Narrow Results

Suppose that we are given a set of n elements d of which have a property called defective. A group test can check for any subset, called a pool, whether it contains a defective. It is known that a nearly optimal number of O(d log (n/d)) pools in two stages (where tests within a stage are done in parallel) are sufficient, but then the searcher must know d in advance. Here we explore group testing strategies that use a nearly optimal number of pools and a few stages although d is not known beforehand. We prove a lower bound of Ω(log d/log log d) stages and more general pools versus stages tradeoff. This is almost tight, since O(log d) stages are sufficient for a strategy with O(d log n) pools. As opposed to this negative result, we devise a randomized strategy using O(d log (n/d)) pools in three stages, with any desired success probability 1-ϵ. With some additional measures even two stages are enough. Open questions concern the optimal constant factors and practical implications. A related problem motivated by biological network analysis is to learn hidden vertex covers of a small size k in unknown graphs by edge group tests. (Does a given subset of vertices contain an edge?) We give a one-stage strategy using O(k³log n) pools, with any parameterized algorithm for vertex cover enumeration as a decoder. During the course of this work we also provide a classification of types of randomized search strategies in general.

Group testing is the problem of finding d defectives in a set of n elements, by asking carefully chosen subsets (pools) whether they contain defectives. Strategies are preferred that use both a small number of tests close to the information-theoretic lower bound d log₂ n, and a small constant number of stages, where tests in every stage are done in parallel, in order to save time. They should even work if d is not known in advance. In fact, one can succeed with O(d log n) queries in two stages, if certain tests are randomized and a constant failure probability is allowed. An essential ingredient of such strategies is to get an estimate of d within a constant factor. This problem is also interesting in its own right. It can be solved with O(log n) randomized group tests of a certain type. We prove that Ω(log n) tests are also necessary, if elements for the pools are chosen independently. The proof builds upon an analysis of the influence of tests on the searcher's ability to distinguish between any two candidate numbers with a constant ratio. The next challenge is to get optimal constant factors in the O(log n) test number, depending on the prescribed error probability and the accuracy of d. We give practical methods to derive upper bound tradeoffs and conjecture that they are already close to optimal. One of them uses a linear programming formulation.

High-throughput (HT) screening and combinatorial searches for the discovery, development and optimization of functional materials have been widely accepted in many new materials discovery. Dry powder HT library synthesis have advantages such as using same powder materials in lab as in production, and avoiding the use of additives and/or solvents which could be harmful for cells. The VaryDose dry powder dispensing technology was adapted in this work to dispense nanobioceramic powders in quantities as low as 0.1 mg per dispensing. Nanocalcium phosphate biomaterials, including hydroxyapatite (HA) and β-tricalcium phosphate (β-TCP), were selected to demonstrate the library fabrication. The dispensing unit design and the effect of the dispensing parameters on dosage control and uniformity are discussed.