Approximate Set Union via Approximate Randomization

Abstract

We develop a randomized approximation algorithm for the size of set union problem $| A_1\cup A_2\cup \ldots \cup A_{m}|$, which is given a list of sets $A_1,\,\ldots ,\,A_{m}$ with approximate set size $m_i$ for $A_i$ with $m_i\in ((1-\beta _L)|A_i|$, $(1+\beta _R)|A_i|)$, and biased random generators with probability $\mathrm{Prob}\left( x=\mathrm{RandomElement}(A_i)\right) \in \left[ {1-\alpha _L\over |A_i|},\, {1+\alpha _R\over |A_i|}\right]$ for each input set $A_i$ and element $x\in A_i,$ where $i=1,\, 2,\, \ldots ,\,m$ and $\alpha _L,\, \alpha _R,\, \beta _L,\,\beta _R\in (0,\,1)$. The approximation ratio for $| A_1\cup A_2\cup \ldots \cup A_m|$ is in the range $[(1-\epsilon )(1-\alpha _L)(1-\beta _L),\, (1+\epsilon )(1+\alpha _R)$ $(1+\beta _R)]$ for any $\epsilon \in (0,\,1).$ The complexity of the algorithm is measured by both time complexity and round complexity. One round of the algorithm has non-adaptive accesses to those $\mathrm{RandomElement}(A_i)$ functions $1\le i\le m$, and membership queries ($x\in A_i$?) to input sets $A_i$ with $1\le i\le m$. Our algorithm gives an approximation scheme with ${O}(m\cdot (\log m)^{7})$ running time and ${O}(\log m)$ rounds in contrast to the existing algorithm  [18] that needs $\varOmega (m)$ rounds in the worst case with ${O}((1+\epsilon )m/\epsilon ^2)$ running time, where $m$ is the number of sets. Our algorithm gives a flexible tradeoff with time complexity ${O}\left( m^{1+\xi }\right)$ and round complexity ${O}\left( {1\over \xi }\right)$ for any $\xi \in (0,\,1)$. Our algorithm runs sublinear in time under certain condition that each element in $A_1\cup A_2\cup \ldots \cup A_{m}$ belongs to $m^a$ sets for any fixed $a>0$, to our best knowledge, we have not seen any sublinear results about this problem.