基于循环空间采样的小割集快速计算
Fast Computation of Small Cuts via Cycle Space Sampling
摘要：
我们描述了一种新的基于采样的方法来确定无向图中的割。对于一个图（V，E），它的圈空间是E的所有子集的族，这些子集在每个顶点都有偶数度。我们证明，以高概率，采样的循环空间确定的削减的图。这导致了简单的新的线性时间顺序算法，找到所有的切割边缘和切割对（一组2个边缘，形成一个切割）的图。在图G=（V，E）中的分布式计算模型中，我们的方法对几个问题产生了更快的算法。G的直径用Diam表示，最大度用Delta表示。我们得到简单的O（直径）时间分布算法，找到所有的切割边缘，2-边连接组件，并削减对，匹配或改进以前的时间界限。在自然条件下，这些新算法是普遍最优的-即一个欧米茄（直径）时间下界举行的每一个图。我们得到了一个O（直径+Delta/log V）-时间分布式算法寻找割顶点，这是快于最好的以前的算法时，三角洲，直径= O（sqrt（V））。一个简单的扩展，我们的工作产生的第一个分布式算法与3-边连接组件的次线性时间。基本的分布式算法是蒙特卡洛，但它们可以在不增加渐近复杂性的情况下被拉斯维加斯。在EREW PRAM上的并行计算模型中，我们的方法产生了一个简单的算法，具有最佳的时间复杂度O（log V），用于寻找割对和3-边连接组件。
英文摘要：
We describe a new sampling-based method to determine cuts in an undirected graph. For a graph (V, E), its cycle space is the family of all subsets of E that have even degree at each vertex. We prove that with high probability, sampling the cycle space identifies the cuts of a graph. This leads to simple new linear-time sequential algorithms for finding all cut edges and cut pairs (a set of 2 edges that form a cut) of a graph. In the model of distributed computing in a graph G=(V, E) with O(log V)-bit messages, our approach yields faster algorithms for several problems. The diameter of G is denoted by Diam, and the maximum degree by Delta. We obtain simple O(Diam)-time distributed algorithms to find all cut edges, 2-edge-connected components, and cut pairs, matching or improving upon previous time bounds. Under natural conditions these new algorithms are universally optimal --- i.e. a Omega(Diam)-time lower bound holds on every graph. We obtain a O(Diam+Delta/log V)-time distributed algorithm for finding cut vertices; this is faster than the best previous algorithm when Delta, Diam = O(sqrt(V)). A simple extension of our work yields the first distributed algorithm with sub-linear time for 3-edge-connected components. The basic distributed algorithms are Monte Carlo, but they can be made Las Vegas without increasing the asymptotic complexity. In the model of parallel computing on the EREW PRAM our approach yields a simple algorithm with optimal time complexity O(log V) for finding cut pairs and 3-edge-connected components.
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