The space complexity of deterministic streaming algorithms for approximating the length of the longest increasing subsequence (LIS) in a string of length $n$ has been known to be $\tilde{\Theta}(\sqrt{n})$ for almost two decades. In contrast, the space complexity of this problem for randomized streaming algorithms remains one of the few longstanding open problems in one-pass streaming. In fact, no better than $\Omega(\log n)$ lower bounds are known, and the best upper bounds are no better than their deterministic counterparts.

In this paper, we push the limits of our understanding of the streaming space complexity of the approximate LIS length problem by studying it in the white-box adversarial streaming model. This model is an intermediate model between deterministic and randomized streaming algorithms that has recently attracted attention. In the white-box model, the streaming algorithm can draw fresh randomness when processing each incoming element, but an adversary generating the stream observes all previously used randomness and adaptively chooses the subsequent elements of the stream.

We prove a tight (up to logarithmic factors) $\Omega(\sqrt{n})$ space lower bound for any white-box streaming algorithm that approximates the length of the LIS of a stream of length $n$ to within a factor better than $1.1$. Thus, for this problem, white-box algorithms offer no improvement over deterministic ones.