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This paper presents two algorithms for determining the most reliable paths of a communication network, taking into account both the reliabilities of nodes and links. A network is modeled by a linear graph. The first algorithm adapts a node labeling scheme; second is based on the shortest-path matrix algorithm. Both algorithms can be applied to oriented and nonoriented nonsymetric networks. The numerical examples confirm that both algorithms efficiently handle networks with hundreds of nodes and edges. The total amount of computation, expressed by the number of multiplications and comparisons, increases in the labeling procedure with the square of the number of nodes, and in the matrix method with the cube of the number of nodes.