Does Prior Knowledge Help Detect Collisions?

Suppose you are given a function $f\colon [n] \to [n]$ via (black-box) query access to the function. You are looking to find something local, like a collision (a pair $x \neq y$ s.t.\ $f(x)=f(y)$). The question is whether knowing the `shape' of the function helps you or not (by shape we mean that some permutation of the function is known). Our goal in this work is to characterize all local properties for which knowing the shape may help, compared to an algorithm that does not know the shape. Formally, we investigate the instance optimality of fundamental substructure detection problems in graphs and functions. Here, a problem is considered instance optimal (IO) if there exists an algorithm $A$ for solving the problem which satisfies that for any possible input, the (randomized) query complexity of $A$ is at most a multiplicative constant larger than the query complexity of any algorithm $A'$ for solving the same problem which also holds an unlabeled copy of the input graph or function. We provide a complete characterization of those constant-size substructure detection problems that are IO. Interestingly, our results imply that collision detection is not IO, showing that in some cases an algorithm holding an unlabeled certificate requires a factor of $\Theta(\log n)$ fewer queries than any algorithm without a certificate. We conjecture that this separation result is tight, which would make collision detection an ``almost instance optimal'' problem. In contrast, for all other non-trivial substructures, such as finding a fixed point, we show that the separation is polynomial in $n$.