Heinrich Matzinger

Standard deviation of the longest common subsequence

Let L n be the length of the longest common subsequence of two independent i.i.d. sequences of Bernoulli variables of length n. We prove that the order of the standard deviation of L n is Vn, provided the parameter of the Bernoulli variables is small enough. This validates Watermans conjecture in this situation [Philos. Trans. R. Soc. Lond. Ser. B 344 (1994) 383-390]. The order conjectured by Chvatal and Sankoff [J. Appl. Probab. 12 (1975) 306-315], however, is different.

Reconstruction of sceneries with correlated colors

Matzinger (Random Structure Algorithm 15 (1999a) 196) showed how to reconstruct almost every three color scenery, that is a coloring of the integers with three colors, by observing it along the path of a simple random walk, if this scenery is the outcome of an i.i.d. process. This reconstruction needed among others the transience of the representation of the scenery as a random walk on the three-regular tree T3. Den Hollander (private communication) asked which conditions are necessary to ensure this transience of the representation of the scenery as a random walk on T3 and whether this already suffices to make the reconstruction techniques in Matzinger (1999a) work. In this note we answer the latter question in the affirmative. Also we exhibit a large class of examples where the above-mentioned transience holds true. Some counterexamples show that in some sense the given class of examples is the largest natural class with the property that the representation of the scenery as a random walk is transient.

Reconstructing a two-color scenery by observing it along a simple random walk path

Let {\xi (n)}_{n\in Z} be a two-color random scenery, that is, a random coloring of Z in two colors, such that the \xi (i)s are i.i.d. Bernoulli variables with parameter \tfrac12. Let {S(n)}_{n\in N} be a symmetric random walk starting at 0. Our main result shows that a.s., \xi \circ S (the composition of \xi and S) determines \xi up to translation and reflection. In other words, by observing the scenery \xi along the random walk path S, we can a.s. reconstruct \xi up to translation and reflection. This result gives a positive answer to the question of H. Kesten of whether one can a.s. detect a single defect in almost every two-color random scenery by observing it only along a random walk path.

Reconstructing a piece of scenery with polynomially many observations

Benjamini asked whether the scenery reconstruction problem can be solved using only polynomially many observations. In this article, we answer his question in the affirmative for an i.i.d. uniformly colored scenery on observed along a random walk path with bounded jumps. We assume the random walk is recurrent, can reach every integer with positive probability, and the number of possible single steps for the random walk exceeds the number of colors. For infinitely many l, we prove that a finite piece of scenery of length l around the origin can be reconstructed up to reflection and a small translation from the first p(l) observations with high probability; here p is a polynomial and the probability that the reconstruction succeeds converges to 1 as l-->[infinity].

On the Variance of the Optimal Alignments Score for Binary Random Words and an Asymmetric Scoring Function

We investigate the order of the variance of the optimal alignments (OA) score of two independent iid binary random words having the same length. The letters are equiprobable, but the scoring function is such that one letter has a larger score than the other. In this setting, we prove that the order of variance is linear in the common length. OAs constitute a generalization of longest common subsequences, they can be represented as optimal paths in a two-dimensional last passage percolation setting with dependent weights.

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

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.

The rate of the convergence of the mean score in random sequence comparison

We consider a general class of super-additive scores measuring the similarity of two independent sequences of

Proportion of Gaps and Fluctuations of the Optimal Score in Random Sequence Comparison

n

Closeness to the diagonal for longest common subsequences in random words

i.i.d. letters from a finite alphabet. Our object of interest is the mean score by letter

Optimal alignments of longest common subsequences and their path properties

l_n