A note on randomized streaming space bounds for the longest increasing subsequence problem

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 the lower bound is based have very efficient randomized protocols. The purpose of this note is to guide and alert future researchers working on this very interesting problem.