Diagnosis is an essential subject for the reliability of multiprocessor systems. Under the PMC diagnosis model, Dahbura and Masson [12] proposed a polynomial-time algorithm with time complexity OðN 2:5 Þ to identify all the faulty processors in a system with N processors. In this paper, we present a novel method to diagnose a conditionally faulty system by applying the concept behind the local diagnosis, introduced by Somani and Agarwal [30], and formalized by Hsu and Tan [18]. The goal of local diagnosis is to identify the fault status of any single processor correctly. Under the PMC diagnosis model, we give a sufficient condition to estimate the local diagnosability of a given processor. Furthermore, we propose a helpful structure, called the augmenting star, to efficiently determine the fault status of each processor. For an N-processor system in which every processor has an Oðlog NÞ degree, the time complexity of our algorithm to diagnose any given processor is Oððlog NÞ 2 Þ, provided that each processor can construct an augmenting star structure of full order in time Oððlog NÞ 2 Þ and the time for a processor to test another one is constant. Therefore, the time totals to OðNðlog NÞ 2 Þ for diagnosing the whole system.