Fast hashing with strong concentration bounds

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Previous work on tabulation hashing by PÇtraşcu and Thorup from STOC'11 on simple tabulation and from SODA'13 on twisted tabulation offered Chernoff-style concentration bounds on hash based sums, e.g., the number of balls/keys hashing to a given bin, but under some quite severe restrictions on the expected values of these sums. The basic idea in tabulation hashing is to view a key as consisting of c=O(1) characters, e.g., a 64-bit key as c=8 characters of 8-bits. The character domain ς should be small enough that character tables of size |ς| fit in fast cache. The schemes then use O(1) tables of this size, so the space of tabulation hashing is O(|ς|). However, the concentration bounds by PÇtraşcu and Thorup only apply if the expected sums are g‰ |ς|. To see the problem, consider the very simple case where we use tabulation hashing to throw n balls into m bins and want to analyse the number of balls in a given bin. With their concentration bounds, we are fine if n=m, for then the expected value is 1. However, if m=2, as when tossing n unbiased coins, the expected value n/2 is ≫ |ς| for large data sets, e.g., data sets that do not fit in fast cache. To handle expectations that go beyond the limits of our small space, we need a much more advanced analysis of simple tabulation, plus a new tabulation technique that we call tabulation-permutation hashing which is at most twice as slow as simple tabulation. No other hashing scheme of comparable speed offers similar Chernoff-style concentration bounds.

Original languageEnglish
Title of host publicationSTOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
EditorsKonstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, Julia Chuzhoy
PublisherAssociation for Computing Machinery
Publication date2020
Pages1265-1278
ISBN (Electronic)9781450369794
DOIs
Publication statusPublished - 2020
Event52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 - Chicago, United States
Duration: 22 Jun 202026 Jun 2020

Conference

Conference52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020
LandUnited States
ByChicago
Periode22/06/202026/06/2020
SponsorACM Special Interest Group on Algorithms and Computation Theory (SIGACT)
SeriesProceedings of the Annual ACM Symposium on Theory of Computing
ISSN0737-8017

    Research areas

  • Chernoff bounds, Concentration bounds, Hashing, Sampling, Streaming algorithms

ID: 258498058