Finding and Maintaining Rigid Components

We give the flrst complete analysis that the complexity of flnding and maintaining rigid components of planar bar-and-joint frameworks and arbitrary d-dimensional body-and-bar frameworks, using a family of algorithms called pebble games, is O(n2). To this end, we intro- duce a new data structure problem called union pair- flnd, which maintains disjoint edge sets and supports pair-find queries of whether two vertices are spanned by a set. We present solutions that apply to generalizations of the pebble game algorithms, beyond the original rigidity motivation.