Abstract
We study the quality of weighted shortest paths when a continuous 2-dimensional space is discretized by a weighted triangular tessellation. In order to evaluate how well the tessellation approximates the 2-dimensional space, we study three types of shortest paths: a weighted shortest path \( SP_w (s,t) \), which is a shortest path from s to t in the space; a weighted shortest vertex path \( SVP_w (s,t) \), which is a shortest path where the vertices of the path are vertices of the tessellation; and a weighted shortest grid path \( SGP_w (s,t) \), which is a shortest path whose edges are edges of the tessellation. The ratios \( \frac{\Vert SGP_w (s,t)\Vert }{\Vert SP_w (s,t)\Vert } \), \( \frac{\Vert SVP_w (s,t)\Vert }{\Vert SP_w (s,t)\Vert } \), \( \frac{\Vert SGP_w (s,t)\Vert }{\Vert SVP_w (s,t)\Vert } \) provide estimates on the quality of the approximation.
Given any arbitrary weight assignment to the faces of a triangular tessellation, we prove upper and lower bounds on the estimates that are independent of the weight assignment. Our main result is that \( \frac{\Vert SGP_w (s,t)\Vert }{\Vert SP_w (s,t)\Vert } = \frac{2}{\sqrt{3}} \approx 1.15 \) in the worst case, and this is tight.
Keywords
- Shortest Path
- Tessellation
- Weighted Region Problem
Partially supported by NSERC, project PID2019-104129GB-I00/MCIN/AEI/ 10.13039/501100011033 of the Spanish Ministry of Science and Innovation, and H2020-MSCA-RISE project 734922 - CONNECT.
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Bose, P., Esteban, G., Orden, D., Silveira, R.I. (2022). On Approximating Shortest Paths in Weighted Triangular Tessellations. In: Mutzel, P., Rahman, M.S., Slamin (eds) WALCOM: Algorithms and Computation. WALCOM 2022. Lecture Notes in Computer Science(), vol 13174. Springer, Cham. https://doi.org/10.1007/978-3-030-96731-4_33
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