The "component counting lower bound ll known for deterministic linear search algorithms (LSA's) also holds for their probabilistic versions (PLSA's) for many problems, even if two-sided error is allowed, and if one does not charge for probabilistic choice. This implies lower bounds on PLSA's for e.g. the element distinctness problem (n log n) or the knapsack problem (n 2 ). These results yield the first separations between probabilistic and non-deterministic l ..SA's, because the above problems are nondeterministically much easier. Previous lower bounds for PLSA's either only worked for one-sided error lion the nice side", i.e. on the side where the problems are even non-deterministically hard, or only for probabilistic comparison trees. The proof of the loweb ound differs fundamentally from all known lower bounds for LSA's or PLSA's, because it does not reduce the problem to a combinatorial one but argues extensively about e.g. a non-discrete measure for similarity of sets in R". This lower bound result solves an open problem posed by Manber and Tompa as well as by Snir. Furthermore, a PLSA for n input variables with two-sided error and expected runtime T can be simulated by a (deterministic) LSA in 'fln steps. This proves that the gaps between probabilistic and deterministic LSA's shown by Snir cannot. be too large. As this simulation even holds for algebraic computation trees we show that probabilistic and deterministic versions of this model are polynomially related. This is a weaker version of a result due to the author which shows that in case of LSA's, even the non-deterministic and deterministic versions are polynomially related.
The component counting argument is generalized for lower bounds on twosided error linear PADT in REF .
Nondeterministic versus probabilistic linear search algorithms
26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
Computer Science