I recently implemented kmeans1d—discussed in a prior post—for efficiently performing globally optimal 1D k-means clustering. The implementation utilizes the SMAWK algorithm (Aggarwal et al., 1987), which calculates
argmin(i) for each row i of an arbitrary
n × m totally monotone matrix, in
O(m(1 + lg(n/m))).
I’ve factored out my SMAWK C++ code into the example below. In general, SMAWK works with an implicitly defined matrix, utilizing a function that returns a value corresponding to an arbitrary position in the matrix. An explicitly defined matrix is used in the example for the purpose of illustration.
The program prints the column indices corresponding to the minimum element of each row in a totally monotone matrix. The matrix is from monge.pdf—a course document that I found online.
Aggarwal, Alok, Maria M. Klawe, Shlomo Moran, Peter Shor, and Robert Wilber. “Geometric Applications of a Matrix-Searching Algorithm.” Algorithmica 2, no. 1 (November 1, 1987): 195–208.