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Improved Streaming Algorithm for the Klee’s Measure Problem and Generalizations

Authors Mridul Nandi, N. V. Vinodchandran, Arijit Ghosh, Kuldeep S. Meel, Soumit Pal, Sourav Chakraborty

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Author Details

Mridul Nandi
  • Indian Statistical Institute, Kolkata, India
N. V. Vinodchandran
  • University of Nebraska, Lincoln, USA
Arijit Ghosh
  • Indian Statistical Institute, Kolkata, India
Kuldeep S. Meel
  • University of Toronto, Canada
Soumit Pal
  • Indian Statistical Institute, Kolkata, India
Sourav Chakraborty
  • Indian Statistical Institute, Kolkata, India

Funding

  • Vinodchandran, N. V.: Vinodchandran’s research is partially supported by NSF grants 2130608 and 2342244.
  • Ghosh, Arijit: Arijit Ghosh is partially supported by the Science & Engineering Research Board of the DST, India, through the MATRICS grant MTR/2023/001527.
  • Meel, Kuldeep S.: This work was supported in part by National Research Foundation Singapore under its NRF Fellowship Programme [NRF-NRFFAI1-2019- 0004].

Acknowledgements

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) [RGPIN-2024-05956].

Cite As Get BibTex

Mridul Nandi, N. V. Vinodchandran, Arijit Ghosh, Kuldeep S. Meel, Soumit Pal, and Sourav Chakraborty. Improved Streaming Algorithm for the Klee’s Measure Problem and Generalizations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 26:1-26:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.26

Abstract

Estimating the size of the union of a stream of sets S₁, S₂, …, S_M where each set is a subset of a known universe Ω is a fundamental problem in data streaming. This problem naturally generalizes the well-studied 𝖥₀ estimation problem in the streaming literature, where each set contains a single element from the universe. We consider the general case when the sets S_i can be succinctly represented and allow efficient membership, cardinality, and sampling queries (called a Delphic family of sets). A notable example in this framework is the Klee’s Measure Problem (KMP), where every set S_i is an axis-parallel rectangle in d-dimensional spaces (Ω = [Δ]^d where [Δ] := {1, … ,Δ} and Δ ∈ ℕ). Recently, Meel, Chakraborty, and Vinodchandran (PODS-21, PODS-22) designed a streaming algorithm for (ε,δ)-estimation of the size of the union of set streams over Delphic family with space and update time complexity O((log³|Ω|)/ε² ⋅ log 1/δ) and Õ((log⁴|Ω|)/ε² ⋅ log 1/(δ)), respectively. 
This work presents a new, sampling-based algorithm for estimating the size of the union of Delphic sets that has space and update time complexity Õ((log²|Ω|)/ε² ⋅ log 1/(δ)). This improves the space complexity bound by a log|Ω| factor and update time complexity bound by a log² |Ω| factor. 
A critical question is whether quadratic dependence of log|Ω| on space and update time complexities is necessary. Specifically, can we design a streaming algorithm for estimating the size of the union of sets over Delphic family with space and complexity linear in log|Ω| and update time poly(log|Ω|)? While this appears technically challenging, we show that establishing a lower bound of ω(log|Ω|) with poly(log|Ω|) update time is beyond the reach of current techniques. Specifically, we show that under certain hard-to-prove computational complexity hypothesis, there is a streaming algorithm for the problem with optimal space complexity O(log|Ω|) and update time poly(log(|Ω|)). Thus, establishing a space lower bound of ω(log|Ω|) will lead to break-through complexity class separation results.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sketching and sampling
Keywords
  • Sampling
  • Streaming
  • Klee’s Measure Problem

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