Several efficient, but non-optimal, solutions to the Selection Problem on coarse grained parallel computers have appeared in literature. We consider example of Saukas-Song algorithm; we analyze it without expressing running time terms communication rounds. This shows that while best case algorithm runs asymptotically optimal time, general does not. propose another for selection has expected time.

We first define the coarse-graining of probability measures in terms stochastic kernels. when a measure is part another and say that two coexist if they are both parts single measure. then show any coexist. extend these concepts to observables instruments mention need not discretization an observable as special case have \zeroone next consider finite cases, kernels replaced by matrices. also sa...