Darren Strash

Listing All Maximal Cliques in Sparse Graphs in Near-optimal Time

The degeneracy of an n-vertex graph G is the smallest number d such that every subgraph of G contains a vertex of degree at most d. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron–Kerbosch algorithm and show that it runs in time O(dn3 d/3). We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an n-vertex graph with degeneracy d (when d is a multiple of 3 and n ≥ d + 3) is (n − d)3 d/3. Therefore, our algorithm matches the Θ(d(n − d)3 d/3) worst-case output size of the problem whenever n − d = Ω(n).

Listing all maximal cliques in large sparse real-world graphs

We implement a new algorithm for listing all maximal cliques in sparse graphs due to Eppstein, Loffler, and Strash (ISAAC 2010) and analyze its performance on a large corpus of real-world graphs. Our analysis shows that this algorithm is the first to offer a practical solution to listing all maximal cliques in large sparse graphs. All other theoretically-fast algorithms for sparse graphs have been shown to be significantly slower than the algorithm of Tomita et al. (Theoretical Computer Science, 2006) in practice. However, the algorithm of Tomita et al. uses an adjacency matrix, which requires too much space for large sparse graphs. Our new algorithm opens the door for fast analysis of large sparse graphs whose adjacency matrix will not fit into working memory.

Succinct Greedy Geometric Routing in the Euclidean Plane

We show that greedy geometric routing schemes exist for the Euclidean metric in R 2, for 3-connected planar graphs, with coordinates that can be represented succinctly, that is, with O(logn) bits, where n is the number of vertices in the graph.

Extended dynamic subgraph statistics using h-index parameterized data structures

We present techniques for maintaining subgraph frequencies in a dynamic graph, using data structures that are parameterized in terms of h, the h-index of the graph. Our methods extend previous results of Eppstein and Spiro for maintaining statistics for undirected subgraphs of size three to directed subgraphs and to subgraphs of size four. For the directed case, we provide a data structure to maintain counts for all 3-vertex induced subgraphs in O(h) amortized time per update. For the undirected case, we maintain the counts of size-four subgraphs in O(h^2) amortized time per update. These extensions enable a number of new applications in Bioinformatics and Social Networking research.

Linear-time algorithms for geometric graphs with sublinearly many crossings

We provide linear-time algorithms for geometric graphs with sublinearly many edge crossings. That is, we provide algorithms running in

Finding Near-Optimal Independent Sets at Scale.

O(n)

On Minimizing Crossings in Storyline Visualizations

time on connected geometric graphs having

On the Power of Simple Reductions for the Maximum Independent Set Problem

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Accelerating Local Search for the Maximum Independent Set Problem

vertices and

Finding near-optimal independent sets at scale

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