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arXiv 2004.14891cs.DS
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全动态核心集

Fully-Dynamic Coresets

Monika Henzinger, Sagar Kale

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Added missed important reference. Abstract is shortened
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摘要：

随着输入的规模变得越来越大，核心集（coreset）--小而有代表性的输入摘要--比以往任何时候都更重要。如果任何可行解$S$相对于$C_w$的成本在$S$相对于原始输入的成本的$[1 {\pm} \varepsilon]$之内，则作为输入的子集的加权集合$C_w$是$\varepsilon$ -coreset。我们给出了一个非常一般的技术来计算coresets在全动态设置输入点可以添加或删除。给定一个静态的$\varepsilon$ -coreset算法，运行时间$t(n, \varepsilon, λ)$和计算的大小$s(n, \varepsilon, λ)$的coreset，其中$n$是输入点的数量和$1 {-}λ$是成功的概率，我们给出了一个完全动态的算法，计算的$\varepsilon$ -coreset与最坏情况下的更新时间$O((\log n) \cdot t(s(n, \varepsilon/\log n, λ/n), \varepsilon/\log n, λ/n) )$（这个界限是非正式的），其中成功的概率是$1{-}λ$。我们的技术是一个完全动态的模拟合并和减少技术，适用于插入只设置。虽然我们的空间使用是$O(n)$，我们的工作中存在的自适应对手，我们表明，$Ω(n)$空间是必要的，当对手是自适应的。因此，我们得到完全动态$\varepsilon$ -coreset算法的$k$ -中位数和$k$ -意味着最坏情况下的更新时间$O(\varepsilon^{-2}k^2\log^5 n \log^3 k)$和coreset大小$O(\varepsilon^{-2}k\log n \log^2 k)$忽略$\log \log n$和$\log(1/\varepsilon)$因素，并假设$\varepsilon, λ= Ω(1/$聚$(n))$。这是第一个完全动态的$k$ -中位数和$k$ -均值算法，具有最坏情况更新时间$O($ poly $(k, \log n, \varepsilon^{-1}))$。我们还给出了条件下界更新/查询时间的任何全动态$(4 - δ)$近似算法的$k$ -手段。

英文摘要：

With input sizes becoming massive, coresets -- small yet representative summary of the input -- are relevant more than ever. A weighted set $C_w$ that is a subset of the input is an $\varepsilon$-coreset if the cost of any feasible solution $S$ with respect to $C_w$ is within $[1 {\pm} \varepsilon]$ of the cost of $S$ with respect to the original input. We give a very general technique to compute coresets in the fully-dynamic setting where input points can be added or deleted. Given a static $\varepsilon$-coreset algorithm that runs in time $t(n, \varepsilon, λ)$ and computes a coreset of size $s(n, \varepsilon, λ)$, where $n$ is the number of input points and $1 {-}λ$ is the success probability, we give a fully-dynamic algorithm that computes an $\varepsilon$-coreset with worst-case update time $O((\log n) \cdot t(s(n, \varepsilon/\log n, λ/n), \varepsilon/\log n, λ/n) )$ (this bound is stated informally), where the success probability is $1{-}λ$. Our technique is a fully-dynamic analog of the merge-and-reduce technique that applies to insertion-only setting. Although our space usage is $O(n)$, we work in the presence of an adaptive adversary, and we show that $Ω(n)$ space is required when adversary is adaptive. As a consequence, we get fully-dynamic $\varepsilon$-coreset algorithms for $k$-median and $k$-means with worst-case update time $O(\varepsilon^{-2}k^2\log^5 n \log^3 k)$ and coreset size $O(\varepsilon^{-2}k\log n \log^2 k)$ ignoring $\log \log n$ and $\log(1/\varepsilon)$ factors and assuming that $\varepsilon, λ= Ω(1/$poly$(n))$. These are the first fully-dynamic algorithms for $k$-median and $k$-means with worst-case update times $O($poly$(k, \log n, \varepsilon^{-1}))$. We also give conditional lower bound on update/query time for any fully-dynamic $(4 - δ)$-approximation algorithm for $k$-means.

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。

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