Lossless Prioritized Embeddings

Given metric spaces (X, d) and (Y, ρ) an ordering x1, x2,...,xn of d), embedding f : X → Y is said to have a prioritized distortion α(·), for function if any pair xj, x' distinct points in X, the provided by this at most α(j). If normed space, dimension β(·), f(xj) may β(j) nonzero coordinates. The notion was introduced Filtser current authors [EFN18], where rather general methodology constructing such em-beddings developed. Though enabled [EFN18] come up with many embed-dings, it typically incurs some loss distortion. In other words, worst-case, embeddings obtained via incur which least constant factor off, compared classical counterparts these embeddings. This problematic isometric It also troublesome Matousek's metrics into l∞, parameter k = 1, 2,..., provides 2k−1 O(k log n·n1/k). paper we devise two lossless first one tree l∞ O(log j), matching worst-case guarantee n) Linial et al. [LLR95]. second [MATH HERE] n · n1/k), again 2k − 1 embedding. We provide dimension-prioritized variant Finally, (single) ultra-metric graphs spanning asymptotically optimal