Well-Separated Pairs Decomposition Revisited
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916938089758720 |
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| author | Har-Peled, Sariel Raichel, Benjamin Robson, Eliot W. |
| author_facet | Har-Peled, Sariel Raichel, Benjamin Robson, Eliot W. |
| contents | We revisit the notion of WSPD (i.e., well-separated pairs-decomposition), presenting a new construction of WSPD for any finite metric space, and show that it is asymptotically instance-optimal in size. Next, we describe a new WSPD construction for the weighted unit-distance metric in the plane, and show a bound $O( \varepsilon^{-2} n \log n)$ on its size, improving by a factor of $1/\varepsilon^2$ over previous work. The new construction is arguably simpler and more elegant.
We point out that using WSPD, one can approximate, in near-linear time, the distortion of a bijection between two point sets in low dimensions. As a new application of WSPD, we show how to shortcut a polygonal curve such that its dilation is below a prespecified quantity. In particular, we show a near-linear time algorithm for computing a simple subcurve for a given polygonal curve in the plane so that the new subcurve has no self-intersection. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05997 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-Separated Pairs Decomposition Revisited Har-Peled, Sariel Raichel, Benjamin Robson, Eliot W. Computational Geometry We revisit the notion of WSPD (i.e., well-separated pairs-decomposition), presenting a new construction of WSPD for any finite metric space, and show that it is asymptotically instance-optimal in size. Next, we describe a new WSPD construction for the weighted unit-distance metric in the plane, and show a bound $O( \varepsilon^{-2} n \log n)$ on its size, improving by a factor of $1/\varepsilon^2$ over previous work. The new construction is arguably simpler and more elegant. We point out that using WSPD, one can approximate, in near-linear time, the distortion of a bijection between two point sets in low dimensions. As a new application of WSPD, we show how to shortcut a polygonal curve such that its dilation is below a prespecified quantity. In particular, we show a near-linear time algorithm for computing a simple subcurve for a given polygonal curve in the plane so that the new subcurve has no self-intersection. |
| title | Well-Separated Pairs Decomposition Revisited |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2509.05997 |