Linking Rigid Bodies Symmetrically

The mathematical theory of rigidity of body-bar and body-hinge frameworks provides a useful tool for analyzing the rigidity and flexibility of many articulated structures appearing in engineering, robotics and biochemistry. In this paper we develop a symmetric extension of this theory which permits a rigidity analysis of body-bar and body-hinge structures with point group symmetries. The infinitesimal rigidity of body-bar frameworks can naturally be formulated in the language of the exterior (or Grassmann) algebra. Using this algebraic formulation, we derive symmetry-adapted rigidity matrices to analyze the infinitesimal rigidity of body-bar frameworks with Abelian point group symmetries in an arbitrary dimension. In particular, from the patterns of these new matrices, we derive combinatorial characterizations of infinitesimally rigid body-bar frameworks which are generic with respect to a point group of the form $\mathbb{Z}/2\mathbb{Z}\times \dots \times \mathbb{Z}/2\mathbb{Z}$. Our characterizations are given in terms of packings of bases of signed-graphic matroids on quotient graphs. Finally, we also extend our methods and results to body-hinge frameworks with Abelian point group symmetries in an arbitrary dimension. As special cases of these results, we obtain combinatorial characterizations of infinitesimally rigid body-hinge frameworks with $\mathcal{C}_2$ or $\mathcal{D}_2$ symmetry - the most common symmetry groups found in proteins.Comment: arXiv:1308.6380 version 1 was split into two papers. The version 2 of arXiv:1308.6380 consists of Sections 1 - 6 of the version 1. This paper is based on the second part of the version 1 (Sections 7 and 8

Data-Oblivious Graph Algorithms in Outsourced External Memory

Motivated by privacy preservation for outsourced data, data-oblivious external memory is a computational framework where a client performs computations on data stored at a semi-trusted server in a way that does not reveal her data to the server. This approach facilitates collaboration and reliability over traditional frameworks, and it provides privacy protection, even though the server has full access to the data and he can monitor how it is accessed by the client. The challenge is that even if data is encrypted, the server can learn information based on the client data access pattern; hence, access patterns must also be obfuscated. We investigate privacy-preserving algorithms for outsourced external memory that are based on the use of data-oblivious algorithms, that is, algorithms where each possible sequence of data accesses is independent of the data values. We give new efficient data-oblivious algorithms in the outsourced external memory model for a number of fundamental graph problems. Our results include new data-oblivious external-memory methods for constructing minimum spanning trees, performing various traversals on rooted trees, answering least common ancestor queries on trees, computing biconnected components, and forming open ear decompositions. None of our algorithms make use of constant-time random oracles.Comment: 20 page