Shan-Chyun Ku

Cost-optimal parallel algorithms for the tree bisector and related problems

An edge is a bisector of a simple path if it contains the middle point of the path. Let T=(V,E) be a tree. Given a source vertex s /spl isin/ V, the single-source tree bisector problem is to find, for every vertex /spl upsi/ /spl isin/ V, a bisector of the simple path from s to /spl upsi/. The all-pairs tree bisector problem is to find for, every pair of vertices u, /spl upsi/ /spl isin/ V, a bisector of the simple path from u to /spl upsi/. In this paper, it is first shown that solving the single-source tree bisector problem of a weighted tree has a time lower bound /spl Omega/(n log n) in the sequential case. Then, efficient parallel algorithms are proposed on the EREW PRAM for the single-source and all-pairs tree bisector problems. Two O(log n) time single-source algorithms are proposed. One uses O(n) work and is for unweighted trees. The other uses O(n log n) work and is for weighted trees. Previous algorithms for the single-source problem could achieve the same time O(log n) and the same optimal work, O(n) for unweighted trees and O(n log n) for weighted trees, on the CRCW PRAM. The contribution of our single-source algorithms is the improvement from CRCW to EREW. One all-pairs parallel algorithm is proposed. It requires O(log n) time using O(n/sup 2/) work. All the proposed algorithms are cost-optimal. Efficient tree bisector algorithms have practical applications to several location problems on trees. Using the proposed algorithms, efficient parallel solutions for those problems are also presented.

Efficient Algorithms for Two Generalized 2-Median Problems on Trees

The p-median problem on a tree T is to find a set S of p vertices on T that minimize the sum of distances from Ts vertices to S. For this problem, Tamir [12] had an O(pn 2)-time algorithm, while Gavish and Sridhar [1] had an O(nlog n)-time algorithm for the case of p=2. Wang et al. [13] introduced two generalizations by imposing constraints on the 2-median: one is to limit their distance while the other is to limit their eccentricity, and they had O(n2)-time algorithms for both. We solve both generalizations in O(nlog n) time, matching even the fastest algorithm currently known for the 2-median problem. We also study cases when linear time algorithms exist for the 2-median problem and the two generalizations. For example, we solve all three in linear time when edge lengths and vertex weights are all polynomially bounded integers. Finally, we consider the relaxation of the two generalized problems by allowing 2-medians on any position of edges, instead of just on vertices, and we give O(nlog n)-time algorithms for them.

Efficient algorithms for a constrained k -tree core problem in a tree network

Let T = (V, E) be a free tree in which each vertex has a weight and each edge has a length. Let n = |V|. Given T and parameters k and l, a (k, l)-tree core is a subtree X of T with diameter ≤ l, having k leaves, which minimizes the sum of the weighted distances from all vertices in T to X. In this paper, two efficient algorithms are presented for finding a (k, l)-tree core of T. The first algorithm has O(n2) time complexity for the case that each edge has an arbitrary length. The second algorithm has O(lkn) time complexity for the case that the lengths of all edges are 1. The (k, l)-tree core problem has an application in distributed database systems.

The Conditional Location of a Median Path

In this paper, we study the problem of locating a median path of limited length on a tree under the condition that some existing facilities are already located. The existing facilities may be located at any subset of vertices. Upper and lower bounds are proposed for both the discrete and continuous models. In the discrete model, a median path is not allowed to contain partial edges. In the continuous model, a median path may contain partial edges. The proposed upper bounds for these two models are O(nlog n) and O(nlog n?(n)), respectively. They improve the previous ones from O(nlog2 n) and O(n2), respectively. The proposed lower bounds are both ?(nlog n). The lower bounds show that our upper bound for the discrete model is optimal and the margin for possible improvement on our upper bound for the continuous model is slim.

Parallel algorithms for the tree bisector problem and applications

An edge is a bisector of a simple path if it contains the middle point of the path. In this paper, efficient parallel algorithms are proposed on the EREW PRAM for the single-source and all-pairs tree bisector problems. Two O(log n) time single-source algorithms are proposed. One uses O(n) work and the other uses O(nlog n) work. The one using O(n) work is more efficient but only applicable to unweighted trees. One all-pairs parallel algorithm is proposed. It requires O(log n) time using O(n/sup 2/) work.

Efficient parallel algorithms for optimally locating a k-leaf tree in a tree network

In this paper, an efficient parallel algorithm is proposed for finding a k-tree core of a tree network. The proposed algorithm performs on the EREW PRAM in O(log n log* n) time using O(n) work.

Efficient algorithms for two generalized 2-median problems and the group median problem on trees

The p-median problem on a tree T is to find a set S of p vertices on that minimizes the sum of distances from Ts vertices to S. In this paper, we study two generalizations of the 2-median problem, which are obtained by imposing constraints on the two vertices selected as a 2-median: one is to limit their distance while the other is to limit their eccentricity. Previously, both the best upper bounds of these two generalizations were O(n2) [A. Tamir, D. Perez-Brito, J.A. Moreno-Perez, A polynomial algorithm for the p-centdian problem on a tree, Networks 32 (1998) 255-262; B.-F. Wang, S.-C. Ku, K.-H. Shi, Cost-optimal parallel algorithms for the tree bisector problem and applications, IEEE Transactions on Parallel and Distributed Systems 12 (9) (2001) 888-898]. In this paper, we solve both in O(nlogn) time. We also study cases when linear time algorithms exist for the two generalizations. For example, we solve both in linear time when edge lengths and vertex weights are all polynomially bounded integers. Furthermore, we consider the relaxation of the two generalized problems by allowing 2-medians on any position of edges, instead of just on vertices, and we give O(nlogn)-time algorithms for them. A problem, named the tree marker problem, arises several times in our approaches to the two generalized 2-median problems, and we give an O(nlogn)-time algorithm for this problem. We also use this algorithm to speedup an algorithm of Gupta and Punnen [S.K. Gupta, A.P. Punnen, Group center and group median of a tree, European Journal of Operational Research 65 (1993) 400-406] for the group median problem, improving the running time from O(kn) to O(n+klogn), where k is the number of groups in the input.

Finding the conditional location of a median path on a tree

In this paper, we study the problem of locating a median path of limited length on a tree under the condition that some existing facilities are already located. The existing facilities may be located at any subset of vertices. Upper and lower bounds are proposed for both the discrete and continuous models. In the discrete model, a median path is not allowed to contain partial edges. In the continuous model, a median path may contain partial edges. The proposed upper bounds for these two models are O(nlogn) and O(nlogn@a(n)), respectively. They improve the previous known bounds from O(nlog^2n) and O(n^2), respectively. The proposed lower bounds are both @W(nlogn).

Optimally locating a structured facility of a specified length in a weighted tree network

We propose efficient parallel algorithms on the EREW PRAM for optimally locating in a weighted tree network a tree-shaped facility of a specified length. Two optimization criteria are considered: minimum eccentricity and minimum distance sum. Let n be the number of vertices in the tree network. Both algorithms take O(log nloglog n) time using O(n) work.

An Optimal Simple Parallel Algorithm for Testing Isomorphism of Maximal Outerplanar Graphs

An outerplanar graph is a planar graph that can be imbedded in the plane in such a way that all vertices lie on the exterior face. An outerplanar graph is maximal if no edge can be added to the graph without violating the outerplanarity. In this paper, an optimal parallel algorithm is proposed on the EREW PRAM for testing isomorphism of two maximal outerplanar graphs. The proposed algorithm takes O(logn) time using O(n) work. Besides being optimal, it is very simple. Moreover, it can be implemented optimally on the CRCW PRAM in O(1) time.