Exact Subquadratic Algorithm for Many-to-Many Matching on Planar Point Sets with Integer Coordinates
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908976003678208 |
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| author | Park, Seongbin Oh, Eunjin |
| author_facet | Park, Seongbin Oh, Eunjin |
| contents | In this paper, we study the many-to-many matching problem on planar point sets with integer coordinates: Given two disjoint sets $R,B \subset [Δ]^2$ with $|R|+|B|=n$, the goal is to select a set of edges between $R$ and $B$ so that every point is incident to at least one edge and the total Euclidean length is minimized. In the general case that $R$ and $B$ are point sets in the plane, the best-known algorithm for the many-to-many matching problem takes $\tilde{O}(n^2)$ time. We present an exact $\tilde{O}(n^{1.5} \log Δ)$ time algorithm for point sets in $[Δ]^2$. To the best of our knowledge, this is the first subquadratic exact algorithm for planar many-to-many matching under bounded integer coordinates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_16921 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exact Subquadratic Algorithm for Many-to-Many Matching on Planar Point Sets with Integer Coordinates Park, Seongbin Oh, Eunjin Computational Geometry Data Structures and Algorithms In this paper, we study the many-to-many matching problem on planar point sets with integer coordinates: Given two disjoint sets $R,B \subset [Δ]^2$ with $|R|+|B|=n$, the goal is to select a set of edges between $R$ and $B$ so that every point is incident to at least one edge and the total Euclidean length is minimized. In the general case that $R$ and $B$ are point sets in the plane, the best-known algorithm for the many-to-many matching problem takes $\tilde{O}(n^2)$ time. We present an exact $\tilde{O}(n^{1.5} \log Δ)$ time algorithm for point sets in $[Δ]^2$. To the best of our knowledge, this is the first subquadratic exact algorithm for planar many-to-many matching under bounded integer coordinates. |
| title | Exact Subquadratic Algorithm for Many-to-Many Matching on Planar Point Sets with Integer Coordinates |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2604.16921 |