Max-Throughput for (Conservative) k-of-n Testing

Abstract

We define a variant of k-of-n testing that we call conservative k-of-n testing. We present a polynomialtime, combinatorial algorithm for the problem of maximizing throughput of conservative k-of-n testing, in a parallel setting. This extends previous work of Kodialam and Condon et al., who presented combinatorial algorithms for parallel pipelined filter ordering, which is the special case where k = 1 (or k = n) [4,5,8]. We also consider the problem of maximizing throughput for standard k-of-n testing, and show how to obtain a polynomial-time algorithm based on the ellipsoid method using previous techniques. 1 In an alternative definition of k-of-n testing, the task is to determine whether at least k of the n tests have a value of 1. Symmetric results hold for this definition.MaxThroughput problems are closely related to MinCost problems [6,9]. In the MinCost problem for k-of-n testing, in addition to the probabilities p i , there is a cost c i associated with performing the i th test. The goal is to find a testing strategy (i.e. decision tree) that minimizes the expected cost of testing an individual item. There are polynomial-time algorithms for solving the MinCost problem for standard k-of-n testing [1,3,10,11].Kodialam was the first to study the MaxThroughput k-of-n testing problem, for the special case where k = 1 [8]. He gave a O(n 3 log n) algorithm for the problem. The algorithm is combinatorial, but its correctness proof relies on polymatroid theory. Later, Condon et al. studied the problem, calling it "parallel pipelined filter ordering". They gave two O(n 2 ) combinatorial algorithms, with direct correctness proofs [5].Our Results. In this paper, we extend the previous work by giving a polynomial-time combinatorial algorithm for the MaxThroughput problem for conservative k-of-n testing. Our algorithm can be implemented to run in time O(n 2 ), matching the running time of the algorithms of Condon et al. for 1-of-n testing. More specifically, the running time is O(n(log n + k) + o), where o varies depending on the output representation used; the algorithm can be modified to produce different output representations. We discuss output representations below.The MaxThroughput problem for standard k-of-n testing appears to be fundamentally different from its conservative variant. We leave as an open problem the task of developing a polynomial time combinatorial algorithm for this problem. We show that previous techniques can be used to obtain a polynomial-time algorithm based on the ellipsoid method. This approach could also be used to yield an algorithm, based on the ellipsoid method, for the conservative variant.Output Representation For the type of representation used by Condon et al. in achieving their O(n 2 ) bound, o = O(n 2 ). A more explicit representation has size o = O(n 3 ). We also describe a new, more compact output representation for which o = O(n).In giving running times, we follow Condon et al. and consider only the time taken by the algorithm to produce the output representation. We note, howe...

Sequential Testing of <i>k</i>-out-of-<i>n</i> Systems with Imperfect Tests

“…When only K 0 exists then an optimal policy for an instance of the k-out-of-n testing problem with imperfect tests is constructed from an optimal policy for K 0 -out-of-n conservative testing with values λ i as success probabilities, by interchanging every pair of child nodes at every parent node in its BDT representation. Hellerstein et al (2011) mention the following result for conservative k-out-of-n testing, which follows from a more general result obtained in Boros andÜnlüyurt (1999).…”

“…Computational results for sequencing tests of k-out-of-n systems with general precedence constraints can be found in Wei et al (2013). One specific variant of classic k-out-of-n testing is the so-called conservative k-out-of-n testing, which is defined in the same way, but testing now continues until either k successful tests are observed or until all n tests have been performed (Hellerstein et al, 2011). In this text, we will assume component tests to be independent; Cramer and Kamps (1996) present a generalization in which the failure rate of the untested components is parametrically adjusted based on the number of preceding failures.…”

“…Consider the conservative k-out-of-n testing problem, which is defined similarly as traditional kout-of-n testing (with perfect tests), except that we perform tests either until we have observed k tests with negative outcome, or we have performed all n tests (Hellerstein et al, 2011). In particular, the diagnosis is not interrupted after observing n − k + 1 tests with positive outcome.…”