Expected length of the longest common subsequence for large alphabets

Kiwi, Marcos; Loebl, Martin; Matoušek, Jiřı́

doi:10.1016/j.aim.2004.10.012

Cited by 66 publications

(64 citation statements)

References 18 publications

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“…For equally long sequences (i = j = n), it has been proved that L n,n ≈ γ c n for n ≫ 1, where the averaging is performed over all realizations of the random sequences. The constant γ c is known as the Chvátal-Sankoff constant which, to date, remains undetermined though there exists several bounds [12,14,15], a conjecture due to Steele [13] that γ c = 2/(1 + √ c) and a recent proof [16] that γ c → 2/ √ c as c → ∞. Unfortunately, no exact results are available for the finite size corrections to the leading behavior of the average L n,n , for the variance, and also for the full probability distribution of L n,n .…”

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confidence: 99%

“…(6)- (7) also hold for the LCS model. Note that only the leading behavior of the average L n,n was known before [16] in the c → ∞ limit of the original LCS model. For finite c, while the above mapping to the LIS problem still works, the corresponding permutations of the LIS problem are not generated with equal probability and hence one can no longer use the BDJ results.…”

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confidence: 99%

“…The aim of the LCS problem is to find the longest of such common subsequences. This problem and its variants have been widely studied in biology [4][5][6][7], computer science [8-10,2], probability theory [11][12][13][14][15][16] and more recently in statistical physics [17][18][19]. A particularly important application of the LCS problem is to quantify the closeness between two DNA sequences.…”

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confidence: 99%

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Exact asymptotic results for the Bernoulli matching model of sequence alignment

Majumdar

Nechaev

2005

Phys. Rev. E

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Finding analytically the statistics of the longest common subsequence (LCS) of a pair of random sequences drawn from c alphabets is a challenging problem in computational evolutionary biology. We present exact asymptotic results for the distribution of the LCS in a simpler, yet nontrivial, variant of the original model called the Bernoulli matching (BM) model which reduces to the original model in the c → ∞ limit. We show that in the BM model, for all c, the distribution of the asymptotic length of the LCS, suitably scaled, is identical to the Tracy-Widom distribution of the largest eigenvalue of a random matrix whose entries are drawn from a Gaussian unitary ensemble. In particular, in the c → ∞ limit, this provides an exact expression for the asymptotic length distribution in the original LCS problem.

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Exact asymptotic results for the Bernoulli matching model of sequence alignment

Majumdar

Nechaev

2005

Phys. Rev. E

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“…Although we are interested in the behavior of f ,h when h, we first analyze the behavior for h larger than (in particular when h is substantially larger than 2 /γ), and then show how to use this to obtain bounds for h; a similar approach was used in [5]. For n of size up to roughly 2 /γ we use part (b) of Definition 3 to bound F (n, , h) by the number G(n, ) of string pairs of size n with an lcs length of at least .…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

2008 Proceedings of the Fifth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

2008

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It has long been known [2] that the average length of the longest common subsequence of two random strings of length n over an alphabet of size k is asymptotic to γ k n for some constant γ k depending on k. The value of these constants remains unknown, and a number of papers have proved upper and lower bounds on them. In particular, in [6] we used a modification of methods of [3,4] for determining lower and upper bounds on γ k , combined with large computer computations, to obtain improved bounds on γ 2 . The method of [6] involved a parameter h; empirically, increasing h increased the computation time but gave better upper bounds. Here we show, for arbitrary k, a sufficient condition for a parameterized method to produce a sequence of upper bounds approaching the true value of γ k , and show that a generalization of the method of [6] meets this condition for all k ≥ 2. While [3, 4] do not explicitly discuss how to parameterize their method, which is based on a concept they call domination, to trade off the tightness of the bound vs. the amount of computation, we discuss a very natural parameterization of their method; for the case of alphabet size k = 2 we conjecture but do not prove that it also meets the sufficient condition and hence also yields a sequence of bounds that converges to the correct value of γ 2 . For k > 2, it does not meet our sufficient condition. Thus we leave open the question of whether some method based on the undominated collations of [3,4] gives bounds converging to the correct value for any k ≥ 2.

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“…However, both the use of mathematical apparatus as in [2,3] and numerical modeling (usually with special algorithms) [4,5,6] succeeded to clarify situation only in special cases m = n or α = 2 (see [7]). …”

Section: Introductionmentioning

confidence: 99%

A Formula for the Mean Length of the Longest Common Subsequence

Znamenskij¹

2017

J. Sib. Fed. Univ., Math. phys.

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The expected value E of the longest common subsequence of letters in two random words is considered as a function of the α = |A| of alphabet and of words lengths m and n. It is assumed that each letter independently appears at any position with equal probability. A simple expression for E(α, m, n) and its empirical proof are presented for fixed α and m + n. High accuracy of the formula in a wide range of values is confirmed by numerical simulations.

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Expected length of the longest common subsequence for large alphabets

Cited by 66 publications

References 18 publications

Exact asymptotic results for the Bernoulli matching model of sequence alignment

Exact asymptotic results for the Bernoulli matching model of sequence alignment

2008 Proceedings of the Fifth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

A Formula for the Mean Length of the Longest Common Subsequence

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