Tien-Ching Lin

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=(a1,w1),(a2,w2),…,(an,wn) of n ordered pairs (ai,wi) of numbers ai and width wi>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(n2) consecutive subsequences A(i,j) with width constraint satisfying ℓ≤w(i,j)=∑r=ijwr≤u such that its density

Efficient algorithms for the sum selection problem and k maximum sums problem

d(i^{*},j^{*})=\sum_{r=i^{*}}^{j*}a_{r}/w(i^{*},j^{*})

Geometric Minimum Diameter Minimum Cost Spanning Tree Problem

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 2m) time and O(n+mlog m) space, where

The (1|1)-Centroid Problem on the Plane Concerning Distance Constraints

m=\min\{\lfloor\frac{u-\ell}{w_{\mathrm{min}}}\rfloor,n\}

Finding maximum sum segments in sequences with uncertainty

and wmin=min r=1nwr. As a byproduct, we give another O(n) time and space algorithm for the Maximum-Density Segment Problem.

Spanning Ratio and Maximum Detour of Rectilinear Paths in the L1 Plane

Given a sequence of n real numbers A = a1, a2,..., an and a positive integer k, the Sum Selection Problem is to find the segment A(i,j) = ai , ai+1,..., aj such that the rank of the sum s(i, j) = ∑t=ijat is k over all

Randomized algorithm for the sum selection problem

\frac{n(n-1)}{2}

The (1|1)-Centroid Problem in the Plane with Distance Constraints

segments. We present a deterministic algorithm for this problem that runs in O(n logn) time. The previously best known randomized algorithm for this problem runs in expected O(n logn) time. Applying this algorithm we can obtain a deterministic algorithm for the k Maximum Sums Problem, i.e., the problem of enumerating the k largest sum segments, that runs in O(n logn + k) time. The previously best known randomized and deterministic algorithms for the k Maximum Sums Problem run respectively in expected O(n logn + k) and O(n log2n + k) time in the worst case.

Spanning Ratio and Maximum Detour of Rectilinear Paths in the

In this paper we consider bi-criteria geometric optimization problems, in particular, the minimum diameter minimum cost spanning tree problem and the minimum radius minimum cost spanning tree problem for a set of points in the plane. The former problem is to construct a minimum diameter spanning tree among all possible minimum cost spanning trees, while the latter is to construct a minimum radius spanning tree among all possible minimum cost spanning trees. The graph-theoretic minimum diameter minimum cost spanning tree (MDMCST) problem and the minimum radius minimum cost spanning tree (MRMCST) problem have been shown to be NP-hard. We will show that the geometric version of these two problems, GMDMCST problem and GMRMCST problem are also NP-hard. We also give two heuristic algorithms, one MCST-based and the other MDST-based for the GMDMCST problem and present some experimental results.

Efficient Algorithms for the Sum Selection Problem and K Maximum Sums Problem

In 1982, Drezner proposed the (1|1)-centroid problem on the plane, in which two players, called the leader and the follower, open facilities to provide service to customers in a competitive manner. The leader opens the first facility, and then the follower opens the second. Each customer will patronize the facility closest to him (ties broken in favor of the leaders one), thereby decides the market share of the two players. The goal is to find the best position for the leader’s facility so that his market share is maximized. The best algorithm for this problem is an O(n^2 log n)-time parametric search approach, which searches over the space of possible market share values. In the same paper, Drezner also proposed a general version of (1|1)-centroid problem by introducing a minimal distance constraint R, such that the followers facility is not allowed to be located within a distance R from the leaders. He proposed an O(n^5 log n)-time algorithm for this general version by identifying O(n^4) points as the candidates of the optimal solution and checking the market share for each of them. In this paper, we develop a new parametric search approach searching over the O(n^4) candidate points, and present an O(n^2 log n)-time algorithm for the general version, thereby closing the O(n^3) gap between the two bounds.