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Defense Event

Algorithms and Dynamic Data Structures for Basic Graph Optimization Problems

Ran Duan


 
Thursday, August 04, 2011
3:00pm - 5:00pm
3725 Beyster Bldg.

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About the Event

Graph optimization plays an important role in a wide range of areas such as computer graphics, computational biology, networking applications and machine learning. Among numerous graph optimization problems, some basic problems, such as shortest paths, minimum spanning tree, and maximum matching, are the most fundamental ones. They have practical applications in various fields, and are also building blocks of many other algorithms. Improvements in algorithms for these problems can thus have a great impact both in practice and in theory. In this thesis, we study a number of graph optimization problems. The results are mostly about approximation algorithms solving graph problems, or efficient dynamic data structures which can answer graph queries when a number of changes occur. There are several different models of dynamic graphs. Much of my work focuses on the dynamic subgraph model in which there is a fixed underlying graph and every vertex can be flipped "on" or "off". The queries are based on the subgraph induced by the "on" vertices. Our results make significant improvements to the previous algorithms or structures of these problems. The major results are listed below. - Approximate Matching. We give the first linear time algorithm for computing approximate maximum weighted matching for arbitrarily small approximation ratio. - d-failure Connectivity Oracle. For an undirected graph, we give the first space efficient data structure that can answer connectivity queries between any pair of vertices avoiding d other failed vertices in time polynomial in d and log n. - (Max, Min)-Matrix Multiplication. We give a faster algorithm for the (max, min)-matrix multiplication problem, which has a direct application to the all- pairs bottleneck paths (APBP) problem. Given a directed graph with a capacity on each edge, the APBP problem is to determine, for all pairs of vertices s and t, the path from s to t with maximum flow. - Dual-failure Distance Oracle. We construct the data structure for a given directed graph of nearly quadratic space which can efficiently answer distance and shortest path queries in the presence of two node or link failures. - Dynamic Subgraph Connectivity. We give the first subgraph connectivity structure with worst-case sublinear time bounds for both updates and queries. - Bounded-leg Shortest Path. In a weighted, directed graph an L-bounded leg path is one whose constituent edges have length at most L. We give an algorithm for preprocessing a directed graph in nearly cubic time in order to answer approximate bounded-leg distance and bounded-leg shortest path queries in merely sub-logarithmic time.

Additional Information

Sponsor(s): Seth Pettie

Open to: Public