Arnold Filtser

Efficient determination of the unique decodability of a string

Determining whether an unordered collection of overlapping substrings (called shingles) can be uniquely decoded into a consistent string is a problem common to a broad assortment of disciplines ranging from networking and information theory through cryptography and even genetic engineering and linguistics. We present a new insight that yields an efficient streaming algorithm for determining whether a string of n characters over the alphabet Σ can be uniquely decoded from its two-character shingles; our online algorithm achieves an overall time complexity Θ(n+|Σ|) and space complexity O(|Σ|). As a motivating application, we demonstrate how this algorithm can be adapted to larger, varying-size shingles for (empirically) efficient string reconciliation.

Metric embedding via shortest path decompositions

We study the problem of embedding weighted graphs of pathwidth k into ℓp spaces. Our main result is an O(kmin{1p,12})-distortion embedding. For p=1, this is a super-exponential improvement over the best previous bound of Lee and Sidiropoulos. Our distortion bound is asymptotically tight for any fixed p >1. Our result is obtained via a novel embedding technique that is based on low depth decompositions of a graph via shortest paths. The core new idea is that given a geodesic shortest path P, we can probabilistically embed all points into 2 dimensions with respect to P. For p>2 our embedding also implies improved distortion on bounded treewidth graphs (O((klogn)1p)). For asymptotically large p, our results also implies improved distortion on graphs excluding a minor.