Probabilistic Algorithms and Complexity Classes

The main theme in this dissertation is a computation on a probabilistic Turing machine (PTM), which is a Turing machine with distinguished states called coin-tossing states. In general, it is assumed that a PTM has a fair coin. This is led by that any xed biased coin can simulate another xed biased coin in a constant expected time. We rst consider the computation which will always terminates in bounded time, as opposed to expected time, and show that the coin with probability 1 2 or 1 3 is more expressive coin than the others. The complexity classes PP;BPP and RP are usually de ned via PTMs with a perfect random source. Vazirani and Vazirani proved that RP does not change even if the PTMs only have access to an arbitrary semi-random source. Later, Vazirani proved that BPP does not change, either. We show that PP collapses to BPP if the PTMs only have access to an arbitrary semi-random source. We also show that RP changes to NP while both BPP and PP change to PSPACE if the PTMs only have access to a speci c semi-random source. Next problem is to identify a Boolean function f out of a given set of Boolean functions F by asking for the value of f at adaptively chosen inputs. We consider the cases where F consists of functions obtained from one function g on n inputs by replacing k inputs by given constants, and g is one of the functions disjunction, parity and threshold. For many of the problems we get almost optimal bounds. The maximal path set (MPS) problem is to nd, given an undirected graph G = (V;E), a maximal subset F of E such that the subgraph induced by F is a forest in which each connected component is a path. We design two parallel algorithms for the problem. The rst algorithm is faster and more e cient than the previously known algorithm. The second one is faster, runs on a weaker computation model, and is more e cient for input graphs of bounded degree. We then use the results to design an NC approximation algorithm for a variation of the shortest superstring problem. We also consider the vertex counterpart of the MPS problem. The NP-completeness of several problems are shown in the last chapter. The rst problem is the generalized popular puzzle known as Hi-Q. The second problem is the vertex cover problem on a cubic, planar, and 3-connected graph of girth greater than 3. Problems on a triangulation of the sphere are the last.