A fundamental problem arising in the evolutionary molecular biology is to discover the locations of gene duplications and multiple gene duplication episodes based on the phylogenetic information. The solutions to the MULTIPLE GENE DUPLICATION problems can provide useful clues to place the gene duplication events onto the locations of a species tree and to expose the multiple gene duplication episodes. In this paper, we study two variations of the MULTIPLE GENE DUPLICATION problems: the EPISODE-CLUSTERING (EC) problem and the MINIMUM EPISODES (ME) problem. For the EC problem, we improve the results of Burleigh et al. with an optimal linear-time algorithm. For the ME problem, on the basis of the algorithm presented by Bansal and Eulenstein, we propose an optimal linear-time algorithm.
In this work, we obtain the following new results: -Given a tree T = (V, E) with a length function : E → R and a weight function w : E → R, a positive integer k, and an interval [L, U ], the Weight-Constrained k Longest Paths problem is to find the k longest paths among all paths in T with weights in the interval [L, U ]. We show that the Weight-Constrained k Longest Paths problem has a lower bound Ω(V log V + k) in the algebraic computation tree model and give an O(V log V + k)-time algorithm for it.-Given a sequence A = (a 1 , a 2 , . . . , a n ) of numbers and an interval [L, U ], we define the sum and length of a segment A[i, j] to be a i + a i+1 + · · · + a j and j − i + 1, respectively. The Length-Constrained k Maximum-Sum Segments problem is to find the k maximum-sum segments among all segments of A with lengths in the interval [L, U ]. We show that the Length-Constrained k Maximum-Sum Segments problem can be solved in O
Summary: CNVDetector is a program for locating copy number variations (CNVs) in a single genome. CNVDetector has several merits: (i) it can deal with the array comparative genomic hybridization data even if the noise is not normally distributed; (ii) it has a linear time kernel; (iii) its parameters can be easily selected; (iv) it evaluates the statistical significance for each CNV calling. Availability: CNVDetector (for Windows platform) can be downloaded from
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