The topological complexity of a space X, introduced in 2001 by Farber, is a homotopy invariant of X which measures the complexity of the motion planning problem on X. For example, if X represents the space of all possible states of a robot arm, then the topological complexity of X measures the number of rules required to determine a complete algorithm which dictates how the robot arm will move from any given initial state to any given final state.

The geodesic complexity of a metric space, first introduced and studied by Recio-Mitter, is defined similarly, except that all movements must follow minimizing geodesics. With Davis and Recio-Mitter, we computed the geodesic complexity of configuration spaces of certain trees. Intuitively, our motion planners indicate how multiple points should simultaneously move on a graph without colliding.