Approximate Set Union Via Approximate Randomization
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We develop an randomized approximation algorithm for the size of set union problem A_1∪ A_2∪...∪ A_m, which given a list of sets A_1,...,A_m with approximate set size m_i for A_i with m_i∈((1-β_L)|A_i|, (1+β_R)|A_i|), and biased random generators with Prob(x=(A_i))∈[1-α_L |A_i|,1+α_R |A_i|] for each input set A_i and element x∈ A_i, where i=1, 2, ..., m. The approximation ratio for A_1∪ A_2∪...∪ A_m is in the range [(1-ϵ)(1-α_L)(1-β_L), (1+ϵ)(1+α_R)(1+β_R)] for any ϵ∈ (0,1), where α_L, α_R, β_L,β_R∈ (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(·()^O(1)) running time and O( m) 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(^1+ξ) and round complexity O(1ξ) for any ξ∈(0,1).
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