Optimal (Euclidean) Metric Compression

Near-Optimal (Euclidean) Metric Compression

The metric sketching problem is defined as follows. Given a metric on n points, and > 0, we wish to produce a small size data structure (sketch) that, given any pair of point indices, recovers the distance between the points up to a 1 + distortion. In this paper we consider metrics induced by `2 and `1 norms whose spread (the ratio of the diameter to the closest pair distance) is bounded by Φ >...

Metric and Euclidean TSP

Euclidean vs Graph Metric

The theory of sparse graph limits concerns itself with versions of local convergence and global convergence, see e.g. [41]. Informally, in local convergence we look at a large neighborhood around a random uniformly chosen vertex in a graph and in global convergence we observe the whole graph from afar. In this note rather than surveying the general theory we will consider some concrete examples...

On the Optimal Choice of Quality Metric in Image Compression

There exist many di erent lossy compression meth ods and most of these methods have several tunable parameters In di erent situations di erent methods lead to di erent quality reconstruction so it is impor tant to select in each situation the best compression method A natural idea is to select the compression method for which the average value of some metric d I e I is the smallest possible The...

Near-Optimal Compression for the Planar Graph Metric

The Planar Graph Metric Compression Problem is to compactly encode the distances among k nodes in a planar graph of size n. Two naïve solutions are to store the graph using Opnq bits, or to explicitly store the distance matrix with Opk log nq bits. The only lower bounds are from the seminal work of Gavoille, Peleg, Prennes, and Raz [SODA’01], who rule out compressions into a polynomially smalle...