Résumé : A 2-hop labeling (a.k.a. hub labeling) consists in assigning to each node of a graph asmall subset of nodes called hubs so that any pair of nodes have a common hub lying on a shortest path joining them. Such labelings appeared to provide a very efficient representation of distances in practical road network where surprisingly small hub sets can be computed. A graph parameter called skeleton dimension allows to explain this behaviour. Connecting any two nodes through a common hub can be seen as a 2-hop shortest path in a super-graph of the original graph. A natural extension is to consider more hops and is related to the notion of hopsets introduced in parallel computation of shortest paths. Surprisingly, a 3-hopconstruction leads to a data-structure for representing distances which is asymptotically both smaller and faster than 2-hop labeling in graphs of bounded skeleton dimension. Another natural question is to ask whether 2-hop labelings can be efficient more generally in sparsegraphs. Unfortunately, this is not the case as there exists bounded degree graphs that require quasi-linear average hub set size. The construction of such difficult graphs is related to the construction of dense graphs with n nodes that can be decomposed into n induced matchings as introduced by Ruzsa and Szemerédi in the seventies.
Joint work with Siddharth Gupta, Adrian Kosowski and Przemyslaw Uznanski.