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Poisson Sampling over Acyclic Joins

Liese Bekkers, Frank Neven, Lorrens Pantelis, Stijn Vansummeren

TL;DR

This work proposes an algorithm for Poisson sampling over acyclic joins that is nearly instance-optimal, running in time O(N + k \log N) where N is the size of the input database, and k is the size of the resulting sample.

Abstract

We introduce the problem of Poisson sampling over joins: compute a sample of the result of a join query by conceptually performing a Bernoulli trial for each join tuple, using a non-uniform and tuple-specific probability. We propose an algorithm for Poisson sampling over acyclic joins that is nearly instance-optimal, running in time O(N + k \log N) where N is the size of the input database, and k is the size of the resulting sample. Our algorithm hinges on two building blocks: (1) The construction of a random-access index that allows, given a number i, to randomly access the i-th join tuple without fully materializing the (possibly large) join result; (2) The probing of this index to construct the result sample. We study the engineering trade-offs required to make both components practical, focusing on their implementation in column stores, and identify best-performing alternatives for both. Our experiments on real-world data demonstrate that this pair of alternatives significantly outperforms the repeated-Bernoulli-trial algorithm for Poisson sampling while also demonstrating that the random-access index by itself can be used to competively implement Yannakakis' acyclic join processing algorithm when no sampling is required. This shows that, as far a query engine design is concerned, it is possible to adopt a uniform basis for both classical acyclic join processing and Poisson sampling, both without regret compared to classical join and sampling algorithms.

Poisson Sampling over Acyclic Joins

TL;DR

This work proposes an algorithm for Poisson sampling over acyclic joins that is nearly instance-optimal, running in time O(N + k \log N) where N is the size of the input database, and k is the size of the resulting sample.

Abstract

We introduce the problem of Poisson sampling over joins: compute a sample of the result of a join query by conceptually performing a Bernoulli trial for each join tuple, using a non-uniform and tuple-specific probability. We propose an algorithm for Poisson sampling over acyclic joins that is nearly instance-optimal, running in time O(N + k \log N) where N is the size of the input database, and k is the size of the resulting sample. Our algorithm hinges on two building blocks: (1) The construction of a random-access index that allows, given a number i, to randomly access the i-th join tuple without fully materializing the (possibly large) join result; (2) The probing of this index to construct the result sample. We study the engineering trade-offs required to make both components practical, focusing on their implementation in column stores, and identify best-performing alternatives for both. Our experiments on real-world data demonstrate that this pair of alternatives significantly outperforms the repeated-Bernoulli-trial algorithm for Poisson sampling while also demonstrating that the random-access index by itself can be used to competively implement Yannakakis' acyclic join processing algorithm when no sampling is required. This shows that, as far a query engine design is concerned, it is possible to adopt a uniform basis for both classical acyclic join processing and Poisson sampling, both without regret compared to classical join and sampling algorithms.
Paper Structure (13 sections, 4 theorems, 10 equations, 10 figures, 4 tables)

This paper contains 13 sections, 4 theorems, 10 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Solution Overview
  4. Shredded Random-Access Indexing
  5. Chained Shredding
  6. Unchained Shredding
  7. Position sampling
  8. Experimental Evaluation
  9. Uniform Sampling
  10. Poisson Sampling
  11. Chained vs. unchained
  12. Related Work
  13. Conclusions

Key Result

proposition 1

For every acyclic join query $\hat{Q}$ and any attribute $y$ of $\hat{Q}$ we can compute an equivalent two-phase NSA expression $\mu^*(E)$ such that $y$ is a flat attribute in the output scheme of $E$.

Figures (10)

  • Figure 1: A join tree for the join query of Example \ref{['ex:poisson-query']}.
  • Figure 2: Illustration of nested relations and nested semijoins with $N_2=$$(R(x,y,p) \mathop{\mathrm{\ltimes_{\nu}}}\nolimits S(u,a,x))\mathop{\mathrm{\ltimes_{\nu}}}\nolimits T(v,y)$ and $N_1 = R(x,y,p)$$\mathop{\mathrm{\ltimes_{\nu}}}\nolimits S(u,a,x)$.
  • Figure 3: CSR grouping algorithm.
  • Figure 4: CSR random access.
  • Figure 5: USR Access
  • ...and 5 more figures

Theorems & Definitions (5)

  • proposition 1
  • definition 1
  • proposition 2
  • proposition 3
  • theorem 1