This paper proposes Warshall algorithm-a greedy algorithm for determining connectedness and clustering in matrix-weighted graphs. Built upon a sufficient condition for connectedness, the algorithm reveals a simple but important difference between matrix-weighted and scalar-weighted graphs. In a scalar-weighted graph, once a path exists between two vertices, their connectedness is assured. In a matrix-weighted graph, however, two vertices may belong to two different clusters even if one or several paths exist between them. We propose a heuristic method for determining connectedness between two vertices by exhaustingly considering all simple paths joining them and combining the kernel of these paths via two newly developed matrix logic operators. Our proposed algorithm not only reduces the computational complexity of the brute-force search approach but also enables a distributed implementation. Proofs of correctness, finite step convergence, computational complexity analyses, and numerical examples are provided to establish the validity of the algorithms and to demonstrate their effectiveness.

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