Finding and Maintaining Rigid Components

Audrey Lee, University of Massachusetts Amherst
Ileana Streinu, Smith College
Louis Theran, University of Massachusetts Amherst

Archived as published.

Abstract

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.