Computational geometry: an introduction
Computational geometry: an introduction
The algebraic degree of geometric optimization problems
Discrete & Computational Geometry
The complexity of the free space for a robot moving amidst fat obstacles
Computational Geometry: Theory and Applications
Exact algorithms for partial curve matching via the Fréchet distance
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Better bounds on the union complexity of locally fat objects
Proceedings of the twenty-sixth annual symposium on Computational geometry
Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces
Discrete & Computational Geometry
Median trajectories using well-visited regions and shortest paths
Proceedings of the 19th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
Jaywalking your dog: computing the Fréchet distance with shortcuts
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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The Frechet distance is a well studied and commonly used measure to capture the similarity of polygonal curves. Unfortunately, it exhibits a high sensitivity to the presence of outliers. Since the presence of outliers is a frequently occurring phenomenon in practice, a robust variant of Frechet distance is required which absorbs outliers. We study such a variant here. In this modified variant, our objective is to minimize the length of subcurves of two polygonal curves that need to be ignored (MinEx problem), or alternately, maximize the length of subcurves that are preserved (MaxIn problem), to achieve a given Frechet distance. An exact solution to one problem would imply an exact solution to the other problem. However, we show that these problems are not solvable by radicals over Q and that the degree of the polynomial equations involved is unbounded in general. This motivates the search for approximate solutions. We present an algorithm which approximates, for a given input parameter @d, optimal solutions for the MinEx and MaxIn problems up to an additive approximation error @d times the length of the input curves. The resulting running time is O(n^3@dlog(n@d)), where n is the complexity of the input polygonal curves.