Title : Hub Labeling ( 2 - Hop Labeling ) Name : Daniel Delling 1 , Andrew

Given a directed graph G = (V, A) (with n = |V | and m = |A|) with a length function : A → R + and a pair of vertices s, t, a distance oracle returns the distance dist(s, t) from s to t. A labeling algorithm [18] implements distance oracles in two stages. The preprocessing stage computes a label for each vertex of the input graph. Then, given s and t, the query stage computes dist(s, t) using only the labels of s and t; the query does not explicitly use G and. Hub labeling (HL) (or 2-hop labeling) is a special kind of labeling algorithm. The label L(v) of a vertex v consists of two parts: the forward label L f (v) is a collection of vertices w with their distances dist(v, w) from v, while the backward label L b (v) is a collection of vertices u with their distances dist(u, v) to v. (If the graph is undirected, a single label per vertex suffices.) The vertices in v's label are the hubs of v. The labels must obey the cover property: for any two vertices s and t, the set L f (s) ∩ L b (t) must contain at least one hub that is on the shortest s–t path. Given the labels, HL queries are straightforward: to find dist(s, t), simply find the hub x ∈ L f (s) ∩ L b (t)