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The $k$-segmentation of a video stream is used to partition it into $k$ piecewise-linear segments, so that each linear segment has a meaningful interpretation. Such segmentation may be used to summarize large videos using a small set of images, to identify anomalies within segments and change points between segments, and to select critical subsets for training machine learning models. Exact and approximate segmentation methods for $k$-segmentation exist in the literature. Each of these algorithms occupies a different spot in the trade-off between computational complexity and accuracy. A novel heuristic algorithm is proposed in this paper to improve upon existing methods. It is empirically found to provide accuracies competitive with exact methods at a fraction of the computational expense. The new algorithm is inspired by Lloyd's algorithm for K-Means and Lloyd-Max algorithm for scalar quantization, and is called the LM algorithm for convenience. It works by iteratively minimizing a cost function from any given initialisation; the commonly used $L_2$ cost is chosen in this paper. While the greedy minimization makes the algorithm sensitive to initialisation, the ability to converge from any initial guess to a local optimum allows the algorithm to be integrated into other existing algorithms. Three variants of the algorithm are tested over a large number of synthetic datasets, one being a standalone LM implementation, and two others that combine with existing algorithms. One of the latter two -- LM-enhanced-Bottom-Up segmentation -- is found to have the best accuracy and the lowest computational complexity among all algorithms. This variant of LM can provide $k$-segmentations over data sets with up to a million image frames within several seconds.