## Graphs, matrices, and populations : linear algebraic techniques in theoretical computer science and population genetics

##### Author(s)

Levin, Alex, Ph. D. (Alexander). Massachusetts Institute of Technology
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##### Alternative title

Linear algebraic techniques in theoretical computer science and population genetics

##### Other Contributors

Massachusetts Institute of Technology. Department of Mathematics.

##### Advisor

Bonnie Berger and Jonathan Kelner.

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Show full item record##### Abstract

In this thesis, we present several algorithmic results for problems in spectral graph theory and computational biology. The first part concerns the problem of spectral sparsification. It is known that every dense graph can be approximated in a strong sense by a sparse subgraph, known as a spectral sparsifier of the graph. Furthermore, researchers have recently developed efficient algorithms for computing such approximations. We show how to make these algorithms faster, and also give a substantial improvement in space efficiency. Since sparsification is an important first step in speeding up approximation algorithms for many graph problems, our results have numerous applications. In the second part of the thesis, we consider the problem of inferring human population history from genetic data. We give an efficient and principled algorithm for using single nucleotide polymorphism (SNP) data to infer admixture history of various populations, and apply it to show that Europeans have evidence of mixture with ancient Siberians. Finally, we turn to the problem of RNA secondary structure design. In this problem, we want to find RNA sequences that fold to a given secondary structure. We propose a novel global sampling approach, based on the recently developed RNAmutants algorithm, and show that it has numerous desirable properties when compared to existing solutions. Our method can prove useful for developing the next generation of RNA design algorithms.

##### Description

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013. Cataloged from PDF version of thesis. Includes bibliographical references (pages 149-155).

##### Date issued

2013##### Department

Massachusetts Institute of Technology. Department of Mathematics.##### Publisher

Massachusetts Institute of Technology

##### Keywords

Mathematics.