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 ^[1]: given $n$ even and a function $f:\,\{1,\ldots,n\}\rightarrow\{1,\ldots,n\}$ , 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 $i\in\{1,\ldots,n\}$ . 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 Solution

Solving the 2-to-1 version deterministically requires $n/2%2B1$ queries, and in general distinguishing r-to-1 functions from 1-to-1 functions requires $n/r%2B1$ queries.

This is a straightforward application of the pigeonhole principle: if a function is r-to-1, then after $n/r%2B1$ queries we are guaranteed to have found a collision. If a function is 1-to-1, then no collision exists. Thus, $n/r%2B1$ queries suffice. If we are unlucky, then the first $n/r$ queries could return distinct answers, so $n/r%2B1$ queries is also necessary.

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 $\Theta(\sqrt{n})$ queries.

Quantum Solution

References

^ Scott Aaronson (2004) (PDF). Limits on Efficient Computation in the Physical World. http://www.scottaaronson.com/thesis.pdf.