Rigid components in fixed-lattice and cone frameworks

We study the fundamental algorithmic rigidity problems for generic frameworks periodic with respect to a fixed lattice or a finite-order rotation in the plane. For fixed-lattice frameworks we give an $O(n^2)$ algorithm for deciding generic rigidity and an O(n^3) algorithm for computing rigid components. If the order of rotation is part of the input, we give an O(n^4) algorithm for deciding rigidity; in the case where the rotation's order is 3, a more specialized algorithm solves all the fundamental algorithmic rigidity problems in O(n^2) time.

[1]  Ciprian S. Borcea,et al.  Periodic frameworks and flexibility , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[2]  M. Thorpe,et al.  The flexibility window in zeolites , 2006, Nature materials.

[3]  Michael A. Bender,et al.  The LCA Problem Revisited , 2000, LATIN.

[4]  Igor Rivin,et al.  Enumeration of periodic tetrahedral frameworks. II. Polynodal graphs , 2004 .

[5]  Robert E. Tarjan,et al.  Fast Algorithms for Finding Nearest Common Ancestors , 1984, SIAM J. Comput..

[6]  N. S. Barnett,et al.  Private communication , 1969 .

[7]  Louis Theran,et al.  Generic combinatorial rigidity of periodic frameworks , 2010, 1008.1837.

[8]  Louis Theran,et al.  Generic rigidity of frameworks with crystallographic symmetry , 2011 .

[9]  Harold N. Gabow,et al.  Forests, frames, and games: algorithms for matroid sums and applications , 1988, STOC '88.

[10]  Michael Treacy,et al.  Enumeration of periodic tetrahedral frameworks , 1997 .

[11]  Harold N. Gabow,et al.  Forests, frames, and games: Algorithms for matroid sums and applications , 1992, STOC '88.

[12]  Bernd Schulze,et al.  Symmetric Versions of Laman’s Theorem , 2009, Discret. Comput. Geom..

[13]  Tibor Jordán,et al.  Algorithms for Graph Rigidity and Scene Analysis , 2003, ESA.

[14]  Audrey Lee-St. John,et al.  Finding and Maintaining Rigid Components , 2005, CCCG.

[15]  Audrey Lee-St. John,et al.  Pebble game algorithms and sparse graphs , 2007, Discret. Math..

[16]  Audrey Lee-St. John,et al.  Graded Sparse Graphs and Matroids , 2007, J. Univers. Comput. Sci..

[17]  L. Dale,et al.  FORESTS , 1994, Restoration & Management Notes.

[18]  B. Hendrickson,et al.  An Algorithm for Two-Dimensional Rigidity Percolation , 1997 .