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oai:drops-oai.dagstuhl.de:14173 2024-03-06T10:53:39Z ddc:004 open_access

Non-Mergeable Sketching for Cardinality Estimation Pettie, Seth Wang, Dingyu Yin, Longhui Cardinality Estimation Sketching Cardinality estimation is perhaps the simplest non-trivial statistical problem that can be solved via sketching. Industrially-deployed sketches like HyperLogLog, MinHash, and PCSA are mergeable, which means that large data sets can be sketched in a distributed environment, and then merged into a single sketch of the whole data set. In the last decade a variety of sketches have been developed that are non-mergeable, but attractive for other reasons. They are simpler, their cardinality estimates are strictly unbiased, and they have substantially lower variance. We evaluate sketching schemes on a reasonably level playing field, in terms of their memory-variance product (MVP). E.g., a sketch that occupies 5m bits and whose relative variance is 2/m (standard error √{2/m}) has an MVP of 10. Our contributions are as follows. - Cohen [Edith Cohen, 2015] and Ting [Daniel Ting, 2014] independently discovered what we call the {Martingale transform} for converting a mergeable sketch into a non-mergeable sketch. We present a simpler way to analyze the limiting MVP of Martingale-type sketches. - Pettie and Wang proved that the Fishmonger sketch [Seth Pettie and Dingyu Wang, 2021] has the best MVP, H₀/I₀ ≈ 1.98, among a class of mergeable sketches called "linearizable" sketches. (H₀ and I₀ are precisely defined constants.) We prove that the Martingale transform is optimal in the non-mergeable world, and that Martingale Fishmonger in particular is optimal among linearizable sketches, with an MVP of H₀/2 ≈ 1.63. E.g., this is circumstantial evidence that to achieve 1% standard error, we cannot do better than a 2 kilobyte sketch. - Martingale Fishmonger is neither simple nor practical. We develop a new mergeable sketch called Curtain that strikes a nice balance between simplicity and efficiency, and prove that Martingale Curtain has limiting MVP≈ 2.31. It can be updated with O(1) memory accesses and it has lower empirical variance than Martingale LogLog, a practical non-mergeable version of HyperLogLog. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Seth Pettie and Dingyu Wang and Longhui Yin 2021 Is Part Of LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2021.104 urn:nbn:de:0030-drops-141731 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.104 eng https://creativecommons.org/licenses/by/4.0/legalcode