arXiv 2211.052932022-11cs.DS

流Euclidean Max-Cut：维数与数据约简

Streaming Euclidean Max-Cut: Dimension vs Data Reduction

Xiaoyu Chen, Shaofeng H. -C. Jiang, Robert Krauthgamer

摘要：

最大切割是一个基本问题，已被广泛研究，在各种设置。在动态几何流模型下，以$\mathbb{R}^d$中的一组点为输入，设计了一种欧氏最大割算法，其中输入$X\subseteq [Δ]^d$表示为点的插入和删除序列.先前，Frahling和Sohler [STOC 2005]设计了用于低维区域的$(1+ε)$近似算法，即，它使用空间$\exp(d)$。为了解决这个问题，在高维制度，这是越来越感兴趣的，必须提高对维度$d$的依赖性，理想的空间复杂性$\mathrm{poly}(ε^{-1} d \logΔ)$。Lammersen，Sidiropoulos和Sohler [WADS 2009]证明了Euclidean Max-Cut允许使用目标维度$d' = \mathrm{poly}(ε^{-1})$进行降维。将其与前面提到的使用空间$\exp(d')$的算法相结合，他们得到了一个算法，其整体空间复杂度在$d$中确实是多项式的，但不幸的是在$ε^{-1}$中是指数的。我们设计了另一种方法的\emph{data reduction}，基于重要性采样，并实现空间绑定$\mathrm{poly}(ε^{-1} d \logΔ)$，这是指数更好（在$ε$）比降维方法。为了在流模型中实现该方案，我们采用随机移位的四叉树来构造树嵌入。虽然这是一个众所周知的方法，我们的算法的一个关键特征是，嵌入的失真$O(d\logΔ)$只影响空间复杂度，近似比仍然是$1+ε$。

英文摘要：

Max-Cut is a fundamental problem that has been studied extensively in various settings. We design an algorithm for Euclidean Max-Cut, where the input is a set of points in $\mathbb{R}^d$, in the model of dynamic geometric streams, where the input $X\subseteq [Δ]^d$ is presented as a sequence of point insertions and deletions. Previously, Frahling and Sohler [STOC 2005] designed a $(1+ε)$-approximation algorithm for the low-dimensional regime, i.e., it uses space $\exp(d)$. To tackle this problem in the high-dimensional regime, which is of growing interest, one must improve the dependence on the dimension $d$, ideally to space complexity $\mathrm{poly}(ε^{-1} d \logΔ)$. Lammersen, Sidiropoulos, and Sohler [WADS 2009] proved that Euclidean Max-Cut admits dimension reduction with target dimension $d' = \mathrm{poly}(ε^{-1})$. Combining this with the aforementioned algorithm that uses space $\exp(d')$, they obtain an algorithm whose overall space complexity is indeed polynomial in $d$, but unfortunately exponential in $ε^{-1}$. We devise an alternative approach of \emph{data reduction}, based on importance sampling, and achieve space bound $\mathrm{poly}(ε^{-1} d \logΔ)$, which is exponentially better (in $ε$) than the dimension-reduction approach. To implement this scheme in the streaming model, we employ a randomly-shifted quadtree to construct a tree embedding. While this is a well-known method, a key feature of our algorithm is that the embedding's distortion $O(d\logΔ)$ affects only the space complexity, and the approximation ratio remains $1+ε$.

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摘要：

英文摘要：

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