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The problem of finding a minimum decomposition of a permutation in terms of transpositions with predetermined non-uniform and non-negative costs is addressed. Alternatively, computing the transposition distance between two permutations, where transpositions are endowed with arbitrary non-negative costs, is studied. For such cost functions, polynomial-time, constant-approximation decomposition algorithms are described. For metric-path costs, exact polynomial-time decomposition algorithms are presented. The algorithms in this paper represent a combination of Viterbi-type algorithms and graph-search techniques for minimizing the cost of individual transpositions, and dynamic programing algorithms for finding minimum cost decompositions of cycles. The presented algorithms have a myriad of applications in information theory, bioinformatics, and algebra.