SIAM Journal on Computing


Approximate Shortest Descending Paths

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Article Data

History

Submitted: 20  March  2013
Accepted: 22 November 2013
Published online: 04 March 2014

Publication Data

ISSN (print): 0097-5397
ISSN (online): 1095-7111
CODEN: smjcat

We present an approximation algorithm for the shortest descending path problem. Given a source $s$ and a destination $t$ on a terrain, a shortest descending path from $s$ to $t$ is a path of minimum Euclidean length on the terrain subject to the constraint that the height decreases monotonically as we traverse that path from $s$ to $t$. Given any $\varepsilon \in (0,1)$, our algorithm returns in $O(n^4\log (n/\varepsilon))$ time a descending path of length at most $1+\varepsilon$ times the optimum. This is the first algorithm whose running time is polynomial in $n$ and $\log(1/\varepsilon)$ and independent of the terrain geometry.

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