Determining the unique decodability of a string in linear time

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 three perspectives on this problem: a graph theoretic framework due to Pevzner, an automata theoretic approach from our previous work, and 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 an application, we demonstrate how this algorithm can be adapted to larger, varying-size shingles for (empirically) efficient string reconciliation.