Faster Graph Algorithms Through DAG Compression.

The runtime of graph algorithms such as depth-first search or Dijkstra's algorithm is dominated by the fact that all edges of the graph need to be processed at least once, leading to prohibitive runtimes for large, dense graphs. We introduce a simple data structure for storing graphs (and more general structures) in a compressed manner using directed acyclic graphs (dags). We then show that numerous standard graph problems can be solved in time linear in the size of the dag compression of a graph, rather than in the number of edges of the graph. Crucially, many dense graphs, including but not limited to graphs of bounded twinwidth, have a dag compression of size linear in the number of vertices rather than edges. This insight allows us to improve the previous best results for the runtime of standard algorithms from quasi-linear to linear for the large class of graphs of bounded twinwidth, which includes all cographs, graphs of bounded treewidth, or graphs of bounded cliquewidth.