Dynamic Enumeration of Similarity Joins

Abstract

This paper considers enumerating answers to similarity-join queries under dynamic updates: Given two sets of n points A,B in ℝ^d, a metric ϕ(⋅), and a distance threshold r > 0, report all pairs of points (a, b) ∈ A × B with ϕ(a,b) ≤ r. Our goal is to store A,B into a dynamic data structure that, whenever asked, can enumerate all result pairs with worst-case delay guarantee, i.e., the time between enumerating two consecutive pairs is bounded. Furthermore, the data structure can be efficiently updated when a point is inserted into or deleted from A or B. We propose several efficient data structures for answering similarity-join queries in low dimension. For exact enumeration of similarity join, we present near-linear-size data structures for 𝓁₁, 𝓁_∞ metrics with log^{O(1)} n update time and delay. We show that such a data structure is not feasible for the 𝓁₂ metric for d ≥ 4. For approximate enumeration of similarity join, where the distance threshold is a soft constraint, we obtain a unified linear-size data structure for 𝓁_p metric, with log^{O(1)} n delay and update time. In high dimensions, we present an efficient data structure with worst-case delay-guarantee using locality sensitive hashing (LSH).