### Abstract

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend the previous work and propose the 3D-As-Synchronized-As-Possible (3D-ASAP) algorithm, for the graph realization problem in R^{3}, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch, there corresponds an element of the Euclidean group, Euc(3), of rigid transformations in R^{3}, and the goal was to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-spectral-partitioning (SP)-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a pre-processing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably with similar state-of-the-art localization algorithms.

Original language | English (US) |
---|---|

Pages (from-to) | 21-67 |

Number of pages | 47 |

Journal | Information and Inference |

Volume | 1 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2012 |

### Fingerprint

### Keywords

- Conquer
- Distance geometry
- Divide
- Eigenvectors
- Graph realization
- Rigidity theory
- SDP
- Spectral graph theory
- Synchronization
- The molecule problem

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Analysis
- Applied Mathematics
- Statistics and Probability
- Numerical Analysis

### Cite this

*Information and Inference*,

*1*(1), 21-67. https://doi.org/10.1093/imaiai/ias002

**Eigenvector synchronization, graph rigidity and the molecule problem.** / Cucuringu, Mihai; Singer, Amit; Cowburn, David.

Research output: Contribution to journal › Article

*Information and Inference*, vol. 1, no. 1, pp. 21-67. https://doi.org/10.1093/imaiai/ias002

}

TY - JOUR

T1 - Eigenvector synchronization, graph rigidity and the molecule problem

AU - Cucuringu, Mihai

AU - Singer, Amit

AU - Cowburn, David

PY - 2012/1/1

Y1 - 2012/1/1

N2 - The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend the previous work and propose the 3D-As-Synchronized-As-Possible (3D-ASAP) algorithm, for the graph realization problem in R3, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch, there corresponds an element of the Euclidean group, Euc(3), of rigid transformations in R3, and the goal was to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-spectral-partitioning (SP)-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a pre-processing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably with similar state-of-the-art localization algorithms.

AB - The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend the previous work and propose the 3D-As-Synchronized-As-Possible (3D-ASAP) algorithm, for the graph realization problem in R3, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch, there corresponds an element of the Euclidean group, Euc(3), of rigid transformations in R3, and the goal was to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-spectral-partitioning (SP)-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a pre-processing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably with similar state-of-the-art localization algorithms.

KW - Conquer

KW - Distance geometry

KW - Divide

KW - Eigenvectors

KW - Graph realization

KW - Rigidity theory

KW - SDP

KW - Spectral graph theory

KW - Synchronization

KW - The molecule problem

UR - http://www.scopus.com/inward/record.url?scp=84892391966&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892391966&partnerID=8YFLogxK

U2 - 10.1093/imaiai/ias002

DO - 10.1093/imaiai/ias002

M3 - Article

AN - SCOPUS:84892391966

VL - 1

SP - 21

EP - 67

JO - Information and Inference

JF - Information and Inference

SN - 2049-8772

IS - 1

ER -