Exact solutions to the Weighted Region Problem

[2402.12028] Exact solutions to the Weighted Region Problem Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Computational Geometry arXiv:2402.12028 (cs) [Submitted on 19 Feb 2024 ( v1 ), last revised 9 Apr 2026 (this version, v4)] Title: Exact solutions to the Weighted Region Problem Authors: Sarita de Berg , Guillermo Esteban , Rodrigo I. Silveira , Frank Staals View a PDF of the paper titled Exact solutions to the Weighted Region Problem, by Sarita de Berg and Guillermo Esteban and Rodrigo I. Silveira and Frank Staals View PDF HTML (experimental) Abstract: In this paper, we consider the Weighted Region Problem. In the Weighted Region Problem, the length of a path is defined as the sum of the weights of the subpaths within each region, where the weight of a subpath is its Euclidean length multiplied by a weight $ \alpha \geq 0 $ depending on the region. We study a restricted version of the problem of determining shortest paths through a single weighted rectangular region. We prove that even this very restricted version of the problem is unsolvable within the Algebraic Computation Model over the Rational Numbers (ACMQ). On the positive side, we provide the equations for the shortest paths that are computable within the ACMQ. Additionally, we provide equations for the bisectors between regions of the Shortest Path Map for a source point on the boundary of (or inside) the rectangular region. Subjects: Computational Geometry (cs.CG) Cite as: arXiv:2402.12028 [cs.CG] (or arXiv:2402.12028v4 [cs.CG] for this version) https://doi.org/10.48550/arXiv.2402.12028 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Discrete Mathematics & Theoretical Computer Science, vol. 28:2, Discrete Algorithms (June 22, 2026) dmtcs:15053 Related DOI : https://doi.org/10.46298/dmtcs.15053 Focus to learn more DOI(s) linking to related resources Submission history From: Guillermo Esteban [ view email ] [v1] Mon, 19 Feb 2024 10:36:19 UTC (499 KB) [v2] Thu, 9 Jan 2025 11:18:51 UTC (231 KB) [v3] Fri, 9 Jan 2026 15:47:56 UTC (381 KB) [v4] Thu, 9 Apr 2026 11:38:29 UTC (248 KB) Full-text links: Access Paper: View a PDF of the paper titled Exact solutions to the Weighted Region Problem, by Sarita de Berg and Guillermo Esteban and Rodrigo I. Silveira and Frank Staals View PDF HTML (experimental) TeX Source view license Current browse context: cs.CG < prev | next > new | recent | 2024-02 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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