Fast Algorithms for the Density Finding Problem

Abstract

We study the problem of finding a specific density subsequence of a sequence arising from the analysis of biomolecular sequences. Given a sequence A=(a 1,w 1),(a 2,w 2),…,(a n ,w n ) of n ordered pairs (a i ,w i ) of numbers a i and width w i >0 for each 1≤i≤n, two nonnegative numbers ℓ, u with ℓ≤u and a number δ, the Density Finding Problem is to find the consecutive subsequence A(i *,j *) over all O(n 2) consecutive subsequences A(i,j) with width constraint satisfying ℓ≤w(i,j)=∑ r=i j w r ≤u such that its density $d(i^{*},j^{*})=\sum_{r=i^{*}}^{j*}a_{r}/w(i^{*},j^{*})$is closest to δ. The extensively studied Maximum-Density Segment Problem is a special case of the Density Finding Problem with δ=∞. We show that the Density Finding Problem has a lower bound Ω(nlog n) in the algebraic decision tree model of computation. We give an algorithm for the Density Finding Problem that runs in optimal O(nlog n) time and O(nlog n) space for the case when there is no upper bound on the width of the sequence, i.e., u=w(1,n). For the general case, we give an algorithm that runs in O(nlog 2 m) time and O(n+mlog m) space, where $m=\min\{\lfloor\frac{u-\ell}{w_{\mathrm{min}}}\rfloor,n\}$and w min=min r=1 n w r . As a byproduct, we give another O(n) time and space algorithm for the Maximum-Density Segment Problem

The density maximization problem in graphs

“…A linear time algorithm for the non-uniform case is given by Chung and Lu (2005). Lee et al (2009) show how to select a subsequence whose density is closest to a given density δ in O(n log 2 n) time. Without the upper bound on the length B they present an optimal O(n log n)-time algorithm.…”

Minkowski Sum Selection and Finding

“…The goal is to find a point (x * , y * ) in (P ⊕ Q) Ar≥b such that |f (x * , y * ) − δ| is minimized. This problem originates from the study of the Density Finding problem proposed by Lee et al [12]. The Density Finding problem can be regarded as a specialization of the Minkowski Sum Finding problem with objective function f (x, y) = y x and find applications in recognizing promoters in DNA sequences [11,20].…”

“…-The Minkowski Sum Finding problem with any fixed number of constraints can be solved in optimal O(n log n) time if the objective function f (x, y) is linear or of the form by ax . As a byproduct, we obtain improved algorithms for the Length-Constrained Sum Selection problem [14] and the Density Finding problem [12]. Recently, Lin and Lee [14] proposed an expected O(n log(u − l + 1))-time randomized algorithm for the Length-Constrained Sum Selection problem, where n is the size of the input instance and l, u ∈ N are two given parameters with 1 ≤ l < u ≤ n. In this paper, we obtain a worst-case O(n log(u − l + 1))-time deterministic algorithm for the Length-Constrained Sum Selection problem (see Appendix A).…”

“…Recently, Lin and Lee [14] proposed an expected O(n log(u − l + 1))-time randomized algorithm for the Length-Constrained Sum Selection problem, where n is the size of the input instance and l, u ∈ N are two given parameters with 1 ≤ l < u ≤ n. In this paper, we obtain a worst-case O(n log(u − l + 1))-time deterministic algorithm for the Length-Constrained Sum Selection problem (see Appendix A). Lee, Lin, and Lu [12] showed the Density Finding problem has a lower bound of Ω(n log n) and proposed an O(n log 2 m)-time algorithm for it, where n is the size of the input instance and m is a parameter whose value may be as large as n.…”

The Density Maximization Problem in Graphs

“…A linear time algorithm for the non-uniform case is given by Chung and Lu [4]. Lee et al [12] show how to select a subsequence whose density is closest to a given density δ in O(n log 2 n) time. Without the upper bound on the length B an optimal O(n log n)-time algorithm is given.…”