Approximate Set Union Via Approximate Randomization

We develop an randomized approximation algorithm for the size of set union problem |A1 ∪ A2 ∪ ... ∪ Am|, which given a list of sets A1, ..., Am with approximate set size mi for Ai with mi ∈ ((1− βL)|Ai|, (1 + βR)|Ai|), and biased random generators with Prob(x = RandomElement(Ai)) ∈ [ 1−αL |Ai| , 1+αR |Ai| ] for each input set Ai and element x ∈ Ai, where i = 1, 2, ..., m. The approximation ratio for |A1∪A2∪ ...∪Am| is in the range [(1−ǫ)(1−αL)(1− βL), (1 + ǫ)(1 + αR)(1 + βR)] for any ǫ ∈ (0, 1), where αL, αR, βL, βR ∈ (0, 1). The complexity of the algorithm is measured by both time complexity, and round complexity. The algorithm is allowed to make multiple membership queries and get random elements from the input sets in one round. Our algorithm makes adaptive accesses to input sets with multiple rounds. Our algorithm gives an approximation scheme with O(m · (logm)) running time and O(logm) rounds, where m is the number of sets. Our algorithm can handle input sets that can generate random elements with bias, and its approximation ratio depends on the bias. Our algorithm gives a flexible tradeoff with time complexity O ( m ) and round complexity O (