Partition into heapable sequences, heap tableaux and a multiset extension of Hammersley's process

We investigate partitioning of integer sequences into heapable subsequences (previously defined and established by Mitzenmacher et al). We show that an extension of patience sorting computes the decomposition into a minimal number of heapable subsequences (MHS). We connect this parameter to an interactive particle system, a multiset extension of Hammersley's process, and investigate its expected value on a random permutation. In contrast with the (well studied) case of the longest increasing subsequence, we bring experimental evidence that the correct asymptotic scaling is $\frac{1+\sqrt{5}}{2}\cdot \ln(n)$. Finally we give a heap-based extension of Young tableaux, prove a hook inequality and an extension of the Robinson-Schensted correspondence

When the law of large numbers fails for increasing subsequences of random permutations

Let the random variable $Z_{n,k}$ denote the number of increasing subsequences of length $k$ in a random permutation from $S_n$, the symmetric group of permutations of $\{1,...,n\}$. In a recent paper [Random Structures Algorithms 29 (2006) 277--295] we showed that the weak law of large numbers holds for $Z_{n,k_n}$ if $k_n=o(n^{2/5})$; that is, $$\lim_{n\to\infty}\frac{Z_{n,k_n}}{EZ_{n,k_n}}=1\qquad in probability.$$ The method of proof employed there used the second moment method and demonstrated that this method cannot work if the condition $k_n=o(n^{2/5})$ does not hold. It follows from results concerning the longest increasing subsequence of a random permutation that the law of large numbers cannot hold for $Z_{n,k_n}$ if $k_n\ge cn^{1/2}$, with $c>2$. Presumably there is a critical exponent $l_0$ such that the law of large numbers holds if $k_n=O(n^l)$, with $l<l_0$, and does not hold if $\limsup_{n\to\infty}\frac{k_n}{n^l}>0$, for some $l>l_0$. Several phase transitions concerning increasing subsequences occur at $l=1/2$, and these would suggest that $l_0={1/2}$. However, in this paper, we show that the law of large numbers fails for $Z_{n,k_n}$ if $\limsup_{n\to\infty}\frac{k_n}{n^{4/9}}=\infty$. Thus, the critical exponent, if it exists, must satisfy $l_0\in[{2/5},{4/9}]$.Comment: Published at http://dx.doi.org/10.1214/009117906000000728 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Compressed Spaced Suffix Arrays

Spaced seeds are important tools for similarity search in bioinformatics, and using several seeds together often significantly improves their performance. With existing approaches, however, for each seed we keep a separate linear-size data structure, either a hash table or a spaced suffix array (SSA). In this paper we show how to compress SSAs relative to normal suffix arrays (SAs) and still support fast random access to them. We first prove a theoretical upper bound on the space needed to store an SSA when we already have the SA. We then present experiments indicating that our approach works even better in practice

Growth models, random matrices and Painleve transcendents

The Hammersley process relates to the statistical properties of the maximum length of all up/right paths connecting random points of a given density in the unit square from (0,0) to (1,1). This process can also be interpreted in terms of the height of the polynuclear growth model, or the length of the longest increasing subsequence in a random permutation. The cumulative distribution of the longest path length can be written in terms of an average over the unitary group. Versions of the Hammersley process in which the points are constrained to have certain symmetries of the square allow similar formulas. The derivation of these formulas is reviewed. Generalizing the original model to have point sources along two boundaries of the square, and appropriately scaling the parameters gives a model in the KPZ universality class. Following works of Baik and Rains, and Pr\"ahofer and Spohn, we review the calculation of the scaled cumulative distribution, in which a particular Painlev\'e II transcendent plays a prominent role.Comment: 27 pages, 5 figure