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Diagnosis is an essential subject for the reliability of multiprocessor systems. Under the PMC diagnosis model, Dahbura and Masson proposed a polynomial-time algorithm with time complexity O(N2.5) 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, and formalized by Hsu and Tan. 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 O(log N) degree, the time complexity of our algorithm to diagnose any given processor is O((log N)2), provided that each processor can construct an augmenting star structure of full order in time O((log N)2) and the time for a processor to test another one is constant. Therefore, the time totals to O(N(log N)2) for diagnosing the whole system.