Approximate set union via approximate randomization

We develop a randomized approximation algorithm for the size of set union problem |A1∪A2∪...∪Am|, which is given list sets A1,...,Am with approximate mi Ai mi∈((1−βL)|Ai|, (1+βR)|Ai|), and biased random generators probability Prob(x=RandomElement(Ai))∈[1−αL|Ai|,1+αR|Ai|] each input element x∈Ai, where i=1,2,...,m αL,αR,βL,βR∈(0,1). The ratio |A1∪A2∪...∪Am| in range [(1−ϵ)(1−αL)(1−βL),(1+ϵ)(1+αR) (1+βR)] any ϵ∈(0,1). complexity measured by both time round complexity. One has non-adaptive accesses to those RandomElement(Ai) functions 1≤i≤m, membership queries (x∈Ai?) 1≤i≤m. Our gives an scheme O(m⋅(log⁡m)7) running O(log⁡m) rounds contrast existing [1] that needs Ω(m) worst case O((1+ϵ)m/ϵ2) time, m number sets. flexible tradeoff O(m1+ξ) O(1ξ) ξ∈(0,1). runs sublinear under certain condition A1∪A2∪...∪Am belongs ma fixed a>0, our best knowledge, we have not seen results about this problem. can handle generate elements bias, its depends on bias. prove it #P-hard count lattice points balls, also show there no polynomial intersection n-dimensional balls unless P=NP. As applications algorithm, propose algorithms counting high dimensional maximal coverage balls.