Theory Seminar: Prioritized Metric Structures and Embedding

Arnold Filtser (Ben-Gurion University)
Wednesday, 7.1.2015, 12:30
Taub 401

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 [LLR95], online algorithms [BBMN11], distributed algorithms [KKMPT12] and for computing sparsifiers [ST04]. 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 suit of prioritized metric data structures and embeddings. We show that given a priority ranking $(x_1,x_2,...,x_n)$ of the graph vertices (respectively, metric points) one can devise a metric data structure (respectively, 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) {\em simultaneously}. The worst-case performance of our metric data structures and embeddings is typically asymptotically no worse than their non-prioritized counterparts.

Joint work with Michael Elkin and Ofer Neiman.

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