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A Python package for optimal 1D k-means clustering

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A Python library with an implementation of k-means clustering on 1D data, based on the algorithm in (Xiaolin 1991), as presented in section 2.2 of (Gronlund et al., 2017).

Globally optimal k-means clustering is NP-hard for multi-dimensional data. Lloyd's algorithm is a popular approach for finding a locally optimal solution. For 1-dimensional data, there are polynomial time algorithms. The algorithm implemented here is a O(kn + n log n) dynamic programming algorithm for finding the globally optimal k clusters for n 1D data points.

The code is written in C++, and wrapped with Python.


kmeans1d supports Python 3.x.


kmeans1d is available on PyPI, the Python Package Index.

$ pip3 install kmeans1d

Example Usage

import kmeans1d

x = [4.0, 4.1, 4.2, -50, 200.2, 200.4, 200.9, 80, 100, 102]
k = 4

clusters, centroids = kmeans1d.cluster(x, k)

print(clusters)   # [1, 1, 1, 0, 3, 3, 3, 2, 2, 2]
print(centroids)  # [-50.0, 4.1, 94.0, 200.5]


Tests are in tests/.

# Run tests
$ python3 -m unittest discover tests -v


The code in this repository has an MIT License.



[1] Wu, Xiaolin. "Optimal Quantization by Matrix Searching." Journal of Algorithms 12, no. 4 (December 1, 1991): 663

[2] Gronlund, Allan, Kasper Green Larsen, Alexander Mathiasen, Jesper Sindahl Nielsen, Stefan Schneider, and Mingzhou Song. "Fast Exact K-Means, k-Medians and Bregman Divergence Clustering in 1D." ArXiv:1701.07204 [Cs], January 25, 2017.

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