Graph Sparsification by Edge-Connectivity and Random Spanning Trees

We present new approaches to constructing graph sparsifiers --- weighted subgraphs for which every cut has the same value as the original graph, up to a factor of $(1 \pm \epsilon)$. Our first approach independently samples each edge $uv$ with probability inversely proportional to the edge-connectivity between $u$ and $v$. The fact that this approach produces a sparsifier resolves a question posed by Bencz\'ur and Karger (2002). Concurrent work of Hariharan and Panigrahi also resolves this question. Our second approach constructs a sparsifier by forming the union of several uniformly random spanning trees. Both of our approaches produce sparsifiers with $O(n \log^2(n)/\epsilon^2)$ edges. Our proofs are based on extensions of Karger's contraction algorithm, which may be of independent interest

A Linear-time Algorithm for Sparsification of Unweighted Graphs

Given an undirected graph $G$ and an error parameter $\epsilon > 0$, the {\em graph sparsification} problem requires sampling edges in $G$ and giving the sampled edges appropriate weights to obtain a sparse graph $G_{\epsilon}$ with the following property: the weight of every cut in $G_{\epsilon}$ is within a factor of $(1\pm \epsilon)$ of the weight of the corresponding cut in $G$. If $G$ is unweighted, an $O(m\log n)$-time algorithm for constructing $G_{\epsilon}$ with $O(n\log n/\epsilon^2)$ edges in expectation, and an $O(m)$-time algorithm for constructing $G_{\epsilon}$ with $O(n\log^2 n/\epsilon^2)$ edges in expectation have recently been developed (Hariharan-Panigrahi, 2010). In this paper, we improve these results by giving an $O(m)$-time algorithm for constructing $G_{\epsilon}$ with $O(n\log n/\epsilon^2)$ edges in expectation, for unweighted graphs. Our algorithm is optimal in terms of its time complexity; further, no efficient algorithm is known for constructing a sparser $G_{\epsilon}$. Our algorithm is Monte-Carlo, i.e. it produces the correct output with high probability, as are all efficient graph sparsification algorithms

Faster Worst Case Deterministic Dynamic Connectivity

We present a deterministic dynamic connectivity data structure for undirected graphs with worst case update time $O\left(\sqrt{\frac{n(\log\log n)^2}{\log n}}\right)$ and constant query time. This improves on the previous best deterministic worst case algorithm of Frederickson (STOC 1983) and Eppstein Galil, Italiano, and Nissenzweig (J. ACM 1997), which had update time $O(\sqrt{n})$. All other algorithms for dynamic connectivity are either randomized (Monte Carlo) or have only amortized performance guarantees