Approximate Set Union Via Approximate Randomization

Abstract

We develop an randomized approximation algorithm for the size of set union problem $\arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert$, which given a list of sets $A_1,...,A_m$ with approximate set size $m_i$ for $A_i$ with $m_i\in \left((1-β_L)|A_i|, (1+β_R)|A_i|\right)$, and biased random generators with $Prob(x=\randomElm(A_i))\in \left[{1-α_L\over |A_i|},{1+α_R\over |A_i|}\right]$ for each input set $A_i$ and element $x\in A_i,$ where $i=1, 2, ..., m$. The approximation ratio for $\arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert$ is in the range $[(1-ε)(1-α_L)(1-β_L), (1+ε)(1+α_R)(1+β_R)]$ for any $ε\in (0,1)$, where $α_L, α_R, β_L,β_R\in (0,1)$. 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 $O(\setCount\cdot(\log \setCount)^{O(1)})$ running time and $O(\log m)$ 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 $O\left(\setCount^{1+ξ}\right)$ and round complexity $O\left({1\over ξ}\right)$ for any $ξ\in(0,1)$.