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• W2974196491 abstract "We generalise the results of Bhattacharya et al. (Journal of Computing Systems, 62(1):93-115, 2018) for the list-$k$-means problem defined as -- for a (unknown) partition $X_1, ..., X_k$ of the dataset $X subseteq mathbb{R}^d$, find a list of $k$-center sets (each element in the list is a set of $k$ centers) such that at least one of $k$-center sets ${c_1, ..., c_k}$ in the list gives an $(1+varepsilon)$-approximation with respect to the cost function $min_{textrm{permutation } pi} left[ sum_{i=1}^{k} sum_{x in X_i} ||x - c_{pi(i)}||^2 right]$. The list-$k$-means problem is important for the constrained $k$-means problem since algorithms for the former can be converted to PTAS for various versions of the latter. Following are the consequences of our generalisations: - Streaming algorithm: Our $D^2$-sampling based algorithm running in a single iteration allows us to design a 2-pass, logspace streaming algorithm for the list-$k$-means problem. This can be converted to a 4-pass, logspace streaming PTAS for various constrained versions of the $k$-means problem. - Faster PTAS under stability: Our generalisation is also useful in $k$-means clustering scenarios where finding good centers becomes easy once good centers for a few clusters have been chosen. One such scenario is clustering under stability where the number of such bad clusters is a constant. Using the above idea, we significantly improve the running time of the known algorithm from $O(dn^3) (k log{n})^{poly(frac{1}{beta}, frac{1}{varepsilon})}$ to $O left(dn^3 k^{tilde{O}_{beta varepsilon}(frac{1}{beta varepsilon})} right)$." @default.
• W2974196491 created "2019-09-26" @default.
• W2974196491 creator A5029407896 @default.
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• W2974196491 date "2019-09-16" @default.
• W2974196491 modified "2023-09-27" @default.
• W2974196491 title "Streaming PTAS for Constrained k-Means." @default.
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