Subquadratic-Time Algorithm for Tree Point-Set Embeddings with Prescribed Crossings

Determine whether there exists an algorithm with subquadratic time complexity o(n^2) that, given an n-vertex tree T, a set S of n distinct points in the plane, and an integer χ with 0 ≤ χ ≤ ϑ(T), computes a point-set embedding of T on S with exactly χ edge crossings and constant curve complexity.

Background

The paper presents an O(n2)-time algorithm (Prune–Tangle–Untangle) for constructing, for any χ in [0, ϑ(T)], a point-set embedding of a tree with exactly χ crossings and constant curve complexity, and faster algorithms for paths.

The authors ask whether this quadratic bound can be improved to subquadratic time, even for restricted tree classes such as binary trees.

References

We conclude with some open problems. (1) Is there an o(n2)-time algorithm to compute a point-set embedding of a tree with χ crossings and constant curve complexity? This question is interesting even for binary trees.

— Tangling and Untangling Trees on Point-sets (2508.18535 - Battista et al., 25 Aug 2025) in Section 6, Final Remarks and Open Problems