Faster Algorithms for Rigidity in the Plane

TL;DR

This paper improves the efficiency of algorithms for recognizing Laman graphs, which characterize rigidity in the plane, reducing the running time from quadratic to near-linear using novel modifications.

Contribution

It introduces a simplified and faster algorithm for constructing red-black hierarchies, enabling efficient recognition of Laman graphs in near-linear time.

Findings

Red-black hierarchy construction time reduced to O(n log n)

Recognition of Laman graphs achieved in O(n log n) time

Algorithm improves previous quadratic time methods

Abstract

In [1], a new construction called red-black hierarchy characterizing Laman graphs and an algorithm for computing it were presented. For a Laman graph G=(V,E) with n vertices it runs in O(n^2) time assuming that a partition of (V,E+e) into two spanning trees is given. We show that a simple modification reduces the running time to O(n\log n). The total running time can be reduced O(n\sqrt{n\log n}) using the algorithm by Gabow and Westermann [2] for partitioning a graph into two forests. The existence of a red-black hierarchy is a necessary and sufficient condition for a graph to be a Laman graph. The algorithm for constructing a red-black hierarchy can be then modified to recognize Laman graphs in the same time.