Proceedings of the 17th Canadian Conference on Computational Geometry, CCCG 2005
We give the first complete analysis that the complexity of finding 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- find, 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.
Lee, Audrey; Streinu, Ileana; and Theran, Louis, "Finding and Maintaining Rigid Components" (2005). Computer Science: Faculty Publications, Smith College, Northampton, MA.