Linear Separability in Spatial Databases

TL;DR

This paper introduces efficient algorithms for determining linear separability and computing convex hulls of point sets stored in R-trees, optimizing memory usage and demonstrating practical effectiveness on real and synthetic data.

Contribution

It presents novel algorithms leveraging R-tree structures for linear separability testing and convex hull computation with optimal worst-case time complexity.

Findings

Algorithms run in O(m log m + n log n) time for separability and O(n log n) for convex hulls.

Experimental results show low memory usage and minimal R-tree node access.

Proposed methods are effective on both real and synthetic datasets.

Abstract

Given two point sets $R$ and $B$ in the plane, with cardinalities $m$ and $n$, respectively, and each set stored in a separate R-tree, we present an algorithm to decide whether $R$ and $B$ are linearly separable. Our algorithm exploits the structure of the R-trees, loading into the main memory only relevant data, and runs in $O(m\log m + n\log n)$ time in the worst case. As experimental results, we implement the proposed algorithm and executed it on several real and synthetic point sets, showing that the percentage of nodes of the R-trees that are accessed and the memory usage are low in these cases. We also present an algorithm to compute the convex hull of $n$ planar points given in an R-tree, running in $O(n\log n)$ time in the worst case.