Worst-case and Smoothed Analysis of the ICP Algorithm, with an Application to the k-means Method

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