Collision problem explained

Collision problem should not be confused with Collision detection 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\} → \{ 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 $\backslash frac+1$ queries, and in general distinguishing r-to-1 functions from 1-to-1 functions requires $\backslash frac\; +\; 1$ queries.

This is a straightforward application of the pigeonhole principle: if a function is r-to-1, then after $\backslash frac\; +\; 1$ queries we are guaranteed to have found a collision. If a function is 1-to-1, then no collision exists. Thus, $\backslash frac\; +\; 1$ queries suffice. If we are unlucky, then the first

n/r queries could return distinct answers, so $\backslash frac\; +\; 1$ 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{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

1. Web site: Scott Aaronson . Scott Aaronson. Limits on Efficient Computation in the Physical World . 2004.