Random Sampling in Matroids, with Applications to Graph Connectivity and Minimum Spanning Trees

Random sampling i s a powerful way t o gather informat ion about a group by considering only a small part of it. W e give a paradigm f o r applying this technique to optimization problems, and demonstrate its effectiveness o n matroids. Matroids abstractly model many optimization problems that can be solved by greedy methods, such as the minimum spanning tree ( M S T ) problem. Our results have several applications. W e give an algorithm that uses simple data structures to construct a n M S T in O(m+n logn) t ime (Klein and Tarjan [21] have recently shown that a better choice of parameters makes this algorithm run in O(m + n) t ime). W e give bounds o n the connectivity (minimum cut) of a graph suffering random edge failures. W e give fas t algorithms f o r packing matroid bases, with particular attention to packing spanning trees in graphs.