Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.22219

Linear-Time Transformations Between Connectivity Maintenance and Points Spreading on Linear and Cyclic Domains

Nicolás Honorato-Droguett
View original source ↗Read source document ↗Export metadata ↓

Abstract

Given $n$ points on a line or closed cycle and a threshold $r>0$, the connectivity-maintenance problem is to move the points so that every gap between consecutive points is at most $r$, whereas the points-spreading problem requires every gap to be at least $r$. Li and Wang [CCCG 2015; CGT 2025] and Chen, Gu, Li, and Wang [SWAT 2012; DCG 2013] gave $O(n)$-time algorithms for the cyclic versions of min-max points-spreading and min-max connectivity-maintenance, respectively. Ghadiri and Yazdanbod [CCCG 2016] gave an $O(n\log n)$-time algorithm for the linear version of min-sum points-spreading. In this paper, we show that the two problems can be reduced in linear time to each other for both objectives and on both linear and cyclic domains. As an implication, min-sum connectivity-maintenance is solvable in $O(n\log n)$ time on both domains. Finally, we extend the reduction to points on a line with individual thresholds when their initial order is preserved.

Explore related subjects

cs.CG

Keep this discovery

BibTeXRIS

Nicolás Honorato-Droguett. 2026-08-23. Linear-Time Transformations Between Connectivity Maintenance and Points Spreading on Linear and Cyclic Domains. https://arxiv.org/abs/2608.22219

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Almost Linear Universal Point Sets for Planar Graphs

A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, Devanny, and Eppstein from universal point sets to superpatterns for $213$-avoiding permutations. We represent these permutations by ordered rooted forests and construct a small family of intervals containing every such forest. The result follows from a straightforward bound on the size of the family of intervals. GPT-6 Astra assisted in developing the construction and proof.

cs.CG↗
arXiv2026

Some results on Archdeacon's conjecture for rotation systems

A rotation system on $n$ elements assigns to each element a cyclic order of the other $n-1$ elements. A four-element subset is non-planar if its induced rotation system cannot be realized by a crossing-free drawing of $K_4$. As a combinatorial strengthening of Hill's conjecture on the crossing number of the complete graph, Archdeacon conjectured that every rotation system on $n$ elements has at least $H(n)=\frac{1}{4} \lfloor\frac {n}{2}\rfloor \lfloor\frac{n-1}{2}\rfloor \lfloor\frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$ non-planar four-element subsets. We computationally verify Archdeacon's conjecture for $n\leq 10$ and show that every extremal rotation system in these orders is realizable by a simple drawing. With computer assistance, we prove that every rotation system on $n$ elements has at least $(8/9 - o(1)) H(n)$ non-planar four-element subsets. We also present a proof by hand for a weaker lower bound of $(2/3-o(1)) H(n)$. Finally, extending recent work of Felsner on antipodal pairs in drawings, we show that Archdeacon's conjecture holds for antipodally shellable rotation systems.

cs.CG↗
arXiv2026

The Hyperbolic Surface Distance, Diameter, and Dirichlet Problems

Despite the prominence of hyperbolic surfaces in mathematics, basic algorithmic questions about them, even computing the distance between two points, have remained open, leaving many features of these surfaces inaccessible. The classical machinery assumes a polyhedral structure absent on a smooth surface. We remove these obstacles. We begin with an efficient $O(g^2)$ algorithm for the distance between two points, where $g$ is the genus of the surface. Building on it, we obtain an $O(g^2 \log g)$ method for answering distance queries from a fixed source and, as a consequence, for recentering a Dirichlet domain around an arbitrary point. This understanding of distances on the surface then lets us approximate the diameter to within any $\eps$ in time $O(g^3 \log g / \eps^2)$. We further show that the diameter, a single real number encoding a great deal about the surface, is exactly computable. Its hyperbolic cosine is an algebraic number over the field encoding the coefficients of the hyperbolic isometries defining the surface.

cs.CG↗