On low dimensional local embeddings

We study the problem of embedding metric spaces into low dimensional ℓ_p spaces while faithfully preserving distances from each point to its k nearest neighbors. We show that any metric space can be embedded into ℓ_p^{O(ep log2 k)} with k-local distortion of O((log k)/p). We also show that any ultrametric can be embedded into ℓ_p^{O(log k)/c3} with k-local distortion 1 + ∈. Our embedding results have immediate applications to local Distance Oracles. We show how to preprocess a graph in polynomial time to obtain a data structure of O(nk^1/t log² k) bits, such that distance queries from any node to its k nearest neighbors can be answered with stretch O(t).