Upper bounds for the expected length of a longest common subsequence of two binary sequences

1995 ◽  
Vol 6 (4) ◽  
pp. 449-458 ◽  
Author(s):  
Vlado Dančík ◽  
Mike Paterson
Keyword(s):  
Upper Bounds ◽  
Binary Sequences ◽  
Longest Common Subsequence ◽  
Expected Length ◽  
Common Subsequence

Longest common subsequences of two random sequences

1975 ◽  
Vol 12 (02) ◽  
pp. 306-315 ◽  
Author(s):  
Vacláv Chvátal ◽  
David Sankoff
Keyword(s):  
Monte Carlo ◽  
Molecular Evolution ◽  
Lower Bounds ◽  
Monte Carlo Methods ◽  
Longest Common Subsequence ◽  
Upper And Lower Bounds ◽  
Longest Common Subsequences ◽  
Expected Length ◽  
Common Subsequence ◽  
Limiting Behaviour

Summary Given two random k-ary sequences of length n, what is f(n,k), the expected length of their longest common subsequence? This problem arises in the study of molecular evolution. We calculate f(n,k) for all k, where n ≦ 5, and f(n,2) where n ≦ 10. We study the limiting behaviour of n –1 f(n,k) and derive upper and lower bounds on these limits for all k. Finally we estimate by Monte-Carlo methods f(100,k), f(1000,2) and f(5000,2).

Longest common subsequences of two random sequences

1975 ◽  
Vol 12 (2) ◽  
pp. 306-315 ◽  
Author(s):  
Vacláv Chvátal ◽  
David Sankoff
Keyword(s):  
Monte Carlo ◽  
Molecular Evolution ◽  
Lower Bounds ◽  
Monte Carlo Methods ◽  
Longest Common Subsequence ◽  
Upper And Lower Bounds ◽  
Longest Common Subsequences ◽  
Expected Length ◽  
Common Subsequence ◽  
Limiting Behaviour

SummaryGiven two random k-ary sequences of length n, what is f(n,k), the expected length of their longest common subsequence? This problem arises in the study of molecular evolution. We calculate f(n,k) for all k, where n ≦ 5, and f(n,2) where n ≦ 10. We study the limiting behaviour of n–1f(n,k) and derive upper and lower bounds on these limits for all k. Finally we estimate by Monte-Carlo methods f(100,k), f(1000,2) and f(5000,2).

scholarly journals On a Speculated Relation Between Chvátal–Sankoff Constants of Several Sequences

2009 ◽  
Vol 18 (4) ◽  
pp. 517-532
Author(s):  
M. KIWI ◽  
J. SOTO
Keyword(s):  
Lower Bounds ◽  
Longest Common Subsequence ◽  
Expected Length ◽  
Common Subsequence

It is well known that, when normalized byn, the expected length of a longest common subsequence ofdsequences of lengthnover an alphabet of size σ converges to a constant γσ,d. We disprove a speculation by Steele regarding a possible relation between γ2,dand γ2,2. In order to do that we also obtain some new lower bounds for γσ,d, when both σ anddare small integers.

scholarly journals Large deviations-based upper bounds on the expected relative length of longest common subsequences

2006 ◽  
Vol 38 (03) ◽  
pp. 827-852 ◽  
Author(s):  
Raphael Hauser ◽  
Servet Martínez ◽  
Heinrich Matzinger
Keyword(s):  
High Precision ◽  
Upper Bound ◽  
Large Deviation ◽  
Relative Length ◽  
Random Variable ◽  
Upper Bounds ◽  
Longest Common Subsequence ◽  
Upper And Lower Bounds ◽  
Finite Alphabet ◽  
Common Subsequence

Consider the random variable L n defined as the length of a longest common subsequence of two random strings of length n and whose random characters are independent and identically distributed over a finite alphabet. Chvátal and Sankoff showed that the limit γ=lim n→∞E[L n ]/n is well defined. The exact value of this constant is not known, but various methods for the computation of upper and lower bounds have been discussed in the literature. Even so, high-precision bounds are hard to come by. In this paper we discuss how large deviation theory can be used to derive a consistent sequence of upper bounds, (q m ) m∈ℕ, on γ, and how Monte Carlo simulation can be used in theory to compute estimates, q̂ m , of the q m such that, for given Ξ > 0 and Λ ∈ (0,1), we have P[γ < q̂ < γ + Ξ] ≥ Λ. In other words, with high probability the result is an upper bound that approximates γ to high precision. We establish O((1 − Λ)−1Ξ−(4+ε)) as a theoretical upper bound on the complexity of computing q̂ m to the given level of accuracy and confidence. Finally, we discuss a practical heuristic based on our theoretical approach and discuss its empirical behavior.

Common Subsequences and Supersequences and their Expected Length

1998 ◽  
Vol 7 (4) ◽  
pp. 365-373 ◽  
Author(s):  
VLADO DANČÍK
Keyword(s):  
Longest Common Subsequence ◽  
Expected Length ◽  
Common Subsequence ◽  
Shortest Common Supersequence

Let f(n, k, l) be the expected length of a longest common subsequence of l sequences of length n over an alphabet of size k. It is known that there are constants γ(l)k such that f(n, k, l)→ γ(l)kn as n→∞, and we show that γ(l)k= Θ(k1/l−1) as k→∞. Bounds for the corresponding constants for the expected length of a shortest common supersequence are also presented.

scholarly journals Expected length of the longest common subsequence for large alphabets

2005 ◽  
Vol 197 (2) ◽  
pp. 480-498 ◽  
Author(s):  
Marcos Kiwi ◽  
Martin Loebl ◽  
Jiří Matoušek
Keyword(s):  
Longest Common Subsequence ◽  
Expected Length ◽  
Common Subsequence

scholarly journals Large deviations-based upper bounds on the expected relative length of longest common subsequences

2006 ◽  
Vol 38 (3) ◽  
pp. 827-852 ◽  
Author(s):  
Raphael Hauser ◽  
Servet Martínez ◽  
Heinrich Matzinger
Keyword(s):  
High Precision ◽  
Upper Bound ◽  
Large Deviation ◽  
Relative Length ◽  
Random Variable ◽  
Upper Bounds ◽  
Longest Common Subsequence ◽  
Upper And Lower Bounds ◽  
Finite Alphabet ◽  
Common Subsequence

Consider the random variable Ln defined as the length of a longest common subsequence of two random strings of length n and whose random characters are independent and identically distributed over a finite alphabet. Chvátal and Sankoff showed that the limit γ=limn→∞E[Ln]/n is well defined. The exact value of this constant is not known, but various methods for the computation of upper and lower bounds have been discussed in the literature. Even so, high-precision bounds are hard to come by. In this paper we discuss how large deviation theory can be used to derive a consistent sequence of upper bounds, (qm)m∈ℕ, on γ, and how Monte Carlo simulation can be used in theory to compute estimates, q̂m, of the qm such that, for given Ξ > 0 and Λ ∈ (0,1), we have P[γ < q̂ < γ + Ξ] ≥ Λ. In other words, with high probability the result is an upper bound that approximates γ to high precision. We establish O((1 − Λ)−1Ξ−(4+ε)) as a theoretical upper bound on the complexity of computing q̂m to the given level of accuracy and confidence. Finally, we discuss a practical heuristic based on our theoretical approach and discuss its empirical behavior.

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