We consider scenarios in which long sequences of data are analyzed and subsequences must be traced that are monotone and maximum, according to some measure. A classical example is the online Longest Increasing Subsequence Problem for numeric and alphanumeric data. We extend the problem in two ways: (a) we allow data from any partially ordered set, and (b) we maximize subsequences using much mor...

We study the uctuations, 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. i=1 denote a sequence of i.i.d. random variables with marginal law on the unit square Q = 0; 1] 2. ...

Let λ be a fixed integer, λ ≥ 2. Let s n be any strictly increasing sequence of positive integers satisfying s n ≤ n 15/14+o(1). In this paper we give a version of the large sieve inequality for the sequence λ sn. In particular, we prove that for π(X)(1 + o(1)) primes p, p ≤ X, the numbers λ sn , n ≤ X(log X) 2+ε are uniformly distributed modulo p.

The deterministic space complexity of approximating the length of the longest increasing subsequence of a stream of N integers is known to be Θ̃( √ N). However, the randomized complexity is wide open. We show that the technique used in earlier work to establish the Ω( √ N) deterministic lower bound fails strongly under randomization: specifically, we show that the communication problems on which...

By reviewing Longest Increasing Subsequence (LIS) and Longest Common Subsequence (LCS), the Longest Common Increasing Subsequence (LCIS) problem is explored for two non-random input cases in details. Specifically, we designed two algorithms, one solving the input sequence scenario with the case that one sequence is ordered and duplicate elements are allowed in each of sequences, and the second ...

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...