Engineering an Efficient Reachability Algorithm for Directed Graphs

I declare that I have developed and written the enclosed thesis completely by myself, and have not used sources or means without declaration in the text. A reachability query on a directed graph G asks if there exists a path from a node s to a node t. Answering such queries on large graph like datasets has become an issue in various fields of research and real world applications over the past 20 years. Therefore, answering reachbility queries fast and efficiently has become more and more relevant. Nowadays, most XML [2] files make extensive use of ID [11] and IDREF, which transforms their tree based layout into a more complex directed graph. Therefore, querying for reachability requires a reachability query on a directed graph instead of a simple ancestor query. Similarly, the RDF model [20] relies on directed graphs. Various queries on RDF graphs involve reachability, for example to infer the relationship between objects. Since the semantic web builds on RDF, this topic gains more attention as the semantic web becomes more popular. Network biology uses reachability queries to query for protein-protein interaction on databases like the DIP [33]. Furthermore, reachability plays a role in querying for metabolic pathways on metabolic networks [19] or interaction on gene regulatory networks [3]. Additionally , in the field of model checking [9] reachability queries are needed to check whether a state can reach another state. Similarly, source code analysis uses reachability queries for pointer and dataflow analysis [26, 25]. Since the answer of a reachability query on a directed graph, which contains a cycle covering all nodes is always true, we can reduce a directed graph to its condensation. This is done by calculating the strongly connected components and contracting them into a single node. The condensation is a directed acyclic graph (DAG) which is much smaller in most cases. The two naive algorithms to answer reachability queries for a DAG are either traversing the graph using a Depth-First Search (DFS) or a Breadth-First Search (BFS) or calculating the transitive closure of the DAG. Simply traversing the DAG results in a query time in O(m) whereas storing the transitive closure needs O(n 2) space and has a complexity of O(nm) to compute but can answer a query in O(1). Throughout the past years, numerous different approaches for graph reachability emerged. They often combine the previously mentioned two methods for reachability querying. Thus trading …