# On Approximating Shortest Paths in Weighted Triangular Tessellations

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 13174)

## 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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Correspondence to Guillermo Esteban .

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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