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:
“Et in Arcadia ego.
[I too am in Arcadia.]”
—Anonymous, Anonymous.
Tomb inscription, appearing in classical paintings by Guercino and Poussin, among others. The words probably mean that even the most ideal earthly lives are mortal. Arcadia, a mountainous region in the central Peloponnese, Greece, was the rustic abode of Pan, depicted in literature and art as a land of innocence and ease, and was the title of Sir Philip Sidney’s pastoral romance (1590)
“Coming out, all the way out, is offered more and more as the political solution to our oppression. The argument goes that, if people could see just how many of us there are, some in very important places, the negative stereotype would vanish overnight. ...It is far more realistic to suppose that, if the tenth of the population that is gay became visible tomorrow, the panic of the majority of people would inspire repressive legislation of a sort that would shock even the pessimists among us.”
—Jane Rule (b. 1931)