AI 中文总结
本研究针对对抗性植入着色模型，提出首个亚线性时间植入着色恢复算法，通过对归一化邻接矩阵底部特征空间的亚线性内积访问适配经典谱方法，实现低误差的顶点颜色查询。
AI 中文摘要
我们提出了一种用于植入k着色问题的亚线性算法。给定一个带有植入着色的扩张图G，目标是高效确定给定顶点的颜色类。我们采用David和Feige[STOC 2016]提出的对抗性植入着色模型：对抗方选择一个含n个顶点的d正则谱λ扩张图G，通过将顶点划分为k个等大的部分并删除每个部分内部的所有边，植入一个平衡k着色。该模型推广了Blum和Spencer[J. Algorithms 1995]、Alon和Kahale[STOC 1994]研究的早期随机图模型。我们提出了该模型下首个用于恢复植入着色的亚线性时间算法。该算法的预处理时间和空间复杂度为$\tilde{O}\big(n^{1/2+O(1/\text{log}(d/\boldsymbol{\boldsymbol{\boldsymbol{λ}}))}}\big)$，生成的数据结构可在$\tilde{O}\big(n^{1/2+O(1/\text{log}(d/\boldsymbol{\boldsymbol{\boldsymbol{λ}}))}}\big)$时间内回答颜色查询，所得标记除$O(\boldsymbol{\boldsymbol{\boldsymbol{λ}}/d})$比例的顶点外，均与植入着色一致（最多相差k种颜色的一个排列）。该算法提供了对归一化邻接矩阵底部特征空间的亚线性时间内积访问，使我们能够将Alon和Kahale的经典谱方法适配到亚线性时间场景。
英文摘要
We give a sublinear algorithm for the planted $k$-coloring problem. Given an expander $G$ with a planted coloring, the goal is to efficiently determine the color class of a given vertex. We work in the adversarial planted coloring model of David and Feige [STOC 2016], where an adversary chooses a $d$-regular spectral $λ$-expander $G$ on $n$ vertices and plants a balanced $k$-coloring by partitioning the vertices into $k$ equal parts and deleting all edges within each part. This model generalizes the earlier random graph models studied by Blum and Spencer [J. Algorithms 1995] and Alon and Kahale [STOC 1994]. We give the first sublinear-time algorithm for recovering planted colorings in this model. The algorithm has preprocessing time and space $\widetilde O\left(n^{1/2+O(1/\log(d/λ))}\right)$, and produces a data structure that answers color queries in time $\widetilde O\left(n^{1/2+O(1/\log(d/λ))}\right)$, such that the resulting labeling agrees with the planted coloring on all but an $O(\sqrt{λ/d})$ fraction of vertices, up to a permutation of the $k$ colors. The algorithm gives sublinear-time inner product access to the bottom eigenspace of the normalized adjacency matrix, which allows us to adapt the classical spectral approach of Alon and Kahale in sublinear time.
