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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.01580 |
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| _version_ | 1866918134002221056 |
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| author | Dumitrescu, Adrian Lángi, Zsolt |
| author_facet | Dumitrescu, Adrian Lángi, Zsolt |
| contents | A curve $γ$ that connects $s$ and $t$ has the increasing chord property if $|bc| \leq |ad|$ whenever $a,b,c,d$ lie in that order on $γ$. For planar curves, the length of such a curve is known to be at most $2π/3 \cdot |st|$. Here we examine the question in higher dimensions and from the algorithmic standpoint and show the following:
(I) The length of any $s-t$ curve with increasing chords in $\mathbf{R}^d$ is at most $2 \cdot \left( e/2 \cdot (d+4) \right)^{d-1} \cdot |st|$ for every $d \geq 3$. This is the first bound in higher dimensions.
(II) Given a polygonal chain $P=(p_1, p_2, \dots, p_n)$ in $\mathbf{R}^d$, where $d \geq 4$, $k =\lfloor d/2 \rfloor$, it can be tested whether it satisfies the increasing chord property in $O\left(n^{2-1/(k+1)} {\rm polylog} (n) \right)$ expected time. This is the first subquadratic algorithm in higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01580 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arcs with increasing chords in $\mathbf{R}^d$ Dumitrescu, Adrian Lángi, Zsolt Computational Geometry Discrete Mathematics Combinatorics A curve $γ$ that connects $s$ and $t$ has the increasing chord property if $|bc| \leq |ad|$ whenever $a,b,c,d$ lie in that order on $γ$. For planar curves, the length of such a curve is known to be at most $2π/3 \cdot |st|$. Here we examine the question in higher dimensions and from the algorithmic standpoint and show the following: (I) The length of any $s-t$ curve with increasing chords in $\mathbf{R}^d$ is at most $2 \cdot \left( e/2 \cdot (d+4) \right)^{d-1} \cdot |st|$ for every $d \geq 3$. This is the first bound in higher dimensions. (II) Given a polygonal chain $P=(p_1, p_2, \dots, p_n)$ in $\mathbf{R}^d$, where $d \geq 4$, $k =\lfloor d/2 \rfloor$, it can be tested whether it satisfies the increasing chord property in $O\left(n^{2-1/(k+1)} {\rm polylog} (n) \right)$ expected time. This is the first subquadratic algorithm in higher dimensions. |
| title | Arcs with increasing chords in $\mathbf{R}^d$ |
| topic | Computational Geometry Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2509.01580 |