k-最小值草图的关键压缩极限
Key Compression Limits for $k$-Minimum Value Sketches
AI总结：
本文研究k-最小值草图的压缩极限，证明基于排序和差值编码可实现每个键O(log n)比特的最优期望节省，并提出一种计算高效且接近熵理论最小值的实用压缩方法。
摘要：
k-最小值（k-Minimum Values，KMV）数据草图算法存储通过对数据集中的项目进行哈希生成的最小的 k 个哈希键。我们证明，基于对键进行排序并编码连续差值的压缩方法可以在期望存储节省上提供每个键 O(log n) 比特，其中 n 是数据集中唯一值的数量。我们还证明，对于 n 个随机值中最小的 k 个值，任何形式的压缩所能达到的每个键期望节省 O(log n) 比特是最优的——即该编码方法在所有编码 KMV 草图的方法中接近最优。我们提出了一种执行该压缩的实用方法，证明其计算效率高，并展示其实际平均节省量在基于熵的理论最小值的约百分之五以内。我们验证了我们的方法优于现成的压缩方法，并使用真实和合成数据证明其具有实用性。
英文摘要：
The $k$-Minimum Values (\kmv) data sketch algorithm stores the $k$ least hash keys generated by hashing the items in a dataset. We show that compression based on ordering the keys and encoding successive differences can offer $O(\log n)$ bits per key in expected storage savings, where $n$ is the number of unique values in the data set. We also show that $O(\log n)$ expected bits saved per key is optimal for any form of compression for the $k$ least of $n$ random values -- that the encoding method is near-optimal among all methods to encode a \kmv sketch. We present a practical method to perform that compression, show that it is computationally efficient, and demonstrate that its average savings in practice is within about five percent of the theoretical minimum based on entropy. We verify that our method outperforms off-the-shelf compression methods, and we demonstrate that it is practical, using real and synthetic data.
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