LOSSLESS PRIORITIZED EMBEDDINGS

Given metric spaces (X, d) and (Y, \rho ) and an ordering x1, x2, . . ., xn of (X, d), an embedding f : X \rightarrow Y is said to have a prioritized distortion \alpha (\cdot ), for a function \alpha (\cdot ), if for any pair xj, x^\prime of distinct points in X, the distortion provided by f for this pair is at most \alpha (j). If Y is a normed space, the embedding is said to have prioritized dimension \beta (\cdot ) if f(xj) may have at most \beta (j) nonzero coordinates. The notion of prioritized embedding was introduced by Filtser and the current authors in [M. Elkin, A. Filtser, and O. Neiman, SIAM J. Comput., 47 (2018), pp. 829-858], where a rather general methodology for constructing such embeddings was developed. Though this methodology enabled [M. Elkin, A. Filtser, and O. Neiman, SIAM J. Comput., 47 (2018), pp. 829-858] to come up with many prioritized embeddings, it typically incurs some loss in the distortion. In other words, in the worst case, prioritized embeddings obtained via this methodology incur distortion which is at least a constant factor off compared to the distortion of the classical counterparts of these embeddings. This constant loss is problematic for isometric embeddings. It is also troublesome for Matou\v sek's embedding of general metrics into \ell \infty , which, for a parameter k = 1, 2, . . ., provides distortion 2k - 1 and dimension O(k log n \cdot n^1/k). All logarithms in this paper are base 2. In this paper we devise two lossless prioritized embeddings. The first one is an isometric prioritized embedding of tree metrics into \ell \infty with dimension O(log j), matching the worst-case guarantee of O(log n) of the classical embedding of [N. Linial, E. London, and Y. Rabinovich, Combinatorica, 15 (1995), pp. 215-245]. The second one is a prioritized Matou\v sek embedding of general metrics into \ell \infty , which, for a parameter k = 1, 2, . . ., provides prioritized distortion 2\lceil k _log^log_n^j \rceil - 1 and dimension O(k log n \cdot n^1/k), again matching the worst-case guarantee 2k - 1 in the distortion of the classical Matou\v sek embedding. We also provide a dimension-prioritized variant of Matou\v sek's embedding. Finally, we devise prioritized embeddings of general metrics into a single ultrametric and of general graphs into a single spanning tree, with asymptotically optimal distortion.