Classical Solution
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.
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.
Read more about this topic: Collision Problem
Famous quotes containing the words classical and/or solution:
“Culture is a sham if it is only a sort of Gothic front put on an iron building—like Tower Bridge—or a classical front put on a steel frame—like the Daily Telegraph building in Fleet Street. Culture, if it is to be a real thing and a holy thing, must be the product of what we actually do for a living—not something added, like sugar on a pill.”
—Eric Gill (1882–1940)
“There is a lot of talk now about metal detectors and gun control. Both are good things. But they are no more a solution than forks and spoons are a solution to world hunger.”
—Anna Quindlen (b. 1953)