# The complexity of the Multiple Pattern Matching Problem for random strings

## TL;DR

This paper extends a multiple pattern matching algorithm to handle arbitrary dictionaries over any alphabet size, establishing its optimality by matching upper and lower bounds on complexity, thus improving prior claims.

## Contribution

It generalizes a pattern matching algorithm to arbitrary dictionaries and proves its optimality by deriving matching upper and lower bounds on complexity.

## Key findings

- The algorithm's complexity rate is at most κ_UB * φ(r).
- The lower bound for complexity is at least κ_LB * φ(r).
- The algorithm is proven to be optimal for arbitrary dictionaries.

## Abstract

We generalise a multiple string pattern matching algorithm, recently proposed by Fredriksson and Grabowski [J. Discr. Alg. 7, 2009], to deal with arbitrary dictionaries on an alphabet of size $s$. If $r_m$ is the number of words of length $m$ in the dictionary, and $\phi(r) = \max_m \ln(s\, m\, r_m)/m$, the complexity rate for the string characters to be read by this algorithm is at most $\kappa_{{}_\textrm{UB}}\, \phi(r)$ for some constant $\kappa_{{}_\textrm{UB}}$. On the other side, we generalise the classical lower bound of Yao [SIAM J. Comput. 8, 1979], for the problem with a single pattern, to deal with arbitrary dictionaries, and determine it to be at least $\kappa_{{}_\textrm{LB}}\, \phi(r)$. This proves the optimality of the algorithm, improving and correcting previous claims.