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k-Means and k-Medoids both take an initial random sample from the data as starting centroids for the clustering. This random selection is guided by a random number generator and this number generator again delivers a sequence of random numbers depending on the random seed. On the one hand, this ensures that you will get a sequence of random numbers, on the other hand for the same random seed number, you will always get exactly the same random numbers. Hence, your experiments can be reproduced as long as the same random number seeds are used.
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