compact routing scheme with O(l(n) + log2n/loglogn)-sized headers, $O(\sqrt{n}(l(n) + \log^2{n}/\log{\log{n}}))$-sized routing tables, and an additive stretch of 6. We then investigate two classes of graphs which support exact distance labelings (but do not guarantee constant-sized separators), where we can improve substantially on our general result. In the case of interval graphs, we present a compact routing scheme with O(logn)-sized headers, O(logn)-sized routing tables and additive stretch 1, improving headers and table sizes from a result of [1], which uses O(log3n/loglogn)-bit headers and tables. We also present a compact routing scheme for the related family of circular arc graphs which guarantees O(log2n)-sized headers, O(logn)-sized routing tables and an additive stretch of 1.