An efficient algorithm for testing the generic rigidity of graphs in the plane

Abstract

Given a structure made up of n sites connected by b bars, the problem of recognizing which subsets of sites form rigid units is not a trivial one because of the non-local character of rigidity in central-force systems. Even though this is a very old problem of statics, no simple algorithms are available for it, so the most usual approach has been to solve the elastic equations, which is very time consuming for large systems. Recently an integer algorithm was proposed for this problem in two dimensions, using matching methods from graph theory and Laman's theorem for two-dimensional graphs. The method is relatively simple, but its time complexity grows as in the worst case, and almost as fast on practical cases, so that an improvement is highly desirable. In this paper we describe a further elaboration of that procedure, which relies upon the description of the system as a collection of rigid bodies connected by bars, instead of sites connected by bars. Sets of rigidly connected objects are replaced by a unique body, and this is done recursively as more rigid connections between bodies are discovered at larger scales. As a consequence of this `rescaling transformation', our algorithm has much improved average behaviour, even when its worst-case complexity remains . The time complexity of the body - bar algorithm is found to scale as for the randomly diluted triangular lattice, while the original site - bar version scales as for the same problem.