Prioritized metric structures and embedding

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 (x1, x2, . . ., xn) 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.