Longest biased interval and longest non-negative sum interval

Abstract

Examples of applications to Plasmodium falciparum genomic DNA can be found at the above URL.

Constrained Minkowski Sums: A Geometric Framework for Solving Interval Problems in Computational Biology Efficiently

“…The constraint x 1 1 in the Minkowski sum can be omitted, since we are maximizing ℓ: When the array contains at least one preferred character (the other case is trivial), then the maximum ℓ is at least 1 and it does not matter that we are allowing intervals of zero or negative length. Our (worst-case) running time for this problem thus is O(n), improving upon the (worst-case) trivial running time O(n 2 ) in [2].…”

“…Among all intervals [i, j] with weight L w i + · · · + w j U , find one with the largest density (a i + · · · + a j )/(w i + · · · + w j ). c) The longest biased interval [2]: Given a bias 0 b 1, find an interval [i, j] which has an average (a i + · · · + a j )/( j − i + 1) b and which is as long as possible. Allison [2] uses this problem in the context of "preferred characters", where we additionally have that a i ∈ {0, 1}, as one can use a i as an indicator for whether a character in the array is "preferred" or not.…”

“…c) The longest biased interval [2]: Given a bias 0 b 1, find an interval [i, j] which has an average (a i + · · · + a j )/( j − i + 1) b and which is as long as possible. Allison [2] uses this problem in the context of "preferred characters", where we additionally have that a i ∈ {0, 1}, as one can use a i as an indicator for whether a character in the array is "preferred" or not. d) The length-constrained heaviest segments [16]: Given a bound L, find an interval [i, j] with length at least L which has maximum average (a i + · · · + a j )/( j − i + 1).…”

On the Range Maximum-Sum Segment Query Problem

“…There are many kinds of variants of the maximum-sum segment problem that impose extra constraints on the input or on the output [2,6,7,8,12,13,14,15,18]. For example, Ruzzo and Tompa [15] studied the problem of finding all maximal-sum segments.…”

On the Lower Bound for the Maximum Consecutive Sub-Sums Problem

“…The problem was introduced by Grenader (12) as a special one-dimensional version of its twodimensional counterpart, the Maximum Sum Subarray Problem. It finds applications in pattern matching (37,58), biological sequence analysis (1), and data mining (33). The Maximum Sum Subsequence Problem can be solved in linear time using Kadane's algorithm (12), and sometimes it's also called the Maximum Sum Segment.…”