Non-Uniform and Learning Algorithms for String Matching Problems.

String matching is the fundamental algorithmic problem of deciding whether a string of length $n$ contains a pattern of length $k leq n$. study string matching algorithms from a nonuniform as well as learning theoretic perspective. nonuniform algorithms we consider the problem of designing small circuits that solve string matching under two popular choices of gates: De Morgan and threshold gates. learning algorithms we consider the problem of learning a hidden pattern $sigma$ of length at most $k$ under the classification rule where a string is assigned 1 by the classifier if and only if it contains $sigma$. Perhaps surprisingly both problems seem to have hardly been studied before. We present several upper and lower bounds on the number of gates of circuits solving string matching. depth 2 circuits and certain values of $k$ we demonstrate nearly linear lower bounds for threshold circuits and super polynomial lower bounds for DeMorgan circuits. These lower bounds are nearly tight as we provide almost matching upper bounds. unrestricted (unbounded depth) circuits, we present for certain values of $k$ a lower bound of $Omega(sqrt{n/log n})$ for threshold circuits and a linear lower bound for De Morgan circuits (with unbounded fanin). Our proof for threshold circuits builds on a curious connection between detecting patterns and evaluating Boolean functions when the truth table of the function is given explicitly. For learning algorithms we prove asymptotically tight bounds on the VC dimension and sample complexity and provide an efficient learning algorithm. Several extensions (e.g., classification of 2D images, multiple patterns, infinite alphabets) are discussed as well.