Does Prior Knowledge Help Detect Collisions?
Electron. Colloquium Comput. Complex.(2023)
摘要
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 $n$.
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