String Matching and 1d Lattice Gases

We calculate the probability distributions for the number of occurrences n of of a given l letter word in a random string of k letters. We consider the case in which the letters of the strings belong to an r letter alphabet. Analytical expressions for the distribution are known for the asymptotic regimes (i) k ≫ r ≫ 1 (Gaussian) and k, l → ∞ such that k/r is finite (Compound Poisson). However, it is known that these distributions do now work well in the intermediate regime k & r & 1. We show that the problem of calculating the string matching probability can be cast into a problem of determining the configurational partition function of a 1d lattice gas with interacting particles such that the string matching probability distribution becomes the grand-partition sum of the lattice gas, with the number of particles corresponding to the number of matches on the string. Using this analogy, we perform a virial expansion of the effective equation of state and thereby obtain the probability distribution function. Our result reproduces the behavior of the matching distribution in all regimes, i.e. the asymptotic as well as the intermediate regimes, rather well. We are also able to show analytically how the limiting distributions arise. Our analysis builds on the observation that the effective interactions between the particles consist of a relatively strong core of size l, the word length, followed by a weak, exponentially decaying tail, whose overall strength decreases with increasing l. We find that the asymptotic regimes correspond to the case where the tail of the interactions can be neglected, while in the intermediate regime the effects of the tail needs to be incorporated into the analysis. This is ultimately responsible for the failure of the asymptotic distributions in this regime. Our results are readily generalized to the case where the random strings are generated by more complicated stochastic process such as a non-uniform letter probability distribution or Markov chains. We show that by varying the parameters of the stochastic process, the tails of the effective interactions can be made even more dominant rendering thus the asymptotic approximations less accurate in such a regime. PACS Nos: 02.10.Ox,05.70.Ce,2.50.-r M. Mungan String Matching and 1d Lattice Gases (DRAFT, August 25, 2005) 1