License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2023.1
URN: urn:nbn:de:0030-drops-186545
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Dietzfelbinger, Martin

On Hashing by (Random) Equations (Invited Talk)

LIPIcs-ESA-2023-1.pdf (0.4 MB)


The talk will consider aspects of the following setup: Assume for each (key) x from a set ? (the universe) a vector a_x = (a_{x,0},… ,a_{x,{m-1}}) has been chosen. Given a list Z = (z_i)_{i ∈ [m]} of vectors in {0,1}^r we obtain a mapping φ_Z: ? → {0,1}^r, x ↦ ⟨a_x,Z⟩ := ⨁_{i ∈ [m]} a_{x,i} ⋅ z_i, where ⨁ is bitwise XOR. The simplest way for creating a data structure for calculating φ_Z is to store Z in an array Z[0..m-1] and answer a query for x by returning ⟨ a_x,Z⟩. The length m of the array should be (1+ε)n for some small ε, and calculating this inner product should be fast. In the focus of the talk is the case where for all or for most of the sets S ⊆ ? of a certain size n the vectors a_x, x ∈ S, are linearly independent. Choosing Z at random will lead to hash families of various degrees of independence. We will be mostly interested in the case where a_x, x ∈ ? are chosen independently at random from {0,1}^m, according to some distribution ?. We wish to construct (static) retrieval data structures, which means that S ⊂ ? and some mapping f: S → {0,1}^r are given, and the task is to find Z such that the restriction of φ_Z to S is f. For creating such a data structure it is necessary to solve the linear system (a_x)_{x ∈ S} ⋅ Z = (f(x))_{x ∈ S} for Z. Two problems are central: Under what conditions on m and ? can we expect the vectors a_x, x ∈ S to be linearly independent, and how can we arrange things so that in this case the system can be solved fast, in particular in time close to linear (in n, assuming a reasonable machine model)? Solutions to these problems, some classical and others that have emerged only in recent years, will be discussed.

BibTeX - Entry

  author =	{Dietzfelbinger, Martin},
  title =	{{On Hashing by (Random) Equations}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{1:1--1:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-186545},
  doi =		{10.4230/LIPIcs.ESA.2023.1},
  annote =	{Keywords: Hashing, Retrieval, Linear equations, Randomness}

Keywords: Hashing, Retrieval, Linear equations, Randomness
Collection: 31st Annual European Symposium on Algorithms (ESA 2023)
Issue Date: 2023
Date of publication: 30.08.2023

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