On Hashing by (Random) Equations (Invited Talk)

Author Martin Dietzfelbinger

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Martin Dietzfelbinger
  • Technische UniversitΓ€t Ilmenau, Germany

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Martin Dietzfelbinger. On Hashing by (Random) Equations (Invited Talk). In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik (2023)


The talk will consider aspects of the following setup: Assume for each (key) x from a set 𝒰 (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 Ο†_Z: 𝒰 β†’ {0,1}^r, x ↦ ⟨a_x,Z⟩ := ⨁_{i ∈ [m]} a_{x,i} β‹… z_i, where ⨁ is bitwise XOR. The simplest way for creating a data structure for calculating Ο†_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+Ξ΅)n for some small Ξ΅, 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 βŠ† 𝒰 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 ∈ 𝒰 are chosen independently at random from {0,1}^m, according to some distribution π’Ÿ. We wish to construct (static) retrieval data structures, which means that S βŠ‚ 𝒰 and some mapping f: S β†’ {0,1}^r are given, and the task is to find Z such that the restriction of Ο†_Z to S is f. For creating such a data structure it is necessary to solve the linear system (a_x)_{x ∈ S} β‹… Z = (f(x))_{x ∈ S} for Z. Two problems are central: Under what conditions on m and π’Ÿ 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.

Subject Classification

ACM Subject Classification
  • Theory of computation β†’ Sorting and searching
  • Theory of computation β†’ Randomness, geometry and discrete structures
  • Hashing
  • Retrieval
  • Linear equations
  • Randomness


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