Disjoint spread systems and fault location

TL;DR

This paper investigates the minimal number of nonadaptive tests needed to identify faulty factor-level pairs in complex systems, focusing on cases with at most one fault, by analyzing set partitions with specific properties.

Contribution

It establishes the exact number of tests required for fault localization when at most one fault exists, linking the problem to counting particular set partitions.

Findings

Derived the minimum number of tests for fault detection with a single fault.

Connected fault localization to counting set partitions with distinct classes.

Provided theoretical bounds and exact values for test suite sizes.

Abstract

When $k$ factors each taking one of $v$ levels may affect the correctness or performance of a complex system, a test is selected by setting each factor to one of its levels and determining whether the system functions as expected (passes the test) or not (fails). In our setting, each test failure can be attributed to at least one faulty (factor, level) pair. A nonadaptive test suite is a selection of such tests to be executed in parallel. One goal is to minimize the number of tests in a test suite from which we can determine which (factor, level) pairs are faulty, if any. In this paper, we determine the number of tests needed to locate faults when exactly one (or at most one) pair is faulty. To do this, we address an equivalent problem, to determine how many set partitions of a set of size $N$ exist in which each partition contains $v$ classes and no two classes in the partitions are equal.