Dynamization-based speedup of the testing algorithm for increasing-chord polygonal chains

Ascertain whether the semidynamic fixed-order deletion approach used within the halfspace-emptiness–based randomized algorithm for testing whether an n-vertex polygonal chain in R^d (d ≥ 4) has the increasing chord property can be dynamized (e.g., via the Bentley–Saxe transformation or another technique) to achieve an expected running time of O(n^{2−1/k} polylog n), where k = ⌊d/2⌋, thereby improving upon the current O(n^{2−1/(k+1)} polylog n) bound.

Background

The algorithmic contribution of the paper shows how to test the increasing chord property for a polygonal chain in Rd in expected time O(n{2−1/(k+1)} polylog n) using search-decomposable halfspace emptiness data structures, where k = ⌊d/2⌋.

The authors suggest that a dynamization technique in the style of Bentley–Saxe might allow a further improvement in the exponent, but they emphasize that this is not yet established.

References

It is conceivable that the semidynamic fixed order deletions executed by the algorithm are amenable to a dynamization in the style of Bentley {content} Saxe or to another speedup technique. That may lead to a slightly faster algorithm running in $O\left(n{2-1/k} \polylog(n) \right)$ time, where $k=\lfloor d/2 \rfloor$. This remains to be confirmed.

— Arcs with increasing chords in $\mathbf{R}^d$  (2509.01580 - Dumitrescu et al., 1 Sep 2025) in Section 5, Concluding remarks (item 2)