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We describe an algorithm to compute finite presentations for the mapping class group of a connected, compact, orientable surface, possibly with boundary and punctures. By an inductive process, such an algorithm, starting  from a presentation well known for the mapping class group of the sphere and the torus with ``few'' boundary components and/or punctures, produces a presentation for the mapping class group of any other surface.

Finite presentations for the mapping class group via the ordered complex of curves