在滑动窗口中编辑单调性距离
Edit Distance to Monotonicity in Sliding Windows
摘要：
给定每个与数值相关联的项的流，其单调性的编辑距离是要移除的项的最小数量，使得剩余项相对于该数值不减。在过去的几年中，估计数据流的单调性的编辑距离的空间复杂度变得很好理解。受网络质量监控应用的启发，我们将研究扩展到估计覆盖任何$w \ge 1$流中$w$最新项的滑动窗口单调性的编辑距离。我们给出了一个确定性算法，它可以返回一个$(4+\eps)$的一个因素内使用$O(\frac{1}{\eps^2} \log^2(\eps w))$空间的估计。我们还在两个方向上扩展了研究。首先，我们考虑一个流，其中每个项目都与一个偏序集的值相关联。利用$O(\frac{1}{ε^2} \log ε^2 w \log w)$空间给出了一个随机$(4+ε)$ -近似算法.其次，我们考虑一个无序流，其中每个项目都与创建时间和数值相关联，并且项目可以相对于其创建时间无序。我们的目标是估计编辑距离单调性相对于创建时间的顺序排列的项目的数值。我们表明，任何随机常数近似算法需要线性空间。
英文摘要：
Given a stream of items each associated with a numerical value, its edit distance to monotonicity is the minimum number of items to remove so that the remaining items are non-decreasing with respect to the numerical value. The space complexity of estimating the edit distance to monotonicity of a data stream is becoming well-understood over the past few years. Motivated by applications on network quality monitoring, we extend the study to estimating the edit distance to monotonicity of a sliding window covering the $w$ most recent items in the stream for any $w \ge 1$. We give a deterministic algorithm which can return an estimate within a factor of $(4+\eps)$ using $O(\frac{1}{\eps^2} \log^2(\eps w))$ space. We also extend the study in two directions. First, we consider a stream where each item is associated with a value from a partial ordered set. We give a randomized $(4+ε)$-approximate algorithm using $O(\frac{1}{ε^2} \log ε^2 w \log w)$ space. Second, we consider an out-of-order stream where each item is associated with a creation time and a numerical value, and items may be out of order with respect to their creation times. The goal is to estimate the edit distance to monotonicity with respect to the numerical value of items arranged in the order of creation times. We show that any randomized constant-approximate algorithm requires linear space.
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