I/O-optimal algorithms on grid graphs

Given a graph of which the n vertices form a regular two-dimensional grid, and in which each (possibly weighted and/or directed) edge connects a vertex to one of its eight neighbours, the following can be done in O(scan(n)) I/Os, provided M = Omega(B^2): computation of shortest paths with non-negative edge weights from a single source, breadth-first traversal, computation of a minimum spanning tree, topological sorting, time-forward processing (if the input is a plane graph), and an Euler tour (if the input graph is a tree). The minimum-spanning tree algorithm is cache-oblivious. The best previously published algorithms for these problems need Theta(sort(n)) I/Os. Estimates of the actual I/O volume show that the new algorithms may often be very efficient in practice.Comment: 12 pages' extended abstract plus 12 pages' appendix with details, proofs and calculations. Has not been published in and is currently not under review of any conference or journa

Parallel and I/O-efficient randomisation of massive networks using global curveball trades

Graph randomisation is a crucial task in the analysis and synthesis of networks. It is typically implemented as an edge switching process (ESMC) repeatedly swapping the nodes of random edge pairs while maintaining the degrees involved [23]. Curveball is a novel approach that instead considers the whole neighbourhoods of randomly drawn node pairs. Its Markov chain converges to a uniform distribution, and experiments suggest that it requires less steps than the established ESMC [6]. Since trades however are more expensive, we study Curveball’s practical runtime by introducing the first efficient Curveball algorithms: the I/O-efficient EM-CB for simple undirected graphs and its internal memory pendant IM-CB. Further, we investigate global trades [6] processing every node in a single super step, and show that undirected global trades converge to a uniform distribution and perform superior in practice. We then discuss EM-GCB and EMPGCB for global trades and give experimental evidence that EM-PGCB achieves the quality of the state-of-the-art ESMC algorithm EM-ES [15] nearly one order of magnitude faster

Certifying Induced Subgraphs in Large Graphs

We introduce I/O-effiient certifying algorithms for bipartite graphs, as well as for the classes of split, threshold, bipartite chain, and trivially perfect graphs. When the input graph is a member of the respective class, the certifying algorithm returns a certificate that characterizes this class. Otherwise, it returns a forbidden induced subgraph as a certificate for non-membership. On a graph with n vertices and m edges, our algorithms take I/Os in the worst case for split, threshold and trivially perfect graphs. In the same complexity bipartite and bipartite chain graphs can be certified with high probability. We provide implementations for split and threshold graphs and provide a preliminary experimental evaluation