Abstract
We propose an algorithm for finding a \documentclass[12pt]{minimal}
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\begin{document}$$(1+\varepsilon )$$\end{document}-approximate shortest path through a weighted 3D simplicial complex \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal T$$\end{document}. 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 \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal T$$\end{document}. Let \documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} be some arbitrary constant. Let \documentclass[12pt]{minimal}
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\begin{document}$$\kappa $$\end{document} be the size of the largest connected component of tetrahedra whose aspect ratios exceed \documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document}. There exists a constant C dependent on \documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} but independent of \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal T$$\end{document} such that if \documentclass[12pt]{minimal}
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\begin{document}$$\kappa \le \frac{1}{C}\log \log n + O(1)$$\end{document}, the running time of our algorithm is polynomial in n, \documentclass[12pt]{minimal}
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\begin{document}$$1/\varepsilon $$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$\log (NW)$$\end{document}. If \documentclass[12pt]{minimal}
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\begin{document}$$\kappa = O(1)$$\end{document}, the running time reduces to \documentclass[12pt]{minimal}
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\begin{document}$$O(n \varepsilon ^{-O(1)}(\log (NW))^{O(1)})$$\end{document}.