掩护时间，毯子时间，和马修一家
The cover time, the blanket time, and the Matthews bound
摘要：
图G的覆盖时间C是从最差顶点开始的随机游动覆盖G中所有顶点的期望时间。类似地，覆盖时间B是在固定分布所建议的恒定次数因子内访问所有顶点的预期时间。(Our定义会比这个稍微强一点）。显然，当图被覆盖时，所有顶点都被覆盖，因此C <= B。毯式时间是由Winkler和Zuckerman在马尔可夫估计和分布式计算中的应用而引入的。他们证明了B =O（C），并证明了B=O（C ln n）.本文利用Matthews定理引入了另一个参数M，证明了M/2 <= C <= B = O（（MLln ln n）^2）.因此，时间复杂度为O（n）。对于从平稳分布开始的覆盖时间C（π），下界仍然有效。我们还证明了存在一个多项式时间算法来在2倍内逼近M，并且在O（（ln ln n）^2）倍内逼近C，从而改进了Matthews的O（ln n）的上界。
英文摘要：
The cover time C of a graph G is the expected time for a random walk starting from the worst vertex to cover all vertices in G. Similarly, the blanket time B is the expected time to visit all vertices within a constant factor of number of times suggested by the stationary distribution. (Our definition will be slightly stronger than this.) Obviously, all vertices are covered when the graph is blanketed, and hence C <= B. The blanket time is introduced by Winkler and Zuckerman motivated by applications in Markov estimation and distributed computing. They conjectured B =O(C) and proved B=O(C ln n ). In this paper, we introduce another parameter M motivated by Matthews' theorem and prove M/2 <= C <= B = O(( M ln ln n)^2). In particular, B = O(C (ln ln n)^2). The lower bound is still valid for the cover time C(π) starting from the stationary distribution. We also show that there is a polynomial time algorithm to approximate M within a factor of 2 and so does for C within a factor of O((ln ln n)^2), improving previous bound of O(ln n) of Matthews'.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
