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 solutions

Deterministic

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

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

References

Scott Aaronson (2004). "Limits on Efficient Computation in the Physical World" (PDF).

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[1] Scott Aaronson (2004). "Limits on Efficient Computation in the Physical World" (PDF).