Backtracking Candidate Elimination: A One-Pass Algorithm for the Chip Testing Problem

Abstract

In the chip testing problem, we are given $n$ chips, strictly more than half of which are good. Chips can test one another in pairs; a good chip always reports the status of the other chip correctly, whereas a bad chip may report arbitrarily and adversarially. The goal is to identify a single chip that is guaranteed to be good. The problem originates in system-level fault diagnosis and is closely related to the "knights and spies" puzzle. The standard textbook solution is a halving recursion that tests disjoint pairs in rounds and keeps one chip from each consistent pair. We present the Backtracking Candidate Elimination (BCE) algorithm, a sequential alternative that scans the chips once while maintaining a current candidate and a stack of retained chips. Every chip is tested at most once as the incoming chip; when a test is inconclusive the candidate and the incoming chip are discarded together, and the algorithm backtracks to the most recently retained chip. BCE uses at most $n-1$ tests and $O(n)$ time, needs no parity case analysis, and works online. Its correctness follows from two invariants: the retained chips all have the same type, and every discarded pair contains at least one bad chip. We explain how BCE can be viewed as the Boyer-Moore majority vote algorithm with its counter replaced by a stack of physical witnesses, and why that replacement is needed. We also give an early termination rule and a variant for the weaker model of one-directional tests.