Optimal Bounds-Only Pruning for Spatial AkNN Joins

TL;DR

This work addresses exact Euclidean AkNN joins on partitioned, unindexed spatial datasets typical of data warehouses. It introduces AllPointsCloser, a three-bound bounds-only proximity test that prune partitions early by ensuring all points in an origin partition are closer to all points in a evaluation partition than to any point in a basis partition, and proves its optimality. An optimized O(R) version and a generalized partial-order interpretation enable efficient, direction-aware pruning and loading order. The approach provides practical gains in pruning before loading data or building indexes, enhancing performance for spatial data warehouses employing partition statistics.

Abstract

We propose a bounds-only pruning test for exact Euclidean AkNN joins on partitioned spatial datasets. Data warehouses commonly partition large tables and store row group statistics for them to accelerate searches and joins, rather than maintaining indexes. AkNN joins can benefit from such statistics by constructing bounds and localizing join evaluations to a few partitions before loading them to build spatial indexes. Existing pruning methods are overly conservative for bounds-only spatial data because they do not fully capture its directional semantics, thereby missing opportunities to skip unneeded partitions at the earliest stages of a join. We propose a three-bound proximity test to determine whether all points within a partition have a closer neighbor in one partition than in another, potentially occluded partition. We show that our algorithm is both optimal and efficient.

Figures (5)

• Figure 1: Point-to-Bound Distance Measures. Minimum and maximum distance to a partition shown, as well as the special case maximum distance for $k=1$ on an AABB (MinMaxDist).
• Figure 2: Bound-to-Bound Distance Measures
• Figure 3: Spatial partition layout with three neighbor search partitions P1-P3. One partition, P3, can be pruned without visitation because another partition, P2, will have all points closer to O. Only our proposed three-partition test identifies this.
• Figure 4: Geometric View of Pruning Test. Circles are drawn around each corner point of the origin partition to the nearest point in the basis partition; any geometry fully within their intersection must be closer than the basis partition.
• Figure 5: Visualization of used distance functions. \ref{['fig:img1']} and \ref{['fig:img3']} show existing point-to-bound functions. \ref{['fig:img2']} shows our difference-of-distances function internally used by AllPointsCloser.

Theorems & Definitions (12)

• theorem 1: All-Points Proximity Theorem
• proof
• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof
• lemma 4
• proof