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Optimal Non-Adaptive Cell Probe Dictionaries and Hashing

Kasper Green Larsen, Rasmus Pagh, Giuseppe Persiano, Toniann Pitassi, Kevin Yeo, Or Zamir

TL;DR

The paper resolves the static dictionary and membership problems in the non-adaptive cell-probe model by giving a simple data structure that stores $n$ key–value pairs in $s$ cells of $w=\Theta(\log u)$ bits and answers queries in $t=O\left(\dfrac{\log(u/n)}{\log(s/n)}\right)$ probes, together with matching lower bounds. It shows that non-adaptivity incurs a fundamental time-space trade-off and establishes an adaptivity gap: a single round of adaptivity suffices to achieve constant-time queries, while non-adaptive schemes require super-constant time. The construction leverages $(\le n)$-non-contractive expanders and Hall's marriage theorem, and its Hashing extension yields an optimal non-adaptive evaluation of $n$-wise independent hash functions in the cell-probe model using similar probe counts. Although non-explicit, these results imply that explicit expanders with suitable parameters would enable RAM implementations, and they highlight the pivotal role of expanders and dispersers in non-adaptive data-structure design.

Abstract

We present a simple and provably optimal non-adaptive cell probe data structure for the static dictionary problem. Our data structure supports storing a set of n key-value pairs from [u]x[u] using s words of space and answering key lookup queries in t = O(lg(u/n)/ lg(s/n)) nonadaptive probes. This generalizes a solution to the membership problem (i.e., where no values are associated with keys) due to Buhrman et al. We also present matching lower bounds for the non-adaptive static membership problem in the deterministic setting. Our lower bound implies that both our dictionary algorithm and the preceding membership algorithm are optimal, and in particular that there is an inherent complexity gap in these problems between no adaptivity and one round of adaptivity (with which hashing-based algorithms solve these problems in constant time). Using the ideas underlying our data structure, we also obtain the first implementation of a n-wise independent family of hash functions with optimal evaluation time in the cell probe model.

Optimal Non-Adaptive Cell Probe Dictionaries and Hashing

TL;DR

The paper resolves the static dictionary and membership problems in the non-adaptive cell-probe model by giving a simple data structure that stores key–value pairs in cells of bits and answers queries in probes, together with matching lower bounds. It shows that non-adaptivity incurs a fundamental time-space trade-off and establishes an adaptivity gap: a single round of adaptivity suffices to achieve constant-time queries, while non-adaptive schemes require super-constant time. The construction leverages -non-contractive expanders and Hall's marriage theorem, and its Hashing extension yields an optimal non-adaptive evaluation of -wise independent hash functions in the cell-probe model using similar probe counts. Although non-explicit, these results imply that explicit expanders with suitable parameters would enable RAM implementations, and they highlight the pivotal role of expanders and dispersers in non-adaptive data-structure design.

Abstract

We present a simple and provably optimal non-adaptive cell probe data structure for the static dictionary problem. Our data structure supports storing a set of n key-value pairs from [u]x[u] using s words of space and answering key lookup queries in t = O(lg(u/n)/ lg(s/n)) nonadaptive probes. This generalizes a solution to the membership problem (i.e., where no values are associated with keys) due to Buhrman et al. We also present matching lower bounds for the non-adaptive static membership problem in the deterministic setting. Our lower bound implies that both our dictionary algorithm and the preceding membership algorithm are optimal, and in particular that there is an inherent complexity gap in these problems between no adaptivity and one round of adaptivity (with which hashing-based algorithms solve these problems in constant time). Using the ideas underlying our data structure, we also obtain the first implementation of a n-wise independent family of hash functions with optimal evaluation time in the cell probe model.
Paper Structure (7 sections, 9 theorems, 11 equations)

This paper contains 7 sections, 9 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Adaptivity and Membership
  3. Hash Functions with High Independence
  4. Non-Adaptive Dictionaries
  5. Hashing
  6. Lower Bound for Non-Adaptive Dictionaries
  7. Conclusion and Open Problems

Key Result

Theorem 1

For any $s = \Omega(n)$, there is a non-adaptive static cell probe data structure for the dictionary problem, storing $n$ key-value pairs $(x_i,y_i) \in [u] \times [u]$ using $s$ memory cells of $w = \Theta(\lg u)$ bits and answering queries in $t = O(\lg(u/n)/\lg(s/n))$ probes.

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4: Hall's Marriage Theorem
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 5 more