Abstract
Let us be given a set of water samples where possibly some are contaminated with a chemical substance. We wish to find these ”defective” samples and their concentrations. Assume that there is an indicator available that can only detect whether the concentration is at least a known threshold, but we may split, merge, and dilute different samples. This problem is a generalized version of the well-known group testing problem and is also related to learning Boolean threshold functions.
First we consider the problem of approximating the concentration in a single sample by a threshold indicator only. We present a practical and efficient strategy that achieves a close approximation using a small number of tests and auxiliary operations. This strategy is formulated as a pebble moving game on a suitable graph.
Then we give principal asymptotic complexity results for finding, in a set of samples, those samples where the concentration exceeds a given limit. The results for the number of tests are asymptotically optimal, furthermore we propose a way for the efficient use of the other resources.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
I.Althöfer, E.Triesch: Edge search in graphs and hypergraphs of bounded rank, Discrete Math. 115 (1993), 1–9
J.A.Aslam, A.Dhagat: Searching in the presence of linearly bounded errors, 23th ACM STOC 1991, 486–493
A.Bar-Noy, F.K.Hwang, I.Kessler, S.Kutten: A new competitive algorithm for group testing, Discrete Applied Math. 52 (1994)
E.Boros, P.L.Hammer, T.Ibaraki, K.Kawakami: Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle, SIAM J. Computing, to appear
P.Damaschke: A tight upper bound for group testing in graphs, Discrete Applied Math. 48 (1994), 101–109
P.Damaschke: A parallel algorithm for nearly optimal edge search, Info. Proc. Letters 56 (1995), 233–236
P. Damaschke: Searching for faulty leaves in binary trees, in: M. Nagl (ed.), Graph-Theoretic Concepts in Computer Science — 21st Int. Workshop WG'95, Aachen 1995, LNCS 1017 (Springer), 265–274
D.Z.Du, F.K.Hwang: Competitive group testing, Discrete Applied Math. 45 (1993)
K.Makino, T.Ibaraki: A fast and simple algorithm for identifying 2-monotonic positive Boolean functions, Proc. 6th ISAAC'95, LNCS 1004, Springer 1995, 291–300
D.Z.Du, H.Park: On competitive group testing, SIAM J. Comp. 23 (1994), 1019–1025
D.Z.Du, G.L.Xue, S.Z.Sun, S.W.Cheng: Modifications of competitive group testing, SIAM J. Comp. 23 (1994), 82–96
R.Reischuk: Einführung in die Komplexitätstheorie (in German), Teubner, Stuttgart 1990
E.Triesch: A group testing problem for hypergraphs of bounded rank, 1995, submitted
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Damaschke, P. (1997). The algorithmic complexity of chemical threshold testing. In: Bongiovanni, G., Bovet, D.P., Di Battista, G. (eds) Algorithms and Complexity. CIAC 1997. Lecture Notes in Computer Science, vol 1203. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62592-5_73
Download citation
DOI: https://doi.org/10.1007/3-540-62592-5_73
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-62592-6
Online ISBN: 978-3-540-68323-0
eBook Packages: Springer Book Archive