In this paper we provide an algorithm that generates a graph with given degree sequence uniformly at random. Provided that $\Delta^4=O(m)$, where $\Delta$ is the maximal degree and $m$ is the number of edges,the algorithm runs in expected time $O(m)$. Our algorithm significantly improves the previously most efficient uniform sampler, which runs in expected time $O(m^2\Delta^2)$ for the same family of degree sequences. Our method uses a novel ingredient which progressively relaxes restrictions on an object being generated uniformly at random, and we use this to give fast algorithms for uniform sampling of graphs with other degree sequences as well. Using the same method, we also obtain algorithms with expected run time which is (i) linear for power-law degree sequences in cases where the previous best was $O(n^{4.081})$, and (ii) $O(nd+d^4)$ for $d$-regular graphs when $d=o(\sqrt n)$, where the previous best was $O(nd^3)$.
翻译:在本文中, 我们提供一种算法, 以任意方式生成一个具有给定度序列的图表 。 只要$\ Delta ⁇ 4=O( m)$, 美元是最大度, 美元是边缘数, 算法以预期的时间运行 $( m) 美元。 我们的算法大大改进了以前最有效的统一取样器, 以预期的时间运行 $( m) 2\ Delta ⁇ 2) 美元, 同一等级序列 。 我们的方法使用一种新型的成分, 逐步放松对一个物体的随机生成限制, 我们用它来提供快速的算法, 用其他度序列来统一地样图。 使用同一方法, 我们还获得一个具有预期运行时间的算法序列( 一) 线性算法级序列 $( n) $( n) 0. 481} 美元, 和 ( 二) 美元( + d) 美元( 4) 普通图, 美元为美元=o( sqrt n) 美元, 美元是前一个最佳的 O( n) 美元。