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**1 - 3**of**3**### Homogeneous String Segmentation using Trees and Weighted Independent Sets

, 2007

"... We divide a string into k segments, each with only one sort of symbols, so as to minimize the total number of exceptions. Motivations come from machine learning and data mining. For binary strings we develop a linear-time algorithm for any k. Key to efficiency is a special-purpose data structure, ca ..."

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We divide a string into k segments, each with only one sort of symbols, so as to minimize the total number of exceptions. Motivations come from machine learning and data mining. For binary strings we develop a linear-time algorithm for any k. Key to efficiency is a special-purpose data structure, called W-tree, which reflects relations between repetition lengths of symbols. For non-binary strings we give a nontrivial dynamic programming algorithm. Our problem is equivalent to finding weighted independent sets with certain size constraints, either in paths (binary case) or special interval graphs (general case). We also show that this problem is FPT in bounded-degree graphs.

### Computing Maximum-Scoring Segments Optimally

"... Given a sequence of length n, the problem studied in this paper is to find a set of k disjoint subsequences of consecutive elements such that the total sum of all elements in the set is maximized. This problem arises in the analysis of DNA sequences. The previous best known algorithm requires Θ(nα(n ..."

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Given a sequence of length n, the problem studied in this paper is to find a set of k disjoint subsequences of consecutive elements such that the total sum of all elements in the set is maximized. This problem arises in the analysis of DNA sequences. The previous best known algorithm requires Θ(nα(n, n)) time in the worst case, where α(n, n) is the inverse Ackermann function. We present a linear-time algorithm, which is optimal, for this problem. 1