Diagnosis is an essential subject for the reliability of a multiprocessor system. Under the comparison diagnosis model, Sengupta and Dahbura proposed a polynomial-time algorithm with time complexity O(N-5) to identify all the faulty processors for a given syndrome in a system with N processors. In this paper, we present a novel idea on system diagnosis called node diagnosability. The node diagnosability can be viewed as a local strategy toward system diagnosability. There is a strong relationship between the node diagnosability and the traditional diagnosability. For this local sense, we focus more on a single processor and require only identifying the status of this particular processor correctly. Under the comparison diagnosis model, we propose a sufficient condition to determine the node diagnosability of a given processor. Furthermore, we propose a useful local structure called an extended star to guarantee the node diagnosability and provide an efficient algorithm to determine the faulty or fault-free status of each processor based on this structure. For a multiprocessor system with total number of processors N, the time complexity of our algorithm to diagnose a given processor is O(log N) and that to diagnose all the faulty processors is O(N log N) under the comparison model, provided that there is an extended star structure at each processor and that the time for looking up the testing result of a comparator in the syndrome table is constant.