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How many users have been here for a long time? Efficient solutions for counting long aggregated visits

Peyman Afshani, Rezaul Chowdhury, Inge Li Gørtz, Mayank Goswami, Francesco Silvestri, Mariafiore Tognon

TL;DR

This work tackles the problem of counting distinct users with long aggregated visits across queried region sets in mobility data, introducing two formulations: a general $(k,r)$-CLAV and a geometric Geometric-CLAV variant. It develops exact data structures with space-time tradeoffs, along with linear-space sampling and sketch-based approximations, and establishes unconditional and conditional lower bounds that guide feasibility. For geometry-aware instances, it provides tabulation and a colored dominance counting-based data structure to achieve near-optimal performance under standard hardness assumptions. Overall, the results offer scalable methods and fundamental limits for counting long aggregated visits in massive mobility datasets, enabling efficient analytics without enumerating users.

Abstract

This paper addresses the Counting Long Aggregated Visits problem, which is defined as follows. We are given $n$ users and $m$ regions, where each user spends some time visiting some regions. For a parameter $k$ and a query consisting of a subset of $r$ regions, the task is to count the number of distinct users whose aggregate time spent visiting the query regions is at least $k$. This problem is motivated by queries arising in the analysis of large-scale mobility datasets. We present several exact and approximate data structures for supporting counting long aggregated visits, as well as conditional and unconditional lower bounds. First, we describe an exact data structure that exhibits a space-time tradeoff, as well as efficient approximate solutions based on sampling and sketching techniques. We then study the problem in geometric settings where regions are points in $\mathbb{R}^d$ and queries are hyperrectangles, and derive exact data structures that achieve improved performance in these structured spaces.

How many users have been here for a long time? Efficient solutions for counting long aggregated visits

TL;DR

This work tackles the problem of counting distinct users with long aggregated visits across queried region sets in mobility data, introducing two formulations: a general -CLAV and a geometric Geometric-CLAV variant. It develops exact data structures with space-time tradeoffs, along with linear-space sampling and sketch-based approximations, and establishes unconditional and conditional lower bounds that guide feasibility. For geometry-aware instances, it provides tabulation and a colored dominance counting-based data structure to achieve near-optimal performance under standard hardness assumptions. Overall, the results offer scalable methods and fundamental limits for counting long aggregated visits in massive mobility datasets, enabling efficient analytics without enumerating users.

Abstract

This paper addresses the Counting Long Aggregated Visits problem, which is defined as follows. We are given users and regions, where each user spends some time visiting some regions. For a parameter and a query consisting of a subset of regions, the task is to count the number of distinct users whose aggregate time spent visiting the query regions is at least . This problem is motivated by queries arising in the analysis of large-scale mobility datasets. We present several exact and approximate data structures for supporting counting long aggregated visits, as well as conditional and unconditional lower bounds. First, we describe an exact data structure that exhibits a space-time tradeoff, as well as efficient approximate solutions based on sampling and sketching techniques. We then study the problem in geometric settings where regions are points in and queries are hyperrectangles, and derive exact data structures that achieve improved performance in these structured spaces.
Paper Structure (18 sections, 11 theorems, 6 equations, 1 table, 4 algorithms)

This paper contains 18 sections, 11 theorems, 6 equations, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. $(k,r)$-CLAV
  3. Geometric-CLAV
  4. Previous works
  5. Exact solutions to the $(k,r)$-CLAV problem
  6. Lower bounds for the $(k,r)$-CLAV problem
  7. Upper bound for the $(k,r)$-CLAV problem
  8. Approximate solutions for the $(k,r)$-CLAV problem
  9. Approximation via Sampling
  10. Approximation via Sketch
  11. Geometric-CLAV
  12. An Unconditional Space Lower Bound
  13. A Conditional Lower Bound from Fast-Query SetDisjointness
  14. A Conditional Lower Bound for $d=2$ using the Boolean-Matrix-Multiplication Conjecture
  15. Upper Bounds for Geometric-CLAV
  16. ...and 3 more sections

Key Result

Theorem 2.1

Assume Conjecture 1 holds. Then any data structure for the $(k,r)$-CLAV problem that uses $S$ space and has query time $T$ must satisfy $ST^{r}=\widetilde{\Omega}(N^{r})$.

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2: Geometric-CLAV
  • Conjecture 1: goldstein2017conditional
  • Theorem 2.1
  • proof
  • Conjecture 2
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • ...and 17 more