We consider the problem of :nding the longest increasing subsequence in a sliding window over a given sequence (LISW). We propose an output-sensitive data structure that solves this problem in time O(n log log n+OUTPUT) for a sequence of n elements. This data structure substantially improves over the na?@ve generalization of the longest increasing subsequence algorithm and in fact produces an o...

In this paper we give parallel solutions to the problem of finding the Longest Increasing Subsequence of a given sequence of n integers. First, we show the existence of a simple dynamic programming solution. Its running time is Θ(n) and its space requirement is Θ(n). Then, we continue by showing that it is possible to develop two parallel solutions based on optical bus system of n processors, o...

We study the fluctuations, in the large deviations regime, of the longest increasing subsequence of a random i.i.d. sample on the unit square. In particular, our results yield the precise upper and lower exponential tails for the length of the longest increasing subsequence of a random permutation. §.

The interplay between two-dimensional percolation growth models and one-dimensional particle processes has always been a fruitful source of interesting mathematical phenomena. In this paper we develop a connection between the construction of Busemann functions in the Hammersley last-passage percolation model with i.i.d. random weights, and the existence, ergodicity and uniqueness of equilibrium...

We give a survey of some of the recent results on certain two-dimensional random growth models and their relation to random matrix theory, in particular to the Tracy-Widom distribution for the largest eigenvalue. The problems are related to that of finding the length of the longest increasing subsequence in a random permutation. We also give a new approach to certain results for the Schur measu...

We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single “time” (fixed y) distribution is the Tracy-Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has ...

We consider the concentration of measure for n iid, two-dimensional random variables, under the conditioning that they form a record. Under mild conditions, we show that all random variables tend to concentrate, as n ! 1, around limiting curves, which are the solutions of an appropriate variational problem. We also show that the same phenomenum occurs, without the records conditioning, for the ...