Lossless Prioritized Embeddings
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Given metric spaces (X,d) and (Y,ρ) and an ordering x_1,x_2,...,x_n of (X,d), an embedding f: X → Y is said to have a prioritized distortion α(·), if for any pair x_j,x' of distinct points in X, the distortion provided by f for this pair is at most α(j). If Y is a normed space, the embedding is said to have prioritized dimension β(·), if f(x_j) may have nonzero entries only in its first β(j) coordinates. The notion of prioritized embedding was introduced by EFN15, where a general methodology for constructing such embeddings was developed. Though this methodology enables EFN15 to come up with many prioritized embeddings, it typically incurs some loss in the distortion. This loss is problematic for isometric embeddings. It is also troublesome for Matousek's embedding of general metrics into ℓ_∞, which for a parameter k = 1,2,..., provides distortion 2k-1 and dimension O(k n · n^1/k). In this paper we devise two lossless prioritized embeddings. The first one is an isometric prioritized embedding of tree metrics into ℓ_∞ with dimension O( j). The second one is a prioritized Matousek's embedding of general metrics into ℓ_∞, which provides prioritized distortion 2 k j n - 1 and dimension O(k n · n^1/k), again matching the worst-case guarantee 2k-1 in the distortion of the classical Matousek's embedding. We also provide a dimension-prioritized variant of Matousek's embedding. Finally, we devise prioritized embeddings of general metrics into (single) ultra-metric and of general graphs into (single) spanning tree with asymptotically optimal distortion.
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