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Novel conditions are derived that guarantee convergence of the sum-product algorithm (also known as loopy belief propagation or simply belief propagation (BP)) to a unique fixed point, irrespective of the initial messages, for parallel (synchronous) updates. The computational complexity of the conditions is polynomial in the number of variables. In contrast with previously existing conditions, our results are directly applicable to arbitrary factor graphs (with discrete variables) and are shown to be valid also in the case of factors containing zeros, under some additional conditions. The conditions are compared with existing ones, numerically and, if possible, analytically. For binary variables with pairwise interactions, sufficient conditions are derived that take into account local evidence (i.e., single-variable factors) and the type of pair interactions (attractive or repulsive). It is shown empirically that this bound outperforms existing bounds.