Hsiao-Fei Liu

CNVDetector: locating copy number variations using array CGH data

UNLABELLED 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 http:www.csie.ntu.edu.tw/~kmchao/tools/CNVDetector/. The manual of CNVDetector is also available.

Linear-Time Algorithms for the Multiple Gene Duplication Problems

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.

A tight analysis of the Katriel–Bodlaender algorithm for online topological ordering

Katriel and Bodlaender [Irit Katriel, Hans L. Bodlaender, Online topological ordering, ACM Transactions on Algorithms 2 (3) (2006) 364-379] modify the algorithm proposed by Alpern et al. [Bowen Alpern, Roger Hoover, Barry K. Rosen, Peter F. Sweeney, F. Kenneth Zadeck, Incremental evaluation of computational circuits, in: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 1990, pp. 32-42] for maintaining the topological order of the n nodes of a directed acyclic graph while inserting m edges and prove that their algorithm runs in O(min{m^3^/^2logn,m^3^/^2+n^2logn}) time and has an @W(m^3^/^2) lower bound. In this paper, we give a tight analysis of their algorithm by showing that it runs in time @Q(m^3^/^2+mn^1^/^2logn).

MINKOWSKI SUM SELECTION AND FINDING

Let P,Q ⊆ ℝ2 be two n-point multisets and Ar ≥ b be a set of λ inequalities on x and y, where A ∈ ℝλ×2, , and b ∈ ℝλ. Define the constrained Minkowski sum(P ⊕ Q)Ar≥b as the multiset {(p + q)|p ∈ P, q ∈ Q,A(p + q) ≥ b}. Given P, Q, Ar ≥ b, an objective function f : ℝ2 → ℝ, and a positive integer k, the MINKOWSKI SUM SELECTION problem is to find the kth largest objective value among all objective values of points in (P ⊕ Q)Ar≥b. Given P, Q, Ar ≥ b, an objective function f : ℝ2 → ℝ, and a real number δ, the MINKOWSKI SUM FINDING problem is to find a point (x*, y*) in (P ⊕ Q)Ar≥b such that |f(x*,y*) - δ| is minimized. For the MINKOWSKI SUM SELECTION problem with linear objective functions, we obtain the following results: (1) optimal O(n log n)-time algorithms for λ = 1; (2) O(n log2 n)-time deterministic algorithms and expected O(n log n)-time randomized algorithms for any fixed λ > 1. For the MINKOWSKI SUM FINDING problem with linear objective functions or objective functions of the form , we construct optimal O(n log n)-time algorithms for any fixed λ ≥ 1. As a byproduct, we obtain improved algorithms for the LENGTH-CONSTRAINED SUM SELECTION problem and the DENSITY FINDING problem.

Minkowski Sum Selection and Finding

Let P,Q ⊆ ℝ2 be two n-point multisets and Ar ≥ b be a set of λ inequalities on x and y, where A e ℝ λ×2,

Algorithms for computing the length-constrained max-score segments with applications to DNA copy number data analysis

r=[^x_y]

On locating disjoint segments with maximum sum of densities

, and b e ℝλ. Define the constrained Minkowski sum (P ⊕ Q)Ar ≥ b as the multiset {(p + q) | p e P,q e Q,A(p + q) ≥ b}. Given P, Q, Ar ≥ b, an objective function f:ℝ2→ℝ, and a positive integer k, the Minkowski Sum Selection problem is to find the k th largest objective value among all objective values of points in (P ⊕ Q)Ar ≥ b. Given P, Q, Ar ≥ b, an objective function f:ℝ2→ℝ, and a real number δ, the Minkowski Sum Finding problem is to find a point (x *,y *) in (P ⊕ Q)Ar ≥ b such that |f(x *,y *) - δ| is minimized. For the Minkowski Sum Selection problem with linear objective functions, we obtain the following results: (1) optimal O(nlogn) time algorithms for λ= 1; (2) O(nlog2 n) time deterministic algorithms and expected O(nlogn) time randomized algorithms for any fixed λ> 1. For the Minkowski Sum Finding problem with linear objective functions or objective functions of the form

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

f(x,y)=\frac{by}{ax}

Optimal algorithms for the average-constrained maximum-sum segment problem

, we construct optimal O(nlogn) time algorithms for any fixed λ ≥ 1. As a byproduct, we obtain improved algorithms for the Length-Constrained Sum Selection problem and the Density Finding problem.

An optimal algorithm for querying tree structures and its applications in bioinformatics

Given a sequence of n real numbers A = (a1, a2,..., an), two integers L and U with 1 ≤ L ≤ U ≤ n, and a score function f : IR+ × IR → IR, the LENGTH-CONSTRAINED MAX-SCORE SEGMENT PROBLEM is to find a segment A[i, j] = (ai, ai+1,..., aj) maximizing f(j - i + 1, Σh=ij ah) subject to j - i + 1 ∈ [L, U]. In this paper, we solve the LENGTH-CONSTRAINED MAX-SCORE SEGMENT PROBLEM for the case where the given score function f(l, w) = w/r√l for any constant r > 1. Our algorithm runs in O(n T(L1/2)/L1/2) time, where T(n′) is the time required to solve the all-pairs shortest paths problem on a graph of n′ nodes. By the latest result of Chan [7], T(n′) = O(n′3 (log log n′)3/(log n′)2), so our algorithm runs in subquadratic time O(nL (log log L)3/(log L)2). Lipson et al. [21] studied a more restricted case where the score function f(l,w) = w/2√l and there are no length constraints, i.e., L = 1 and U = n. They also showed how to apply their algorithm to analyzing DNA copy number data. However, their algorithm takes Ω(n2) time in the worst situation. Since the length lower bound L for the case considered by Lipson et al. is a constant, our algorithm solves it in O(n) time.