Collision problem

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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.

[edit] Classical Solution

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

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