Exploiting Hopsets: Improved Distance Oracles for Graphs of Constant Highway Dimension and Beyond For fixed h geq 2, we consider the task of adding to a graph G a set of weighted shortcut edges on the same vertex set, such that the length of a shortest hhop path between any pair of vertices in the augmented graph is exactly the same as the original distance between these vertices in G. A set of shortcut edges with this property is called an exact hhopset and may be applied in processing distance queries on graph G. In particular, a 2hopset directly corresponds to a distributed distance oracle known as a hub labeling. In this work, we explore centralized distance oracles based on 3hopsets and display their advantages in several practical scenarios. In particular, for graphs of constant highway dimension, and more generally for graphs of constant skeleton dimension, we show that 3hopsets require exponentially fewer shortcuts per node than any previously described distance oracle while incurring only a quadratic increase in the query decoding time, and actually offer a speedup when compared to simple oracles based on a direct application of 2hopsets. Finally, we consider the problem of computing minimumsize hhopset (for any h geq 2) for a given graph G, showing a polylogarithmicfactor approximation for the case of unique shortest path graphs. When h3, for a given bound on the space used by the distance oracle, we provide a construction of hopsets achieving polylog approximation both for space and query time compared to the optimal 3hopset oracle given the space bound.

Reliable Hubs for PartiallyDynamic AllPairs Shortest Paths in Directed Graphs We give new partiallydynamic algorithms for the allpairs shortest paths problem in weighted directed graphs. Most importantly, we give a new deterministic incremental algorithm for the problem that handles updates in widetildeO(mn43logWepsilon) total time (where the edge weights are from 1,W) and explicitly maintains a (1epsilon)approximate distance matrix. For a fixed epsilonu003e0, this is the first deterministic partially dynamic algorithm for allpairs shortest paths in directed graphs, whose update time is o(n2) regardless of the number of edges. Furthermore, we also show how to improve the stateoftheart partially dynamic randomized algorithms for allpairs shortest paths Baswana et al. STOCu002702, Bernstein STOCu002713 from Monte Carlo randomized to Las Vegas randomized without increasing the running time bounds (with respect to the widetildeO(cdot) notation).  :121,results are obtained by giving new algorithms for the problem of dynamically maintaining hubs, that is a set of widetildeO(nd) vertices which hit a shortest path between each pair of vertices, provided it has hoplength Omega(d). We give new subquadratic deterministic and Las Vegas algorithms for maintenance of hubs under either edge insertions or deletions.