Sampling an Edge in Sublinear Time Exactly and Optimally

T. Eden, Shyam Narayanan, Jakub Tvetek
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Abstract

Sampling edges from a graph in sublinear time is a fundamental problem and a powerful subroutine for designing sublinear-time algorithms. Suppose we have access to the vertices of the graph and know a constant-factor approximation to the number of edges. An algorithm for pointwise $\varepsilon$-approximate edge sampling with complexity $O(n/\sqrt{\varepsilon m})$ has been given by Eden and Rosenbaum [SOSA 2018]. This has been later improved by T\v{e}tek and Thorup [STOC 2022] to $O(n \log(\varepsilon^{-1})/\sqrt{m})$. At the same time, $\Omega(n/\sqrt{m})$ time is necessary. We close the problem, by giving an algorithm with complexity $O(n/\sqrt{m})$ for the task of sampling an edge exactly uniformly.
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在亚线性时间内精确和最优地采样边缘
在亚线性时间内对图进行边采样是设计亚线性时间算法的一个基本问题,也是一个强大的子程序。假设我们可以访问图的顶点,并且知道边数的常数因子近似值。Eden和Rosenbaum [SOSA 2018]给出了一种复杂度为$O(n/\sqrt{\varepsilon m})$的逐点$\varepsilon$ -近似边缘采样算法。这后来由T \v{e} tek和Thorup [STOC 2022]改进为$O(n \log(\varepsilon^{-1})/\sqrt{m})$。同时,$\Omega(n/\sqrt{m})$时间是必要的。我们通过给出一个复杂度为$O(n/\sqrt{m})$的算法来完成精确均匀采样边缘的任务,从而解决了这个问题。
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