Communication Complexity of Collision

The Collision problem is to decide whether a given list of numbers $(x_1,\dots ,x_n)\in [n{]}^n$ is $1$-to-$1$ or $2$-to-$1$ when promised one of them is the case. We show an $n^{\mathrm{\Omega }(1)}$ randomised communication lower bound for the natural two-party version of Collision where Alice holds the first half of the bits of each $x_i$ and Bob holds the second half. As an application, we also show a similar lower bound for a weak bit-pigeonhole search problem, which answers a question of Itsykson and Riazanov (CCC 2021).