Prioritized Metric Structures and Embedding

Abstract

Metric data structures (distance oracles, distance labeling schemes, routing schemes) and low-distortion embeddings provide a powerful algorithmic methodology, which has been successfully applied for approximation algorithms [N. Linial, E. London, and Y. Rabinovich, Combinatorica, 15 (1995), pp. 215--245], online algorithms [N. Bansal et al., Proceedings of the 52th Annual IEEE Symposium on Foundations of Computer Science, FOCS '08, IEEE Computer Society, Washington, DC, 2011, pp. 267--276], distributed algorithms [M. Khan et al., Distrib. Comput., 25 (2012), pp. 189--205], and for computing sparsifiers [Y. Shavitt and T. Tankel, IEEE/ACM Trans. Netw., 12 (2004), pp. 993--1006]. However, this methodology appears to have a limitation: the worst-case performance inherently depends on the cardinality of the metric, and one could not specify in advance which vertices/points should enjoy a better service (i.e., stretch/distortion, label size/dimension) than that given by the worst-case guarantee. In this paper we alleviate this limitation by devising a suite of prioritized metric data structures and embeddings. We show that given a priority ranking $(x_1,x_2,\ldots,x_n)$ of the graph vertices (resp., metric points) one can devise a metric data structure (resp., embedding) in which the stretch (resp., distortion) incurred by any pair containing a vertex $x_j$ will depend on the rank $j$ of the vertex. We also show that other important parameters, such as the label size and (in some sense) the dimension, may depend only on $j$. In some of our metric data structures (resp., embeddings) we achieve both prioritized stretch (resp., distortion) and label size (resp., dimension) simultaneously. The worst-case performance of our metric data structures and embeddings is typically asymptotically no worse than of their nonprioritized counterparts.