k-means++: the advantages of careful seeding
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Improved smoothed analysis of the k-means method
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Approximate clustering without the approximation
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
The Planar k-Means Problem is NP-Hard
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
Smoothed analysis of probabilistic roadmaps
Computational Geometry: Theory and Applications
k-means requires exponentially many iterations even in the plane
Proceedings of the twenty-fifth annual symposium on Computational geometry
Smoothed analysis: an attempt to explain the behavior of algorithms in practice
Communications of the ACM - A View of Parallel Computing
Smoothed Analysis of Balancing Networks
ICALP '09 Proceedings of the 36th Internatilonal Collogquium on Automata, Languages and Programming: Part II
Automatic color based reassembly of fragmented images and paintings
IEEE Transactions on Image Processing
The planar k-means problem is NP-hard
Theoretical Computer Science
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We show a worst-case lower bound and a smoothed upper bound on the number of iterations performed by the Iterative Closest Point (ICP) algorithm. First proposed by Besl and McKay, the algorithm is widely used in computational geometry where it is known for its simplicity and its observed speed. The theoretical study of ICP was initiated by Ezra, Sharir and Efrat, who bounded its worst-case running time between \Omega(n log n) and O(n^{2}d)^{d}. We substantially tighten this gap by improving the lower bound to \Omega(n/d)^{d+1}. To help reconcile this bound with the algorithm's observed speed, we also show the smoothed complexity of ICP is polynomial, independent of the dimensionality of the data. Using similar methods, we improve the best known smoothed upper bound for the popular k-means method to n^{O(k)}, once again independent of the dimension.