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COL-Trees: Efficient Hierarchical Object Search in Road Networks

Tenindra Abeywickrama, Muhammad Aamir Cheema, Sabine Storandt

TL;DR

COL-Tree introduces a hierarchical, landmark-based index for efficient exact object search on road networks, addressing AkNN, kFN, and range queries where Euclidean heuristics fail. The core ideas are the Subgraph-Landmark Tree (SUL-Tree) for road-indexing and the Compacted Object Landmark Tree (COL-Tree) for compact object indexing, enabling tight lower- and upper-bounds to guide hierarchical traversal. The approach leverages the convexity-preserving properties of common aggregates (e.g., sum, max) to retrieve candidates incrementally and terminate early, achieving up to four orders of magnitude faster query times in experiments while incurring modest pre-processing overhead. Empirically, COL-Tree outperforms state-of-the-art landmark-based and Voronoi-based methods across AkNN, kFN, and range queries on large-scale road networks, with practical benefits for real-time location-based services and planning tasks.

Abstract

Location-based services rely heavily on efficient methods that search for relevant points-of-interest (POIs) near a given location. A k Nearest Neighbor (kNN) query is one such example that finds the k closest POIs from an agent's location. While most existing techniques focus on retrieving nearby POIs for a single agent, these search heuristics do not translate to many other applications. For example, Aggregate k Nearest Neighbor (AkNN) queries require POIs that are close to multiple agents. k Farthest Neighbor (kFN) queries require POIs that are the antithesis of nearest. Such problems naturally benefit from a hierarchical approach, but existing methods rely on Euclidean-based heuristics, which have diminished effectiveness in graphs such as road networks. We propose a novel data structure, COL-Tree (Compacted Object-Landmark Tree), to address this gap by enabling efficient hierarchical graph traversal using a more accurate landmark-based heuristic. We then present query algorithms that utilize COL-Trees to efficiently answer AkNN, kFN, and other queries. In our experiments on real-world and synthetic datasets, we demonstrate that our techniques significantly outperform existing approaches, achieving up to 4 orders of magnitude improvement. Moreover, this comes at a small pre-processing overhead in both theory and practice.

COL-Trees: Efficient Hierarchical Object Search in Road Networks

TL;DR

COL-Tree introduces a hierarchical, landmark-based index for efficient exact object search on road networks, addressing AkNN, kFN, and range queries where Euclidean heuristics fail. The core ideas are the Subgraph-Landmark Tree (SUL-Tree) for road-indexing and the Compacted Object Landmark Tree (COL-Tree) for compact object indexing, enabling tight lower- and upper-bounds to guide hierarchical traversal. The approach leverages the convexity-preserving properties of common aggregates (e.g., sum, max) to retrieve candidates incrementally and terminate early, achieving up to four orders of magnitude faster query times in experiments while incurring modest pre-processing overhead. Empirically, COL-Tree outperforms state-of-the-art landmark-based and Voronoi-based methods across AkNN, kFN, and range queries on large-scale road networks, with practical benefits for real-time location-based services and planning tasks.

Abstract

Location-based services rely heavily on efficient methods that search for relevant points-of-interest (POIs) near a given location. A k Nearest Neighbor (kNN) query is one such example that finds the k closest POIs from an agent's location. While most existing techniques focus on retrieving nearby POIs for a single agent, these search heuristics do not translate to many other applications. For example, Aggregate k Nearest Neighbor (AkNN) queries require POIs that are close to multiple agents. k Farthest Neighbor (kFN) queries require POIs that are the antithesis of nearest. Such problems naturally benefit from a hierarchical approach, but existing methods rely on Euclidean-based heuristics, which have diminished effectiveness in graphs such as road networks. We propose a novel data structure, COL-Tree (Compacted Object-Landmark Tree), to address this gap by enabling efficient hierarchical graph traversal using a more accurate landmark-based heuristic. We then present query algorithms that utilize COL-Trees to efficiently answer AkNN, kFN, and other queries. In our experiments on real-world and synthetic datasets, we demonstrate that our techniques significantly outperform existing approaches, achieving up to 4 orders of magnitude improvement. Moreover, this comes at a small pre-processing overhead in both theory and practice.
Paper Structure (37 sections, 3 theorems, 7 equations, 19 figures, 8 tables, 3 algorithms)

This paper contains 37 sections, 3 theorems, 7 equations, 19 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Limitations of Existing Techniques
  3. Contributions
  4. Preliminaries
  5. Data Structures
  6. Road Network Index: Subgraph-Landmark Tree (SUL-Tree)
  7. Object Index: Compacted Object Landmark Tree (COL-Tree)
  8. Lower-Bound Heuristic for Graph Traversal
  9. SUL-Tree Landmark Selection Policy
  10. Tightening Lower-Bounds with Tree Depth
  11. Query Algorithm for AkNN Search
  12. Object Distance Lists and Convexity
  13. Search Algorithm
  14. Query Algorithm for kFN Search
  15. Search Algorithm
  16. ...and 22 more sections

Key Result

Lemma 1

Given a monotonic aggregate function $agg$ and lower-bounds distance $LB(q_i,p)$ from each query vertex $q_i \in Q$ to object $p$, the aggregation of the lower-bound distances gives a lower-bound aggregate distance on the true aggregate distance. I.e., $LB_{agg}(Q,p) = agg(LB(q_0,p),\ldots,LB(q_{|Q|

Figures (19)

  • Figure 1: Subgraph Landmark Tree (SUL-Tree)
  • Figure 2: Compacted Object-Landmark Tree (COL-Tree)
  • Figure 3: Hierarchical Subgraph Vertex Ordering
  • Figure 4: Lower-Bound Distance to Subgraph Borders
  • Figure 5: A$k$NN query performance on different real-world POI sets
  • ...and 14 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof