Approximate Set Union Via Approximate Randomization

TL;DR

This paper introduces a randomized approximation algorithm for estimating the size of the union of multiple sets, accommodating biased random element generation and providing a tradeoff between time and round complexity.

Contribution

It presents a novel approximation scheme that handles biased random sampling and offers adjustable tradeoffs between computational resources.

Findings

Achieves an approximation ratio within a specified range considering bias.

Runs in nearly linear time with respect to the number of sets, with logarithmic rounds.

Supports biased random element generation with quantifiable impact on accuracy.

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 $m_i\in \left((1-\beta_L)|A_i|, (1+\beta_R)|A_i|\right)$, and biased random generators with $Prob(x=\randomElm(A_i))\in \left[{1-\alpha_L\over |A_i|},{1+\alpha_R\over |A_i|}\right]$ for each input set $A_i$ and element $x\in A_i,$ where $i=1, 2, ..., m$. The approximation ratio for $\arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert$ is in the range $[(1-\epsilon)(1-\alpha_L)(1-\beta_L), (1+\epsilon)(1+\alpha_R)(1+\beta_R)]$ for any $\epsilon\in (0,1)$, where $\alpha_L, \alpha_R, \beta_L,\beta_R\in (0,1)$. 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 $O\left(\setCount^{1+\xi}\right)$ and round complexity $O\left({1\over \xi}\right)$ for any $\xi\in(0,1)$.