Truly subquadratic detection of three collinear points

Determine whether three collinear points among a set of n points in the plane can be found in truly subquadratic time.

Background

The paper obtains faster algorithms for problems equivalent to 3SUM, APSP, or Exact Triangle, but it does not improve problems that are only known to be 3SUM-hard or APSP-hard because the known reductions have the opposite direction. Detecting three collinear points is given as a representative geometric problem whose best known running time remains quadratic.

References

For instance, it is now open whether one can find three collinear points among $n$ points in the plane in truly subquadratic time.

— Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs  (2610.06783 - Alman et al., 5 Oct 2026) in Section 1, paragraph “More new algorithms”

More broadly, the complexity of the many problems that are only known to be 3SUM-hard or APSP-hard, rather than equivalent to $3$SUM or APSP, is now open (the gray boxes of Figure~\ref{fig:web}); we get no faster algorithms for them, since the reductions go the other way.

— Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs  (2610.06783 - Alman et al., 5 Oct 2026) in Section 8, Section “Conclusion,” paragraph “Fine-grained complexity”