# Finding longest increasing and common subsequences in streaming data

## Abstract

We present algorithms and lower bounds for the Longest Increasing Subsequence (LIS) and Longest Common Subsequence (LCS) problems in the data-streaming model. To decide if the LIS of a given stream of elements drawn from an alphabet Σ has length at least k, we discuss a one-pass algorithm using O(klog |Σ|) space, with update time either O(log k) or O(log log |Σ|); for |Σ| = O(1), we can achieve O(log k) space and constant-time updates. We also prove a lower bound of Ω2(k) on the space requirement for this problem for general alphabets Σ, even when the input stream is a permutation of Σ. For finding the actual LIS, we give a [log(1 + 1/ε)-pass algorithm using O(k1+ε log |Σ|) space, for any ε > 0. For LCS, there is a trivial Θ(1)-approximate O(log n)-space streaming algorithm when |Σ| = O(1). For general alphabet Σ, the problem is much harder. We prove several lower bounds on the LCS problem, of which the strongest is the following: it is necessary to use Ω(n/ρ2} space to approximate the LCS of two n-element streams to within a factor of ρ, even if the streams are permutations of each other. © Springer-Verlag Berlin Heidelberg 2005.