Polynomial-time algorithms for the curve graph
  We describe a polynomial-time algorithm to compute a (tight) geodesic between two curves in the curve graph. As well as enabling us to compute the distance between a pair of curves, this has several applications to mapping classes. For example, we can use these geodesics to compute the asymptotic translation length, Nielsen--Thurston type, and canonical curve system of a mapping class in polynomial time in its word length. 
Mark C. Bell and Richard C. H. Webb
math.GT