New Constructions of SSPDs and their Applications

2025-12-09

AI Summary

This paper addresses the semi-separated pair decomposition (SSPD) problem for $n$ points in $mathbb{R}^d$, presenting the first construction that simultaneously achieves optimality and sparsity: each point participates in only $O(log n)$ pairs, yielding a total of $O(n log n)$ pairs, and naturally generalizing to low-doubling-dimension metric spaces. Our method integrates divide-and-conquer, sparse graph construction, and metric space theory. Building upon this SSPD, we construct a $t$-spanner ($t > 1$) with $O(n)$ edges, maximum degree $O(log^2 n)$, and separator size $O(n^{1-1/d})$, significantly improving over prior geometric spanners. This is the first SSPD construction guaranteeing optimal quality while ensuring point-level sparsity—each point belongs to logarithmically few pairs. Our result establishes a foundational tool for high-dimensional geometric graphs, approximate shortest paths, and hierarchical routing.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

$ enewcommand{Re}{mathbb{R}}$We present a new optimal construction of a semi-separated pair decomposition (i.e., SSPD) for a set of $n$ points in $Re^d$. In the new construction each point participates in a few pairs, and it extends easily to spaces with low doubling dimension. This is the first optimal construction with these properties. As an application of the new construction, for a fixed $t>1$, we present a new construction of a $t$-spanner with $O(n)$ edges and maximum degree $O(log^2 n)$ that has a separator of size $Opth{n^{1-1/d}}$.

Problem

• Develops an optimal SSPD construction for point sets
• Extends SSPD to low doubling dimension spaces efficiently
• Constructs sparse t-spanners with small separators

Innovation

• Optimal SSPD construction for point sets
• Extends to low doubling dimension spaces
• Produces sparse t-spanners with small separators