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We give offline algorithms for processing a sequence of 2- and 3-edge and vertex connectivity queries in a fully-dynamic undirected graph. While the current best fully-dynamic online data structures for 3-edge and 3-vertex connectivity require O(n^{2/}3) and O(n) time per update, respectively, our per-operation cost is only O(logn) , optimal due to the dynamic connectivity lower bound of Patrascu and Demaine. Our approach utilizes a divide and conquer scheme that transforms a graph into smaller equivalents that preserve connectivity information. This construction of equivalents is closely-related to the development of vertex sparsifiers, and shares important connections to several upcoming results in dynamic graph data structures, including online models. 

Current flow closeness centrality (CFCC) has a better discriminating ability than the ordinary closeness centrality based on shortest paths. In this paper, we extend this notion to a group of vertices in a weighted graph, and then study the problem of finding a subset S of k vertices to maximize its CFCC C(S), both theoretically and experimentally. We show that the problem is NP-hard, but propose two greedy algorithms for minimizing the reciprocal of C(S) with provable guarantees using the monotoncity and supermodularity. The first is a deterministic algorithm with an approximation factor (1−kk−1⋅1e) and cubic running time; while the second is a randomized algorithm with a (1−kk−1⋅1e−ϵ)-approximation and nearly-linear running time for any ϵ>0. Extensive experiments on model and real networks demonstrate that our algorithms are effective and efficient, with the second algorithm being scalable to massive networks with more than a million vertices. 

Motivated by the study of matrix elimination orderings in combinatorial scientific computing, we utilize graph sketching and local sampling to give a data structure that provides access to approximate fill degrees of a matrix undergoing elimination in O(polylog(n)) time per elimination and query. We then study the problem of using this data structure in the minimum degree algorithm, which is a widely-used heuristic for producing elimination orderings for sparse matrices by repeatedly eliminating the vertex with (approximate) minimum fill degree. This leads to a nearly-linear time algorithm for generating approximate greedy minimum degree orderings. Despite extensive studies of algorithms for elimination orderings in combinatorial scientific computing, our result is the first rigorous incorporation of randomized tools in this setting, as well as the first nearly-linear time algorithm for producing elimination orderings with provable approximation guarantees. While our sketching data structure readily works in the oblivious adversary model, by repeatedly querying and greedily updating itself, it enters the adaptive adversarial model where the underlying sketches become prone to failure due to dependency issues with their internal randomness. We show how to use an additional sampling procedure to circumvent this problem and to create an independent access sequence. Our technique for decorrelating the interleaved queries and updates to this randomized data structure may be of independent interest.