Logarithmic factor in WSPD size for planar weighted unit-distance graphs

Determine whether the logarithmic factor O(log n) in the size bound O(ε^{-2} n log n) for a 1/ε–well-separated pairs decomposition (WSPD) of the shortest-path metric on connected weighted unit-distance graphs G = (P, E) in the plane is necessary, or whether one can construct a 1/ε–WSPD of size O(ε^{-2} n) for all such instances (i.e., remove the log n factor).

Background

The paper introduces a new construction of a 1/ε–WSPD for the shortest-path metric on connected weighted unit-distance graphs in the plane, achieving a size bound of O(ε{-2} n log n). This improves the previous O(ε{-4} n log n) bound of Gao and Zhang by a factor of 1/ε2.

The construction uses multiple resolution levels up to O(log n), contributing the logarithmic factor in the size bound. The authors explicitly note that it remains unknown whether this O(log n) term is inherent or an artifact of the analysis, leaving a gap between the new upper bound and potential lower bounds. The open problem asks to close this gap by either removing the logarithmic term or proving its necessity.

References

Theorem{wspd_u_d} improves over the bound of Gao and Zhang by a factor of $1/\varepsilon2$. The question whether the $O(\log n)$ should be in the bound is still open.

— Well-Separated Pairs Decomposition Revisited  (2509.05997 - Har-Peled et al., 7 Sep 2025) in Remark (item a) after Theorem wspd_u_d, Section “WSPD for weighted unit-distance graphs”