Published February 5, 2026 | Version v1
Preprint Open

Conditional Lower Bounds for Grid Pathfinding Oracles

Authors/Creators

Description

This paper proves that, assuming the Strong Exponential Time Hypothesis (SETH), any distance oracle for n×n grid graphs with O(1) query time requires Ω(n^(2-o(1))) preprocessing time. The proof proceeds by reduction from the Orthogonal Vectors (OV) problem via an explicit gadget construction.

The main result establishes a fundamental space-time tradeoff: achieving constant-time distance queries on grids is provably expensive in preprocessing. This provides theoretical justification for the design of practical pathfinding algorithms like Jump Point Search (JPS+), which achieve O(n²) preprocessing with O(path length) query time.

Files

Conditional_Lower_Bounds_for_Grid_Pathfinding_Oracles.pdf

Files (295.4 kB)

Additional details

References

  • Williams, R. (2015). Hardness of easy problems: Basing hardness on popular conjectures such as the Strong Exponential Time Hypothesis. Proceedings of IPEC 2015. https://doi.org/10.4230/LIPIcs.IPEC.2015.356
  • Backurs, A., & Indyk, P. (2015). Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). Proceedings of STOC 2015. https://doi.org/10.1145/2746539.2746612
  • Thorup, M., & Zwick, U. (2005). Approximate distance oracles. Journal of the ACM, 52(1), 1-24. https://doi.org/10.1145/1044731.1044732
  • Harabor, D., & Grastien, A. (2011). Online graph pruning for pathfinding on grid maps. Proceedings of AAAI 2011.
  • Harabor, D., & Grastien, A. (2014). Improving jump point search. Proceedings of ICAPS 2014.
  • Geisberger, R., Sanders, P., Schultes, D., & Delling, D. (2008). Contraction hierarchies: Faster and simpler hierarchical routing in road networks. Proceedings of WEA 2008. https://doi.org/10.1007/978-3-540-68552-4_24