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This paper proposes a new algorithm that enables observability analysis (restoration and identification of observable islands) and identification of critical measurements and critical sets for observable or unobservable systems. The proposed algorithm is based on path graph concepts and triangular factorization of the Jacobian matrix, and has the following suitable characteristics for real-time operation: it is fast, simple, easy to implement and does not require solutions of any algebraic equation. Small numerical examples, using a 7 bus test system, that show the application of the proposed algorithm are presented. The IEEE 14, 30 and 57 bus systems with different measurement scenarios were used to evaluate the performance of the proposed algorithm.