Abstract
We show that it is NP-hard to decide the Fréchet distance between (i) non-intersecting polygons with holes embedded in the plane, (ii) 2d terrains, and (iii) self-intersecting simple polygons in 2d, which can be unfolded in 3d. The only previously known NP-hardness result for 2d surfaces was based on self-intersecting polygons with an unfolding in 4d. In contrast to this old result, our NP-hardness reductions are substantially simpler.
As a positive result we show that the Fréchet distance between polygons with one hole can be computed in polynomial time.
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Buchin, K., Buchin, M., Schulz, A. (2010). Fréchet Distance of Surfaces: Some Simple Hard Cases. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6347. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15781-3_6
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DOI: https://doi.org/10.1007/978-3-642-15781-3_6
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