String Matching: Communication, Circuits, and Learning

TL;DR

This paper investigates the complexity of string matching across communication, circuit, and learning models, providing bounds and evidence for optimality and gaps in various regimes.

Contribution

It offers new bounds on communication and circuit complexity for string matching and establishes optimal learning bounds for pattern recognition.

Findings

Near-optimal bounds for small pattern lengths in communication complexity

Circuit complexity bounds for threshold and DeMorgan gates

Optimal VC dimension and sample complexity for learning patterns

Abstract

String matching is the problem of deciding whether a given $n$-bit string contains a given $k$-bit pattern. We study the complexity of this problem in three settings. Communication complexity. For small $k$, we provide near-optimal upper and lower bounds on the communication complexity of string matching. For large $k$, our bounds leave open an exponential gap; we exhibit some evidence for the existence of a better protocol. Circuit complexity. We present several upper and lower bounds on the size of circuits with threshold and DeMorgan gates solving the string matching problem. Similarly to the above, our bounds are near-optimal for small $k$. Learning. We consider the problem of learning a hidden pattern of length at most $k$ relative to the classifier that assigns 1 to every string that contains the pattern. We prove optimal bounds on the VC dimension and sample complexity of this problem.