Approximating shortest paths in weighted square and hexagonal meshes Article Swipe
Prosenjit Bose
,
Guillermo Esteban
,
David Orden
,
Rodrigo I. Silveira
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2404.07562
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2404.07562
Continuous 2-dimensional space is often discretized by considering a mesh of weighted cells. In this work we study how well a weighted mesh approximates the space, with respect to shortest paths. We consider a shortest path $ \mathit{SP_w}(s,t) $ from $ s $ to $ t $ in the continuous 2-dimensional space, a shortest vertex path $ \mathit{SVP_w}(s,t) $ (or any-angle path), which is a shortest path where the vertices of the path are vertices of the mesh, and a shortest grid path $ \mathit{SGP_w}(s,t) $, which is a shortest path in a graph associated to the weighted mesh. We provide upper and lower bounds on the ratios $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $, $ \frac{\lVert \mathit{SVP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $, $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SVP_w}(s,t)\rVert} $ in square and hexagonal meshes, extending previous results for triangular grids. These ratios determine the effectiveness of existing algorithms that compute shortest paths on the graphs obtained from the grids. Our main results are that the ratio $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $ is at most $ \frac{2}{\sqrt{2+\sqrt{2}}} \approx 1.08 $ and $ \frac{2}{\sqrt{2+\sqrt{3}}} \approx 1.04 $ in a square and a hexagonal mesh, respectively.
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- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2404.07562
- https://arxiv.org/pdf/2404.07562
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4394781812
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- OpenAlex ID
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https://openalex.org/W4394781812Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2404.07562Digital Object Identifier
- Title
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Approximating shortest paths in weighted square and hexagonal meshesWork title
- Type
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preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
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2024Year of publication
- Publication date
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2024-04-11Full publication date if available
- Authors
-
Prosenjit Bose, Guillermo Esteban, David Orden, Rodrigo I. SilveiraList of authors in order
- Landing page
-
https://arxiv.org/abs/2404.07562Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2404.07562Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
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https://arxiv.org/pdf/2404.07562Direct OA link when available
- Concepts
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Combinatorics, Shortest path problem, Vertex (graph theory), Hexagonal tiling, Path (computing), Hexagonal crystal system, Mathematics, Space (punctuation), Square (algebra), Discrete mathematics, Graph, Physics, Grid, Computer science, Geometry, Crystallography, Operating system, Chemistry, Programming languageTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.hexagonal | 126, 184 |
| abstract_inverted_index.Continuous | 0 |
| abstract_inverted_index.algorithms | 141 |
| abstract_inverted_index.associated | 94 |
| abstract_inverted_index.continuous | 49 |
| abstract_inverted_index.triangular | 132 |
| abstract_inverted_index.considering | 7 |
| abstract_inverted_index.discretized | 5 |
| abstract_inverted_index.\frac{\lVert | 109, 114, 119, 161 |
| abstract_inverted_index.approximates | 23 |
| abstract_inverted_index.2-dimensional | 1, 50 |
| abstract_inverted_index.effectiveness | 138 |
| abstract_inverted_index.respectively. | 186 |
| abstract_inverted_index.\mathit{SP_w}(s,t) | 37 |
| abstract_inverted_index.\mathit{SGP_w}(s,t) | 84 |
| abstract_inverted_index.\mathit{SVP_w}(s,t) | 57 |
| abstract_inverted_index.\mathit{SP_w}(s,t)\rVert} | 111, 116, 163 |
| abstract_inverted_index.\mathit{SVP_w}(s,t)\rVert} | 121 |
| abstract_inverted_index.\frac{2}{\sqrt{2+\sqrt{2}}} | 169 |
| abstract_inverted_index.\frac{2}{\sqrt{2+\sqrt{3}}} | 175 |
| abstract_inverted_index.\mathit{SGP_w}(s,t)\rVert}{\lVert | 110, 120, 162 |
| abstract_inverted_index.\mathit{SVP_w}(s,t)\rVert}{\lVert | 115 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 4 |
| citation_normalized_percentile |