Algorithms for finding the weight-constrained k longest paths in a tree and the length-constrained k maximum-sum segments of a sequence

Authors:
Hsiao-Fei Liu;Kun-Mao Chao
Affiliations:
Department of Computer Science and Information Engineering, National Taiwan University, Taipei 106, Taiwan;Department of Computer Science and Information Engineering, National Taiwan University, Taipei 106, Taiwan and Graduate Institute of Biomedical Electronics and Bioinformatics, National Taiwan Univ ...
Venue:
Theoretical Computer Science
Year:
2008

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Abstract

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-ConstrainedkLongest 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-ConstrainedkLongest Paths problem has a lower bound @W(VlogV+k) in the algebraic computation tree model and give an O(VlogV+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-ConstrainedkMaximum-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-ConstrainedkMaximum-Sum Segments problem can be solved in O(n+k) time.