Longest common subsequences and the Bernoulli matching model: numerical   work and analyses of the r-reach simplification
  The expected length of longest common subsequences is a problem that has been in the literature for at least twenty five years. Determining the limiting constants \gamma_k appears to be quite difficult, and the current best bounds leave much room for improvement. Boutet de Monvel explores an independent version of the problem he calls the Bernoulli Matching model. He explores this problem and its relation to the longest common subsequence problem. This paper continues this pursuit by focusing on a simplification we term r-reach. For the string model, L_r(u,v) is the longest common subsequence of u and v given that each matched pair of letters is no more than r letters apart. 
Jonah Blasiak