John Hempel has suggested a simple method for finding reasonably efficient paths between loops in the curve complex. Recall that the curve complex is the simplicial complex whose vertices are isotopy classes of essential simple closed curves in a given surface and whose simplices bound collections of pairwise disjoint (pairwise non-isotopic) loops. A path in this complex is a sequence of loops in the given surface such that consecutive loops are disjoint. It’s not too hard to show that this complex is connected. A geodesic between two vertices is a path of minimal length.
This algorithm might generalize to surfaces with boundary, but I’m going to assume we have a closed surface. We start with two loops, say a and z that have been isotoped to intersect minimally and whose complement is a collection of disks. (If a component of the complement contains a non-trivial loop then they are distance one or two and finding a geodesic is trivial.) Each component of the complement is a polygon such that the edges of the polygon alternate between arcs of a and arcs of z. Put a vertex at the center of each such polygon and draw an edge from the vertex to each edge of the polygon that comes from an arc of z. We connect the edges along arcs of z to form a graph.
The vertices in squares in the complement are valence two so we can erase these vertices and think of only the vertices in the larger polygons. The resulting graph turns out to be a spine for the complement in our surface of the loop a. Any simple closed edge path determines an essential loop in the complement of a and Hempel suggests we let b be any such loop. I want to be a little more cautious. Since there are a finite number of simple closed edge paths, we might as well pick one that intersects z minimally. In fact, if we do this then we will have chosen a loop in the complement of a that minimizes the intersection number with z over all loops in the complement of a. (This is a reasonably straightforward exercise.) We can then repeat the process with b and z and so on until we get to a loop that is distance two from z.
On principle, this process should give us a very efficient path. The question is, does it yield a geodesic? Hempel has shown that it does for very short paths (I think he said it works for distance three or four). Now, there is already an algorithm for computing geodesics in the curve complex, due to work of Kenneth Shakleton. However, Hempel’s algorithm is extremely simple. The algorithm points in the direction that locally seems best, always choosing a loop that is distance one from a and intersects z minimally. It would be very interesting to see if such a simple idea could be effective for understanding an object as complex and unforgiving as the curve complex.