Finding Articulation Points of Large Graphs in Linear Time

The complexity analysis of many classical graph algorithms is often based on the assumption of random access to the input data. But as the size of the graph increases, it becomes impossible to maintain the graph in memory, and the cost to just access the graph will dominate the running time. The streaming model was proposed as a practical framework for studying the complexity of large problems, e.g., the space efficient algorithms to compute frequency moments in a data stream by Alon, Matias, and Szegedy. More recently, an alternative model called semi-streaming was introduced by Muthukrishnan as a more realistic framework for studying the complexity of graph problems. In this model, the algorithm scans all the edges of graph sequentially in a few passes and the amount of working space is limited to $\O (n \polylog n)$, where $n$ is the number of vertices. The space limitation is reasonable because many important graph properties such as connectivity and bipartiteness require $\Omega(n \log n)$ space. With these restrictions, some fundamental algorithms such as BFS and DFS are believed difficult to be solved in a small number of passes. As BFS and DFS are not efficient in the semi-streaming model, many classical graph algorithms that require BFS or DFS as a subroutine are no longer easy under this setting. In this talk, we study the problem of finding all the bridges and articulation points in a graph. Typically, bridges and articulation points are found through DFS in linear time. We propose an algorithm working in the semi-streaming model to solve this problem in linear time within a single pass. Besides, we show that the space lower bound under this streaming setting is \Omega (n \log n) bits, which shows the optimality of our algorithm both in the number of passes, time, and space. Examination Committee: William Steiger (Chair), Mario Szegedy, Shan Muthukrishnan and Alex Borgida