Finding longest increasing and common subsequences in streaming data

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 α bet has length at least k , we discuss a one-pass algorithm using O ( k log α betsize ) space, with update time either O (log k ) or O (log log α betsize ); for α betsize = O (1), we can achieve O (log k ) space and constant-time updates. We also prove a lower bound of Ω( k ) on the space requirement for this problem for general alphabets α bet , even when the input stream is a permutation of α bet . For finding the actual LIS, we give a ⌈log (1 + 1/ɛ)-pass algorithm using O ( k 1+ɛ log α betsize ) space, for any ɛ > 0. For LCS, there is a trivial Θ(1)-approximate O (log n )-space streaming algorithm when α betsize = O (1). For general alphabets α bet , 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 /ρ 2 ) 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.