Hung-I Yu

Improved algorithms for the minmax-regret 1-center and 1-median problems

In this article, efficient algorithms are presented for the minmax-regret 1-center and 1-median problems on a general graph and a tree with uncertain vertex weights. For the minmax-regret 1-center problem on a general graph, we improve the previous upper bound from <i>O</i>(<i>mn</i>² log <i>n</i>) to <i>O</i>(<i>mn</i> log <i>n</i>). For the problem on a tree, we improve the upper bound from <i>O</i>(<i>n</i>²) to <i>O</i>(<i>n</i> log² <i>n</i>). For the minmax-regret 1-median problem on a general graph, we improve the upper bound from <i>O</i>(<i>mn</i>² log <i>n</i>) to <i>O</i>(<i>mn</i>² + <i>n</i>³ log <i>n</i>). For the problem on a tree, we improve the upper bound from <i>O</i>(<i>n</i> log² <i>n</i>) to <i>O</i>(<i>n</i> log <i>n</i>).

Improved algorithms for the minmax-regret 1-center problem

This paper studies the problem of finding the 1-center on a graph where vertex weights are uncertain and the uncertainty is characterized by given intervals. It is required to find a minmax-regret solution, which minimizes the worst-case loss in the objective function. Averbakh and Berman had an O(mn2log n)-time algorithm for the problem on a general graph. On a tree, the time complexity of their algorithm becomes O(n2). In this paper, we improve these two bounds to O(mnlog n) and O(nlog2n), respectively.

The broadcast median problem in heterogeneous postal model

We propose the problem of finding broadcast medians in heterogeneous networks. A heterogeneous network is represented by a graph G=(V,E), in which each edge has a weight that denotes the communication time between its two end vertices. The overall delay of a vertex v∈V(G), denoted as b(v,G), is the minimum sum of the communication time required to send a message from v to all vertices in G. The broadcast median problem consists of finding the set of vertices v∈V(G) with minimum overall delay b(v,G) and determining the value of b(v,G). In this paper, we consider the broadcast median problem following the heterogeneous postal model. Assuming that the underlying graph G is a general graph, we show that computing b(v,G) for an arbitrary vertex v∈V(G) is NP-hard. On the other hand, assuming that G is a tree, we propose a linear time algorithm for the broadcast median problem in heterogeneous postal model.

Improved algorithms for the minmax regret 1-median problem

This paper studies the problem of finding the 1-median on a graph where vertex weights are uncertain and the uncertainty is characterized by given intervals. It is required to find a minmax regret solution, which minimizes the worst-case loss in the objective function. Averbakh and Berman had an O(mn2log n)-time algorithm for the problem on a general graph, and had an O(nlog2n)-time algorithm on a tree. In this paper, we improve these two bounds to O(mn2 + n3log n) and O(nlog n), respectively.

The multi-service center problem☆

Abstract We propose a new type of network location problem for placing multiple but distinct facilities, called the p-service center problem. In this problem, we are to locate p facilities in the graph, each of which provides distinct service required by all vertices. For each vertex, its p-service distance is the summation of its weighted distances to the p facilities. The objective is to minimize the maximum value among the p-service distances of all vertices. In this paper, we show that the p-service center problem on a general graph is NP-hard, and propose a simple approximation algorithm of factor p / c for any integer constant c. Moreover, we study the basic case p = 2 on paths and trees. When the underlying network is a path, we solve the 2-service center problem in O ( n ) time, where n is the number of vertices. When the underlying network is a tree, we give an O ( n 3 ) -time algorithm for the case of nonnegative weights, an O ( n log ⁡ n ) -time algorithm for the case of positive weights, and an O ( n ) -time algorithm for the case of uniform weights.

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

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.

The Multi-Service Center Problem

We propose a new type of multiple facilities location problem, called the p-service center problem. In this problem, we are to locate p facilities in the graph, each of which provides distinct service required by all vertices. For each vertex, its p-service distance is the summation of its weighted distances to the p facilities. The objective is to minimize the maximum value among the p-service distances of all vertices.

Finding maximum sum segments in sequences with uncertainty

In this paper, we propose to study the famous maximum sum segment problem on a sequence consisting of n uncertain numbers, where each number is given as an interval characterizing its possible value. Given two integers L and U , a segment of length between L and U is called a potential maximum sum segment if there exists a possible assignment of the uncertain numbers such that, under the assignment, the segment has maximum sum over all segments of length between L and U . We define the maximum sum segment with uncertainty problem, which consists of two sub-problems: (1) reporting all potential maximum sum segments; (2) counting the total number of those segments. For L =1 and U =n , we propose an O (n +K )-time algorithm and an O (n )-time algorithm, respectively, where K is the number of potential maximum sum segments. For general L and U , we give an O (n (U −L ))-time algorithm for either problem.

A faster query algorithm for the text fingerprinting problem

Let S be a string over a finite, ordered alphabet Σ. For any substring S′ of S, the set of distinct characters contained in S′ is called its fingerprint. The text fingerprinting problem consists of constructing a data structure for the string S in advance, so that on given any input set C ⊆ Σ of characters, we can answer the following queries efficiently: (1) determine if C represents a fingerprint of some substrings in S; (2) find all maximal substrings of S whose fingerprint is equal to C. The best results known so far solved these two queries in Θ(|Σ|) and Θ(|Σ|+K) time, respectively, where K is the number of maximal substrings. In this paper, we propose a new data structure that improves the time complexities of the two queries to O(|C| log(|Σ|/|C|)) and O(|C| log(|Σ|/|C|) + K) time, respectively, where the term |C| log(|Σ|/|C|) is always bounded by Θ(|Σ|). This result answers the open problem proposed by Amir et al. [A. Amir, A. Apostolico, G.M. Landau, G. Satta, Efficient text fingerprinting via Parikh mapping, J. Discrete Algorithms 1 (2003) 409- 421]. In addition, our data structure uses less storage than the existing solutions.