Collision problem

Template:Short description Template:Distinguish 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 $f:\{1,\dots ,n\}\to \{1,\dots ,n\}$, 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 $i\in \{1,\dots ,n\}$. 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 $\frac{n}{2}+1$ queries, and in general distinguishing r-to-1 functions from 1-to-1 functions requires $\frac{n}{r}+1$ queries.

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

Quantum solution

The BHT algorithm, which uses Grover's algorithm, solves this problem optimally by only making $O(n^{1/3})$ queries to f.