Asymptotic lower bound of O(n^2)

Question

Are there problems in P that have a proven asymptotic lower bound of O(n^2) or higher? (n is the number of bits a problem instance can be represented by). This is not a homework question, just curiosity.

In P? What's P? And yes, there are, for example bubble sort is `O(n ^ 2)`. — , Jun 22 '13 at 19:12
@H2CO3 (1) [P](http://en.wikipedia.org/wiki/P_%28complexity%29) as in "P=NP?" I suppose. (2) Please don't confuse problems and algorithms. — , Jun 22 '13 at 19:14
@H2CO3- The question asks whether there are problems that *require* quadratic time to solve, rather than whether there are *algorithms* that require quadratic time to finish. It's much harder to show that a problem can't be solved in subquadratic time. — templatetypedef, Jun 22 '13 at 19:29
@delnan Algorithms are ways to decide problems. It's just a different perspective. — G. Bach, Jun 22 '13 at 21:25
@G.Bach Exactly, they are closely related but not the same thing. As templatetypedef explained above, the complexity of a problem is not necessary the complexity of one particular algorithm that solves that problem. — , Jun 22 '13 at 21:27

score 6 · Accepted Answer · answered Jun 22 '13 at 19:12

6

Yes, the time hierarchy theorem implies the existence of such problems. They're arguably not natural because they involve diagonalizing over all O(n^2)-time algorithms.

answered Jun 22 '13 at 19:12

David Eisenstat

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score 2 · Answer 2 · answered Jun 23 '13 at 08:35

3SUM comes to mind. There's a quadratic lower bound known for a certain linear decision-tree model due to Jeff Erickson. (There are some barely-subquadratic algorithms for 3SUM in the literature for various models of computation. But I haven't looked at them and I don't know how they work.)

Asymptotic lower bound of O(n^2)

2 Answers2