We give improved algorithms for constructing minimum directed and undirected cycle bases in graphs. For general graphs, the new algorithms are Monte Carlo and have running time O(m ω ), where ω is the exponent of matrix multiplication. The previous best algorithm had running timeÕ(m 2 n). For planar graphs, the new algorithm is deterministic and has running time O(n 2 ). The previous best algorithm had running time O(n 2 log n). A key ingredient to our improved running times is the insight that the search for minimum bases can be restricted to a set of candidate cycles of total length O(nm).

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