Algorithms for the Problems of Length-Constrained Heaviest Segments

We present algorithms for length-constrained maximum sum segment and maximum density segment problems, in particular, and the problem of finding length-constrained heaviest segments, in general, for a sequence of real numbers. Given a sequence of n real numbers and two real parameters L and U (L 6 U), the maximum sum segment problem is to find a consecutive subsequence, called a segment, of length at least L and at most U such that the sum of the numbers in the subsequence is maximum. The maximum density segment problem is to find a segment of length at least L and at most U such that the density of the numbers in the subsequence is the maximum. For the first problem with nonuniform width there is an algorithm with time and space complexities in O(n). We present an algorithm with time complexity in O(n) and space complexity in O(U). For the second problem with non-uniform width there is a combinatorial solution with time complexity in O(n) and space complexity in O(U). We present a simple geometric algorithm with the same time and space complexities. We extend our algorithms to respectively solve the length-constrained k maximum sum segments problem in O(n + k) time and O(max{U, k}) space, and the length-constrained k maximum density segments problem in O(nmin{k, U − L}) time and O(U + k) space. We present extensions of our algorithms to find all the length-constrained segments having user specified sum and density in O(n + m) and O(n log(U − L) + m) times respectively, where m is the number of output. Previously, there was no known algorithm with non-trivial result for these problems. We indicate the extensions of our algorithms to higher dimensions. All the algorithms can be extended in a straight forward way to solve the problems with non-uniform width and non-uniform weight. The algorithms have potential applications in different areas of biomolecular sequence analysis including finding CG-rich regions, TA and CGdeficient regions, CpG islands and regions rich in periodical three-base patterns, post processing sequence alignment, annotating multiple sequence alignments, and computing length constrained ungapped local alignment. They also have applications in other areas such as pattern recognition, digital image processing and data mining. ? Research supported by an NSERC discovery grant awarded to this author. ar X iv :1 10 8. 49 72 v1 [ cs .C G ] 2 5 A ug 2 01 1 2 Md. Shafiul Alam and Asish Mukhopadhyay