To load balance in Internet, we need more valid routing paths to share load in the case of no long-term routing loops to be introduced. It is acknowledged to adopt near or relaxed best routing to extend the number of available paths in multi-path routing. However, it is difficult to determine the degree of approximation or relaxed. A new distributed algorithm (which is called similarity multi-paths routing, SMR) for the dynamic computation of multiple paths from source to destination in a computer network is presented in this paper. SMR uses similarity principle to computes similarity coefficient between the shortest path and other paths, and then makes use of similarity coefficient to estimate the degree of approximation. Simulations show us it is robust for SMR to select near or relaxed best paths. Based on SMR, we also propose a traffic balancing algorithm. Its average performance is analyzed by simulation and compared against Equal Cost Multi-path (ECMP).