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Abstract. Given a graph G = (V, E) with non-negative edge lengths whose vertices are assigned a label from L = {λ 1 , . . . , λ }, we construct a compact distance oracle that answers queries of the form: "What is δ (v, λ)?", where v ∈ V is a vertex in the graph, λ ∈ L a vertex label, and δ(v, λ) is the distance (length of a shortest path) between v and the closest vertex labeled λ in G. We formalize this natural problem and provide a hierarchy of approximate distance oracles that require subquadratic space and return a distance of constant stretch. We also extend our solution to dynamic oracles that handle label changes in sublinear time.

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