Final doctoral examination and defense of dissertation of Yu Gao, May 9, 2022

Title: Analysis and Maintenance of Graph Laplacians via Random Walks

Thesis draft:

Date: Monday, May 9, 2022

Time: 09:00am EST


Yu Gao
Ph.D. Candidate
Algorithms, Combinatorics, and Optimization
School of Computer Science
Georgia Institute of Technology


Dr. Richard Peng (Advisor, School of Computer Science, Georgia Institute of Technology)
Dr. Matthew Baker (School of Mathematics, Georgia Institute of Technology)
Dr. Zvi Galil (School of Computer Science, Georgia Institute of Technology)
Dr. Jonathan Kelner (Department of Mathematics and CSAIL, Massachusetts Institute of Technology)
Dr. Mohit Singh (School of Industrial & Systems Engineering, Georgia Institute of Technology)


Dr. Yitong Yin (Department of Computer Science and Technology, Nanjing University)


Graph Laplacians arise in many natural and artificial contexts. They are linear systems associated with undirected graphs. They are equivalent to electric flows which is a fundamental physical concept by itself and is closely related to other physical models, e.g., the Abelian sandpile model. Many real-world problems can be modeled and solved via Laplacian linear systems, including semi-supervised learning, graph clustering, and graph embedding.
More recently, better theoretical understandings of Laplacians led to dramatic improvements across graph algorithms. The applications include dynamic connectivity problem, graph sketching, and most recently combinatorial optimization. For example, a sequence of papers improved the runtime for maximum flow and minimum cost flow in many different settings.
In this thesis, we present works that the analyze, maintain and utilize Laplacian linear systems in both static and dynamic settings by representing them as random walks. This combinatorial representation leads to better bounds for Abelian sandpile model on grids, the first data structures for dynamic vertex sparsifiers and dynamic Laplacian solvers, and network flows on planar as well as general graphs.