## The unassigned distance geometry problem

## Abstract

Studies of distance geometry problems (DGP) have focused on cases where the vertices at the ends of all or most of the given distances are known or assigned, which we call assigned distance geometry problems (aDGPs). In this contribution we consider the unassigned distance geometry problem (uDGP) where the vertices associated with a given distance are unknown, so the graph structure has to be discovered. uDGPs arises when attempting to find the atomic structure of molecules and nanoparticles using X-ray or neutron diffraction data from non-crystalline materials. Rigidity theory provides a useful foundation for both aDGPs and uDGPs, though it is restricted to generic realizations of graphs, and key results are summarized. Conditions for unique realization are discussed for aDGP and uDGP cases, build-up algorithms for both cases are described and experimental results for uDGP are presented.

- Authors:

- Michigan State Univ., East Lansing, MI (United States). Dept. of Physics and Astronomy
- Brookhaven National Lab. (BNL), Upton, NY (United States). Condensed Matter Physics and Materials Science Dept.
- Columbia Univ., New York, NY (United States). Dept. of Applied Physics and Applied Mathematics

- Publication Date:

- Research Org.:
- Brookhaven National Lab. (BNL), Upton, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1252533

- Alternate Identifier(s):
- OSTI ID: 1336154

- Report Number(s):
- BNL-112665-2016-JA

Journal ID: ISSN 0166-218X; R&D Project: PO011; KC0201060

- Grant/Contract Number:
- SC00112704; AC02-98CH10886

- Resource Type:
- Published Article

- Journal Name:
- Discrete Applied Mathematics

- Additional Journal Information:
- Journal Volume: 204; Journal Issue: C; Journal ID: ISSN 0166-218X

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Nanostructure; Rigid cluster; Percolation; Unassigned distances; LIGA; TRIBOND

### Citation Formats

```
Duxbury, P. M., Granlund, L., Gujarathi, S. R., Juhas, P., and Billinge, S. J. L. The unassigned distance geometry problem. United States: N. p., 2015.
Web. doi:10.1016/j.dam.2015.10.029.
```

```
Duxbury, P. M., Granlund, L., Gujarathi, S. R., Juhas, P., & Billinge, S. J. L. The unassigned distance geometry problem. United States. doi:10.1016/j.dam.2015.10.029.
```

```
Duxbury, P. M., Granlund, L., Gujarathi, S. R., Juhas, P., and Billinge, S. J. L. Thu .
"The unassigned distance geometry problem". United States. doi:10.1016/j.dam.2015.10.029.
```

```
@article{osti_1252533,
```

title = {The unassigned distance geometry problem},

author = {Duxbury, P. M. and Granlund, L. and Gujarathi, S. R. and Juhas, P. and Billinge, S. J. L.},

abstractNote = {Studies of distance geometry problems (DGP) have focused on cases where the vertices at the ends of all or most of the given distances are known or assigned, which we call assigned distance geometry problems (aDGPs). In this contribution we consider the unassigned distance geometry problem (uDGP) where the vertices associated with a given distance are unknown, so the graph structure has to be discovered. uDGPs arises when attempting to find the atomic structure of molecules and nanoparticles using X-ray or neutron diffraction data from non-crystalline materials. Rigidity theory provides a useful foundation for both aDGPs and uDGPs, though it is restricted to generic realizations of graphs, and key results are summarized. Conditions for unique realization are discussed for aDGP and uDGP cases, build-up algorithms for both cases are described and experimental results for uDGP are presented.},

doi = {10.1016/j.dam.2015.10.029},

journal = {Discrete Applied Mathematics},

number = C,

volume = 204,

place = {United States},

year = {2015},

month = {11}

}

DOI: 10.1016/j.dam.2015.10.029

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