## Efficiently decodable non-adaptive group testing

##### Author(s)

Indyk, Piotr; Ngo, Hung Q.; Rudra, Atri
DownloadIndyk_Efficiently decodable.pdf (310.2Kb)

OPEN_ACCESS_POLICY

# Open Access Policy

Creative Commons Attribution-Noncommercial-Share Alike

##### Terms of use

##### Metadata

Show full item record##### Abstract

We consider the following "efficiently decodable" non-adaptive
group testing problem. There is an unknown string
x 2 f0; 1gn [x is an element of set {0,1} superscript n] with at most d ones in it. We are allowed to test
any subset S [n] [S subset [n] ]of the indices. The answer to the test
tells whether xi = 0 [x subscript i = 0] for all i 2 S [i is an element of S] or not. The objective
is to design as few tests as possible (say, t tests) such that
x can be identifi ed as fast as possible (say, poly(t)-time).
Efficiently decodable non-adaptive group testing has applications
in many areas, including data stream algorithms and
data forensics.
A non-adaptive group testing strategy can be represented
by a t x n matrix, which is the stacking of all the
characteristic vectors of the tests. It is well-known that if
this matrix is d-disjunct, then any test outcome corresponds
uniquely to an unknown input string. Furthermore, we know
how to construct d-disjunct matrices with t = O(d2 [d superscript 2] log n)
efficiently. However, these matrices so far only allow for a
"decoding" time of O(nt), which can be exponentially larger
than poly(t) for relatively small values of d.
This paper presents a randomness efficient construction
of d-disjunct matrices with t = O(d2 [d superscript 2] log n) that can be decoded
in time poly(d) [function composed of] t log2 t + O(t2) [t log superscript 2 t and O (t superscript 2)]. To the best of our
knowledge, this is the first result that achieves an efficient decoding
time and matches the best known O(d2 log n) [O (d superscript 2 log n)] bound
on the number of tests. We also derandomize the construction,
which results in a polynomial time deterministic construction
of such matrices when d = O(log n= log log n).
A crucial building block in our construction is the
notion of (d,l)-list disjunct matrices, which represent the
more general "list group testing" problem whose goal is to
output less than d + l positions in x, including all the (at
most d) positions that have a one in them. List disjunct
matrices turn out to be interesting objects in their own right
and were also considered independently by [Cheraghchi,
FCT 2009]. We present connections between list disjunct
matrices, expanders, dispersers and disjunct matrices. List
disjunct matrices have applications in constructing (d,l)-
sparsity separator structures [Ganguly, ISAAC 2008] and in
constructing tolerant testers for Reed-Solomon codes in the
data stream model.
1 Introduction

##### Date issued

2010-01##### Department

Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science##### Journal

ACM-SIAM Symposium on Discrete Algorithms (SODA). Proceedings, 21st, 2010

##### Publisher

Society for Industrial and Applied Mathematics / Association for Computing Machinery

##### Citation

Indyk, Piotr, Hung Q. Ngo and Atri Rudra. "Efficiently Decodable Non-adaptive Group Testing" ACM-SIAM Symposium on Discrete Algorithms, 21st, January 17-19, 2010 Hyatt Regency Austin, Austin, Texas.

Version: Author's final manuscript

##### ISSN

1071-9040