Complexity, the Changing Minimum and Closest Pair

1 Las Vegas and Monte Carlo algorithms Definition 1.1 A Las Vegas algorithm!Las Vegas algorithm is a randomized algorithms that always return the correct result. The only variant is that it's running time might change between executions. An example for a Las Vegas algorithm is the QuickSort algorithm. Definition 1.2 Monte Carlo algorithm!Monte Carlo algorithm is a randomized algorithm that might output an incorrect result. However, the probability of error can be diminished by repeated executions of the algorithm. The MinCut algorithm was an example of a Monte Carlo algorithm. If you do now know what are those things, you should read about them. Some of that is covered in the randomized algorithms book, and some other stuff is covered in any basic text on complexity theory. Definition 1.3 The class P consists of all languages L that have a polynomial time algorithm A, such that for any input Σ * , • x ∈ L ⇒ A(x) accepts. Definition 1.4 The class NP consists of all languages L that have a polynomial time algorithm A, such that for any input Σ * , • x ∈ L ⇒ then ∃y ∈ Σ * , A(x, y) accepts, where |y| (i.e. the length of y) is bounded by a polynomial in |x|. Definition 1.5 For a complexity class C, we define the complementary class co-C as the set of languages whose complement is in the class C.