We give offline algorithms for processing a sequence of 2 and 3edge and vertex connectivity queries in a fullydynamic undirected graph. While the current best fullydynamic online data structures for 3edge and 3vertex connectivity require O(n^{2/}3) and O(n) time per update, respectively, our peroperation 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 closelyrelated to the development of vertex sparsifiers, and shares important connections to several upcoming results in dynamic graph data structures, including online models.
Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers
We present a general framework of designing efficient dynamic approximate algorithms for optimization on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers, gives data structures that maintain approximate solutions in sublinear update and query time. We illustrate the applicability of our paradigm to the following problems.
(1) A fullydynamic algorithm that approximates allpair
maximumflows/minimumcuts up to a nearly logarithmic factor in $\tilde{O}(n^{2/3})$ amortized time against an oblivious adversary, and $\tilde{O}(m^{3/4})$ time against an adaptive adversary.
(2) An incremental data structure that maintains $O(1)$approximate shortest path in $n^{o(1)}$ time per operation, as well as fully dynamic approximate allpair shortest path and transshipment in $\tilde{O}(n^{2/3+o(1)})$ amortized time per operation.
(3) A fullydynamic algorithm that approximates allpair effective resistance up to an $(1+\eps)$ factor in $\tilde{O}(n^{2/3+o(1)} \epsilon^{O(1)})$ amortized update time per operation.
The key tool behind result (1) is the dynamic maintenance of an algorithmic construction due to Madry [FOCS' 10], which partitions a graph into a collection of simpler graph structures (known as jtrees) and approximately captures the cutflow and metric structure of the graph. The $O(1)$approximation guarantee of (2) is by adapting the distance oracles by [ThorupZwick JACM `05]. Result (3) is obtained by more »
 Award ID(s):
 1846218
 Publication Date:
 NSFPAR ID:
 10253473
 Journal Name:
 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 1619, 2020
 Page Range or eLocationID:
 1135 to 1146
 Sponsoring Org:
 National Science Foundation
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