Many applications of data partitioning (clustering) have been well studied in bioinformatics. Consider, for instance, a set N of organisms (elements) based on DNA marker data. A partition divides all elements in N into two or more disjoint clusters that cover all elements, where a cluster contains a non-empty subset of N. Different partitioning algorithms may produce different partitions. To compute the distance and find the consensus partition (also called consensus clustering) between two or more partitions are important and interesting problems that arise frequently in bioinformatics and data mining, in which different distance functions may be considered in different partition algorithms. In this article, we discuss the k partition-distance problem. Given a set of elements N with k partitions of N, the k partition-distance problem is to delete the minimum number of elements from each partition such that all remaining partitions become identical. This problem is NP-complete for general k > 2 partitions, and no algorithms are known at present. We design the first known heuristic and approximation algorithms with performance ratios 2 to solve the k partition-distance problem in O(k · ρ · |N|) time, where ρ is the maximum number of clusters of these k partitions and |N| is the number of elements in N. We also present the first known exact algorithm in O(ℓ · 2(ℓ)·k(2) · |N|(2)) time, where ℓ is the partition-distance of the optimal solution for this problem. Performances of our exact and approximation algorithms in testing the random data with actual sets of organisms based on DNA markers are compared and discussed. Experimental results reveal that our algorithms can improve the computational speed of the exact algorithm for the two partition-distance problem in practice if the maximum number of elements per cluster is less than ρ. From both theoretical and computational points of view, our solutions are at most twice the partition-distance of the optimal solution. A website offering the interactive service of solving the k partition-distance problem using our and previous algorithms is available (see http://mail.tmue.edu.tw/~yhchen/KPDP.html).
關聯:
Journal of Computational Biology，Vol. 19(4)，P404-417
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