ISAAC 2015: Algorithms and Computation pp 35-45

# Navigating Weighted Regions with Scattered Skinny Tetrahedra

• Siu-Wing Cheng
• Man-Kwun Chiu
• Jiongxin Jin
• Antoine Vigneron
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9472)

## Abstract

We propose an algorithm for finding a $$(1+\varepsilon )$$-approximate shortest path through a weighted 3D simplicial complex $$\mathcal T$$. The weights are integers from the range [1, W] and the vertices have integral coordinates. Let N be the largest vertex coordinate magnitude, and let n be the number of tetrahedra in $$\mathcal T$$. Let $$\rho$$ be some arbitrary constant. Let $$\kappa$$ be the size of the largest connected component of tetrahedra whose aspect ratios exceed $$\rho$$. There exists a constant C dependent on $$\rho$$ but independent of $$\mathcal T$$ such that if $$\kappa \le \frac{1}{C}\log \log n + O(1)$$, the running time of our algorithm is polynomial in n, $$1/\varepsilon$$ and $$\log (NW)$$. If $$\kappa = O(1)$$, the running time reduces to $$O(n \varepsilon ^{-O(1)}(\log (NW))^{O(1)})$$.

## Keywords

Weighted region Shortest path Approximation algorithm

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## Authors and Affiliations

• Siu-Wing Cheng
• 1
• Man-Kwun Chiu
• 2
• 3
• Jiongxin Jin
• 4
• Antoine Vigneron
• 5
1. 1.Department of Computer Science and EngineeringHKUSTHong KongHong Kong
2. 2.National Institute of Informatics (NII)TokyoJapan
3. 3.JST, ERATOKawarabayashi Large Graph ProjectTokyoJapan