We show that the $r$-round communication complexity of the $k$-disjointness problem is precisely $\Theta(k\log^{(r)} k)$ bits. Setting $r=\log^* k$, our upper bound gives an $O(k)$-bits, $\log^* k$-rounds protocol with error probability exponentially small in $k$. This improves the best previous protocol due to Håstad and Wigderson, which ran in $O(\log k)$ rounds and erred with constant probability.
Our lower bound applies even to the simpler task of computing the OR of $k$ independent instances of equality. An important aspect of our work is that the lower bound we get is super-linear in $k$, even though a single instance of equality can be solved with $O(1)$ bits and constant error probability.
We present $L_p$ sampling algorithms using $O(\log^2 n)$ bits of space for $p\in[0,2)$ in the streaming model and show that $\Omega(\log^2 n)$ space is required by any such algorithm.
As an application, we present an $O(\log^2 n)$ space algorithm for finding a duplicate in a stream of length $n+1$ over the alphabet $[n]$, improving on the previous $O(\log^3 n)$ bound due to Gopalan and Radhakrishnan.
We give an $O(\log^2 n)$ space one pass streaming algorithm that finds the period of a stream, provided the stream is periodic. At the core of this algorithm is a new one pass $O(\log n\log m)$ space pattern matching algorithm for finding occurrences of a length $m$ pattern in a text of size $n$. This algorithm uses similar ideas to Porat and Porat’s algorithm in FOCS 2009 but it does not need an offline pre-processing stage and is simpler.
In the second part, we study distance to $p$-periodicity under the Hamming metric and provide space efficient approximation algorithms.