Expected Length of the Longest Common Subsequence for Large Alphabets

We consider the length L of the longest common subsequence of two randomly uniformly and independently chosen n character words over a k-ary alphabet. Subadditivity arguments yield that E[L]/n converges to a constant γ k . We prove a conjecture of Sankoff and Mainville from the early 80’s claiming that \(\gamma_{\kappa}\sqrt{k}\longrightarrow 2\) as \(K \longrightarrow \infty\).

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