Collision Problem

The r-to-1 collision problem is an important theoretical problem in complexity theory, quantum computing, and computational mathematics. The collision problem most often refers to the 2-to-1 version: given $n$ even and a function $f:\,\\rightarrow\$, we are promised that f is either 1-to-1 or 2-to-1. We are only allowed to make queries about the value of $f(i)$ for any $i\in\$. The problem then asks how many such queries we need to make to determine with certainty whether f is 1-to-1 or 2-to-1.

Classical solutions

Deterministic

Solving the 2-to-1 version deterministically requires $\frac+1$ queries, and in general distinguishing r-to-1 functions from 1-to-1 functions requires $\frac + 1$ queries.

This is a straightforward application of the pigeonhole principle: if a function is r-to-1, then after $\frac + 1$ queries we are guaranteed to have found a collision. If a function is 1-to-1, then no collision exists. Thus, $\frac + 1$ queries suffice. If we are unlucky, then the first $n/r$ queries could return distinct answers, so $\frac + 1$ queries is also necessary.

Randomized

If we allow randomness, the problem is easier. By the birthday paradox, if we choose (distinct) queries at random, then with high probability we find a collision in any fixed 2-to-1 function after $\Theta(\sqrt)$ queries.

Quantum solution

The BHT algorithm, which uses Grover's algorithm, solves this problem optimally by only making $O(n^)$ queries to ''f''.
The matching lower bound of $\Omega(n^)$ was proved by Aaronson and Shi using the polynomial method{{cite web , author = Scott Aaronson and Yaoyun Shi, title = Quantum lower bounds for the collision and the element distinctness problems , year = 2004, url = https://doi.org/10.1145/1008731.1008735.

References

Algorithms
Polynomial-time problems