Theses and Dissertations at Montana State University (MSU)
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Item Duplications and deletions in genomes: theory and applications(Montana State University - Bozeman, College of Engineering, 2022) Zou, Peng; Chairperson, Graduate Committee: Binhai ZhuIn computational biology, duplications and deletions in genome rearrangements are important to understand an evolutionary process. In cancer genomics research, intra-tumor genetic heterogeneity is one of the central problems. Gene duplications and deletions are observed occurring rapidly in cancer during tumour formation. Hence, they are recognized as critical mutations of cancer evolution. Understanding these mutations are important to understand the origins of cancer cell diversity which could help with cancer prognostics as well as drug resistance explanation. In this dissertation, first, we prove that the tandem duplication distance problem is NP-complete, even if |sigma| > or = 4, settling a 16-year old open problem. And we obtain some positive results by showing that if one of the input sequences, S, is exemplar, then one can decide if S can be transformed into T using at most k tandem duplications in time 2 O (k 2) + poly(n). Motivated by computing duplication patterns in sequences, a new fundamental problem called the longest letter-duplicated subsequence (LLDS) is investigated. We investigate several variants of this problem. Due to fast mutations in cancer, genome rearrangements on copy number profiles are used more often than genome themselves. We explore the Minimum Copy Number Generation problem. We prove that it is NP-hard to even obtain a constant factor approximation. We also show that the corresponding parameterized version is W[1]-hard. These either improve the previous hardness result or solve an open problem. And we then give a polynomial algorithm for the Copy Number Profile Conforming problem. Finally, we investigate the pattern matching with 1-reversal distance problem. With the known results on Longest Common Extension queries, one can design an O(n+m) time algorithm for this problem. However, we find empirically that this algorithm is very slow for small m. We then design an algorithm based on the Karp-Rabin fingerprints which runs in an expected O(nm) time. The algorithms are implemented and tested on real bacterial sequence dataset. The empirical results shows that the shorter the pattern length is (i.e., when m < 200), the more substrings with 1-reversal distance the bacterial sequences have.Item Finding disjoint dense clubs in an undirected graph(Montana State University - Bozeman, College of Engineering, 2016) Zou, Peng; Chairperson, Graduate Committee: Binhai ZhuFor over a decade, software like Twitter, Facebook and WeChat have changed people's lives by creating social groups and networks easily. They give people a new convenient 'world' where we can share everything that happens around us, and social networks have grown enormously in recent years. In essence, social networks are full of data and have become an indispensable part of our life. Trust is an important feature of the relationship between two users in a social network. With the development of social networks, the trust among its members has become a big issue. In a social network, the trust among its members usually cannot be carried over many users. In the corresponding social network modeled as a graph, a user is denoted by a vertex and an edge between two vertices means that these two users communicate a lot above some threshold and they trust each other. An online social community is usually corresponding to a dense region in such a graph. A complex social network is usually composed of several groups/communities (the regions with a lot of edges), and this characterization of community structure means the appearance of densely connected groups of vertices, with only sparse connections between groups. For analyzing the structure of social networks and the relationship between users, it is important to find disjoint groups/communities with a small diameter and with a decent size, formally called dense clubs in this thesis. We focus on handling this NP-complete problem in this thesis. First, from the parameterized computational complexity point of view, we show that this problem does not admit a polynomial kernel (implying that it is unlikely to apply some reduction rules to obtain a practically small problem size). Then, we focus on the dual version of the problem, i.e., deleting 'd' vertices to obtain some disjoint dense clubs. We show that this dual problem admits a simple FPT algorithm using a bounded search tree method (the running time is still too high for practical datasets). Finally, we combine a simple reduction rule together with some heuristic methods to obtain a practical solution (verified by extensive testing on practical datasets). Empirical results show that this heuristic algorithm is very sensitive to all parameters. This algorithm is suitable on graphs which have a mixture of dense and sparse regions. These graphs are very common in the real world.