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 even and a function , 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 for any . 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.

Bayagbag Condition[edit]

Deterministic[edit]

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

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

Randomized[edit]

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 queries.

Quantum Solution[edit]

The BHT algorithm, which uses Grover's algorithm, solves this problem optimally by only making queries to f.

References[edit]

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