Meyerson 草图自适应代价的紧界
Tight Bounds on the Cost of Adaptivity for the Meyerson Sketch
- Google Research（谷歌研究院）
- Tel Aviv University（特拉维夫大学）
- Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究量化了在线设施选址中Meyerson草图对自适应输入的鲁棒性，证明自适应代价与重放代价之比有紧界，并推广至近似k聚类。
AI中文摘要：
在在线设施选址问题中，点逐个到达，算法必须在到达点处开设设施，或将点路由到现有设施。Meyerson 草图以与到达点到最近开放设施的距离成比例的概率在每个到达点开设设施，并且除了开放中心集合外不需要任何状态。由于其简单性、空间效率以及对离线最优解的强保证，Meyerson 草图已成为流式和在线聚类的核心工具。然而，在许多此类应用中，开放设施集合对生成流的进程可见，该进程可以根据算法过去的随机选择自适应地选择未来点，从而使经典保证失效。在本工作中，我们量化了这种自适应性的影响。我们将草图的适应性生成运行与\textit{ oblivious replay}（即在同一生成序列上使用新硬币的独立执行）进行比较，并研究期望自适应代价与期望重放代价的\textit{自适应比}。我们确定了两个方向的最坏情况比率：自适应性既不能使期望代价或开放设施数量膨胀，也不能使其收缩超过 $O(\log\Delta/\log\log\Delta)$ 因子，其中 $\Delta$ 是输入点的纵横比（最大与最小成对距离之比）。这是渐近紧的，因为实数线上存在确定性生成器，使代价膨胀或收缩 $\Omega(\log\Delta/\log\log\Delta)$ 因子。我们证明这些鲁棒性保证可推广到基于 Meyerson 的近似 $k$ 聚类草图，其草图大小为 $O(k\\,\mathrm{polylog}(n))$。
英文摘要：
In online facility location, points arrive one at a time, and the algorithm must either open a facility at the arriving point or route the point to an existing facility. The Meyerson sketch opens a facility at each arriving point with probability proportional to the point's distance to the closest open facility, and requires no state beyond the set of open centers. Due to its simplicity, space efficiency, and strong guarantees against the offline optimum, the Meyerson sketch has become a workhorse of streaming and online clustering. In many such applications, however, the set of open facilities are visible to the process that generates the stream, which can adaptively select future points based on the algorithm's past random choices, voiding its classical guarantees. In this work, we quantify the effect of such adaptivity. We compare an adaptively generated run of the sketch against an \emph{oblivious replay}, an independent execution, with fresh coins, on the very same generated sequence, and study the \emph{adaptivity ratio} of expected adaptive cost to expected replay cost. We determine the worst-case ratio in both directions: adaptivity can neither inflate nor deflate the expected cost, or the number of open facilities, by more than an $O(\logΔ/\log\logΔ)$ factor, where $Δ$ is the aspect ratio of the input points (the ratio of the largest to the smallest pairwise distance). This is asymptotically tight as there are deterministic generators on the real line that inflate or deflate the cost by an $Ω(\logΔ/\log\logΔ)$ factor. We show that these robustness guarantees carry over to Meyerson-based sketches for approximate $k$ clustering with sketch size $O(k\,\mathrm{polylog}(n))$.
