## Abstract

In this work, we study the problem of estimating the number of defectives d in a set of n items up to a multiplicative factor of Δ>1 using deterministic group testing algorithms. Particularly, given Δ>1 and D>d, we give upper and lower bounds on the number of tests required for the task of generating an estimation dˆ such that d/Δ≤dˆ≤dΔ. In the adaptive settings, we prove that any adaptive deterministic algorithm for estimating d must perform at least Ω((D/Δ^{2})log(n/D)) tests. This extends the same lower bound achieved in [1] for non-adaptive algorithms. In addition, we show that our bound is tight up to a small additive term by constructing an adaptive algorithm for this task. Furthermore, we give a non-constructive proof of an upper bound of O((D/Δ^{2}) (log(n/D)+logΔ)) for the non-adaptive settings. This bound is an improvement over the upper bound O((logD)/(logΔ))Dlogn) from [1], and matches the lower bound up to a small additive term. Moreover, we use existing techniques for building expander regular bipartite graphs, extractors and condensers from [2,3] to construct two polynomial-time non-adaptive algorithms for estimating d. Using the construction from [2], a polynomial non-adaptive algorithm that makes O((D^{1+o(1)}/Δ^{2})⋅logn) tests are defined. The second algorithm exploits the construction from [3] to build another non-adaptive algorithm that performs (D/Δ^{2})⋅Quazipoly (logn) tests. This is the first explicit construction with an almost optimal test complexity.

Original language | English |
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Pages (from-to) | 46-58 |

Number of pages | 13 |

Journal | Theoretical Computer Science |

Volume | 874 |

DOIs | |

State | Published - 12 Jun 2021 |

## Keywords

- Deterministic group testing
- Group testing
- Pooling design

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- General Computer Science