Streaming Algorithm for Determining a Topological Ordering of a Digraph

Finding a topological ordering for a directed graph is one of the fundamental problems in computer science. Several textbook-standard algorithms using linear memory have been discovered and utilized to solve several other problems for many decades, especially in resolving dependencies and solving other graph connectivity problems. However, these algorithms are becoming less practical nowadays as the size of the data we are working with is getting much larger because the data cannot t into memory, prohibiting these algorithms from randomly accessing the input graph data. For this thesis, our goal is to solve the topological ordering problem of a directed graph with a size that is substantially much larger than the size of available memory. Speci cally, we study the problem in the context of the streaming settings in which the set of edges of the graph are provided through a data stream instead of being stored entirely on a random access memory. A recent lower bound result – rising from the analysis of communication complexity – shows that there is no one-pass streaming algorithm for the topological ordering problem that could achieve memory usage that is sublinear in the number of edges of a graph. Our work addresses on the remaining question of whether there is a multi-pass algorithm using sublinear working memory to solve this problem. Our rst result discusses a conjecture based the communication complexity. We study a speci c game of communication between Alice and Bob in which both parties attempt to solve the s–t reachability problem of a 3-layer graph using small memory. The solution to this problem helps us construct a streaming algorithm for topological sorting. Unfortunately, using the notion of additive combinatorics, we are able to construct a counterexample that enforces a large lower bound on the required memory in order to solve the conjectured problem. For the other result, we provide a surprisingly simple steaming algorithm for determining a topological ordering of an n-vertex directed graph; by storing up to k arbitrary incoming edges for each vertex in each pass of the stream, this algorithm achieves Θ(nk ) working memory and uses dn/ke passes on the input, where k is an integer parameter. This result is a nice tradeo between the number of passes of the streaming and the amount of memory usage.