Generating Random Spanning Trees
نویسنده
چکیده
This paper describes a probabilistic algorithm that, given a connected, undirected graph G with n vertices, produces a spanning tree of G chosen uniformly at random among the spanning trees of G. The expected running time is O(n logn) per generated tree for almost all graphs, and O(n3) for the worst graphs. Previously known deterministic algorithms and much more complicated and require O(n3) time per generated tree. A Markov chain is called rapidly mixing if it gets close to the limit distribution in time polynomial in the log of the number of states. Starting from the analysis of the algorithm above we show that the Markov chain on the space of all spanning trees of a given a graph where the basic step is an edge swap is rapidly mixing.
منابع مشابه
Random Walks and Random Spanning Trees
We explore algorithms for generating random spanning trees. We first study an algorithm that was developed independently by David Alduous [1] and Andrei Broder [2]. The algorithm uses a a simple random walk in which edges that correspond to the first visit to vertices are added to the spanning tree. Analysis was inspired by Andrei Broder’s paper. Additionally, we study an algorithm by David Wil...
متن کاملSpanning Tree Size in Random Binary Search Trees
This paper deals with the size of the spanning tree of p randomly chosen nodes in a binary search tree. It is shown via generating functions methods, that for fixed p, the (normalized) spanning tree size converges in law to the Normal distribution. The special case p = 2 reproves the recent result (obtained by the contraction method by Mahmoud and Neininger [Ann. Appl. Probab. 13 (2003) 253–276...
متن کاملCounting the number of spanning trees of graphs
A spanning tree of graph G is a spanning subgraph of G that is a tree. In this paper, we focus our attention on (n,m) graphs, where m = n, n + 1, n + 2, n+3 and n + 4. We also determine some coefficients of the Laplacian characteristic polynomial of fullerene graphs.
متن کاملThe CRT is the scaling limit of unordered binary trees
The Brownian Continuum Random Tree (CRT), introduced by Aldous [2], is a natural object that arises in various situations in Probability Theory. It is known to be the universal scaling limit for conditioned critical Galton-Watson trees with finite variance offspring distribution [4, 21, 12], or of random labeled trees on n vertices (Cayley trees) [2, 10, 1]. Several distinct proofs for the conv...
متن کاملGenerating Random Spanning Trees via Fast Matrix Multiplication
We consider the problem of sampling a uniformly random spanning tree of a graph. This is a classic algorithmic problem for which several exact and approximate algorithms are known. Random spanning trees have several connections to Laplacian matrices; this leads to algorithms based on fast matrix multiplication. The best algorithm for dense graphs can produce a uniformly random spanning tree of ...
متن کامل