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The complexity of the m-means problem and of the k-means problem are analyzed in terms of the complexity of the corresponding linear programs. A linear program, LP, is an optimization problem whose goal is to maximize a linear objective function subject to linear constraints. The complexity of a linear program is typically measured in terms of the size of the smallest set of variables known as a basis of the solution. The basis can be computed in polynomial time by a simple reduction to the assignment problem, or in linear time by the ellipsoid method. The assignment problem is the problem of finding a set of distinct objects to which each object belongs, while minimizing the number of ties. The size of an assignment problem is the number of ties. In the case of the k-means problem, an LP with n variables and m constraints is formulated, and the basis is computed in polynomial time. By contrast, the number of variables in the problem is exponential in the number of clusters, and the number of constraints is also exponential in m. The basis can be computed in exponential time. In this paper, we describe algorithms for approximate k-means clustering that run in polynomial time. The time is linear in the number of data objects, n, and the number of clusters, m. This is O(n^2 log n) and O(n^2) since the number of constraints in the case of deterministic hill climbing and the linear programming relaxation, respectively.

So, let’s take a look at an interesting paper, that was some kind of a “roadmap” for the authors from wikipedia: O(n^2 log n) Algorithms for k-means clustering. A complexity analysis of the classical and deterministic hill climbing algorithm, which aims at finding the global optimum. By Mihajlos, W. R., and J. R. Quinlan. Applied Optics, vol. 28, pp. 4449–4452, 1989.

3. Pandian K, Schmid W, Nicklin M, et al. Global and regional mortality from 302 causes of death for 195 countries in 1990 and 2016: a systematic analysis for the Global Burden of Diseases, Injuries, and Risk Factors Study 2016. Lancet. 2018;391:1704-1712.

Using the two examples above:

Given a fixed number of iterations of the standard algorithm in n-dimensional space, the running time is O(t*k*n*d).

Using a fixed number of restarts, the running time is still O(t*k*n*d) but the complexity improvement is 0(d*n).

Using the best of the two implementations (a fixed number of restarts), the running time is now O(t*k*n*d) + O(d*n), which is O(t*k*n*d) + 0(d*n). The sublinear term of the big-O tells us that the time complexity is in fact

Now, we have to pass through the object only once to find the centroids. If we find the centroids of the object with in parallel, we can generate the centroid (or centroids) with O(k) operations.

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