String Matching: Communication, Circuits, and Learning

Authors Alexander Golovnev, Mika Göös, Daniel Reichman, Igor Shinkar

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Author Details

Alexander Golovnev
  • Harvard University, Cambridge, MA, USA
Mika Göös
  • Institute for Advanced Study, Princeton, NJ, USA
Daniel Reichman
  • Department of Computer Science, Princeton University, NJ, USA
Igor Shinkar
  • School of Computing Science, Simon Fraser University, Burnaby, BC, Canada


We thank Paweł Gawrychowski for his useful feedback and Gy. Turán for sharing [Groeger and Tur{á}n, 1993] with us. We are also very grateful to anonymous reviewers for their insightful comments.

Cite AsGet BibTex

Alexander Golovnev, Mika Göös, Daniel Reichman, and Igor Shinkar. String Matching: Communication, Circuits, and Learning. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Circuit complexity
  • Theory of computation → Boolean function learning
  • string matching
  • communication complexity
  • circuit complexity
  • PAC learning


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