Network Location Problem一直是近年來大家所熱衷研究的題目之一。這個問題有相當廣泛的應用。例如在網際網路上的多媒體伺服器和檔案伺服器等，我們要如何讓它達到最佳化的服務品質（quality of service, QoS）。在這一類問題的品質評估上，一般是以各節點到最近伺服器的距離總和及各節點要到最近伺服器之最遠距離為衡量標準。這類問題所研究的範圍最常見的就是如何找出最佳化的點（例如伺服器)或路徑（例如一條高速公路)。這問題為因應不同的服務類型或特殊狀況而有許多的不同的變型，在論文中我們將有比較詳細的討論。
絕大多數這類問題在一般化網路（general network）上是NP-Complete的問題。因此一般上大家主要的研究對象為在樹狀圖（tree）上的應用。除了因為樹狀圖是一個比較簡單的特例外，樹狀圖也是大家所公認最重要的圖學結構之一。

單源（single-source）和成對（all-pairs）節點之二分邊（tree bisector）問題在很多樹狀圖上的Location Problem演算法中扮演著相當重要的角色，甚至是關鍵的一個步驟。本論文就是找尋樹狀圖上所有成對節點之二分邊的平行演算法及舉出數個這個演算法在樹狀圖上的Location Problem上的應用。這個演算法把問題反過來思考，即為每一個樹狀圖上的邊找出以它為二分邊的兩個節。此演算法可達到最佳化的計算成本（cost-optimal）；成本（cost）=時間（time）□處理器個數（number of processors）。在應用上，此演算法可以很簡單的解決數個現存的樹狀圖上的Location Problem，其中部份更可以達至計算成本的最佳化。

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. The all-pairs tree bisector problem is to find for every pair of vertices u, v in V a bisector of the simple path from u to v. In this thesis, the all-pairs tree bisector problem is considered. We first present a simple sequential algorithm to solve the all-pairs tree bisector problem on weighted trees. The presented algorithm requires O(n^2) time. Then, we propose a cost-optimal parallel algorithm on the EREW PRAM. It requires O(log n) time using O(n^2) work. The problem of optimally locating a service facility in a network has been of considerable interest for many years. Besides being of theoretical interest, efficient algorithms for the all-pairs tree bisector problem have practical applications to several facility location problems on trees. In this thesis, using the proposed all-pairs algorithm, efficient parallel solutions on the EREW PRAM for those problems are also presented.

[1] K. Abrahamson, N. Dadoun, D. G. Kirkpatrick, and T. Przytycka, “A simple parallel tree contraction algorithm,” Journal of Algorithms, vol. 10, pp. 287-302, 1989.
[2] E. A. Albacea, “Parallel algorithm for finding a core of a tree network,” Information Processing Letters, vol. 51, pp. 223-226, 1994.
[3] M. Ajtai, J. Komlos, and E. Szemeredi, “An O(nlog(n)) sorting network,” In Proceedings of the 15-th Annual ACM Symposium on the Theory of Computing, 1983, pp. 1-9.
[4] R. I. Becker and Y. Perl, “Finding the two-core of a tree,” Discrete Applied Mathematics, vol.11, no. 2, pp. 103-113, 1985.
[5] O. Berkman, B. Schieber, and U. Vishkin, “Some doubly logarithmic optimal parallel algorithms based on finding all nearest smaller values,” Technical Report UMIACS-TR-88-79, Institute for Advance Computer Studies, University of Maryland, College Park, MD, 1988.
[6] O. Berkman and U. Vishkin, “Finding level-ancestors in trees,” Journal of Computer and Science, Vol. 48, pp. 231-254, 1994.
[7] R. Cole, “Parallel merge sort,” SIAM Journal on Computing, vol. 17, no. 4, pp. 770-785, 1988.
[8] R. Cole and U. Vishkin, “Approximate parallel scheduling, Part I: The basic technique with applications to optimal parallel list ranking in logarithmic time,” SIAM Journal on Computing, vol. 17, no. 1, pp. 128-142, 1988.
[9] T. H. Cormen, C. E. Leiserson, and R. L. Rivest, Introduction to Algorithms, the MIT Press, 1994.
[10] K. Diks and E. Rytter, “On optimal parallel computations for sequence of brackets,” Theoretical Computer Science, vol. 87, pp. 251-262, 1991.
[11] B. Gavish and S. Sridhar, “Computing the 2-median on tree networks in O(nlog n) time,” Networks, vol. 26, iss. 4, pp. 305-317, 1995.
[12] A. Gibbons and W. Rytter, Efficient Parallel Algorithms, Cambridge University Press, 1988.
[13] S. L. Hakimi, E. F. Schmeichel, and M. Labbe, “On locating path- or tree-shaped facilities on networks, “ Networks, vol. 23, pp. 543-555, 1993.
[14] G. Y. Handler and P. Mirchandani, Location on Networks, MIT Press, Cambridge, MA, 1979.
[15] T. Hayashi, K. Nakano and S. Olariu, “Work time optimal k-merge algorithms on the PRAM,” IEEE Transactions on Parallel and Distributed Systems, vol. 9, no. 3, pp. 275-282, 1998.
[16] J. Jaja, An Introduction to Parallel Algorithms, Addison Wesley, 1992.
[17] O. Kariv and S. L. Hakimi, “An Algorithmic Approach to Network Location Problems. II: the p-Medians,” SIAM Journal on Applied Mathematics, vol. 37, no. 3, 1979.
[18] G. L. Miller and J. Reif, “Parallel tree contraction and its applications,” in Proceedings of the 26-th Annual IEEE Symposium on Foundations of Computer Science, 1985, pp. 478-489.
[19] S. Peng and W.-T. Lo, “A simple optimal algorithm for a core of a tree,” Journal of Parallel and Distributed Computing, vol. 20, pp. 388-392, 1994.
[20] S. Peng and W.-T. Lo, “ The optimal location of a structured facility in a tree network,” Parallel Algorithms and Applications, vol. 2, pp. 43-60, 1994.
[21] S. Peng and W. –T. Lo, “Efficient algorithms for finding a core of a tree with a specified length,” Journal of Algorithms, vol. 20, pp. 445-458, 1996.
[22] F. P. Preparata and M. I. Shamos, Computational Geometry: An Introduction, Springer-Verlag, 1985.
[23] P. V. Rangan, H. M. Vin, and S. Ramanathan, “Designing an on-demand multimedia service,” IEEE Communications Magazine, vol. 30, pp. 56-64, 1992.
[24] A. Tamir, “An O(pn2) algorithm for the p-median and related problems on tree graphs,” Operations Research Letters, vol. 19, iss. 2, pp. 59-64, 1996.
[25] B. C. Tansel, R. L. Francis, and T. J. Lowe, “Location on networks: A survey,” Management Science, vol. 29, pp. 482-511, 1983.
[26] R. E. Tarjan and U. Vishkin, “Finding biconnected components and computing tree functions in logarithmic parallel time,” SIAM Journal on Computing, vol. 14, no. 4, pp. 862-874, 1985.
[27] B.-F. Wang, “Finding a k-tree core and k-tree center of a tree network in parallel,” IEEE Transactions on Parallel and Distributed Systems, vol. 9, no. 2, pp. 186-191, 1998.
[28] B.-F. Wang, “Efficient parallel algorithms for optimally locating a path and a tree of a specified length in a weighted tree network,” Journal of Algorithms, vol. 34, pp. 90-108, 2000.
[29] B.-F. Wang, S.-C. Ku, K.-H. Shi, T.-K. Hung, and P.-S. Liu, “Parallel algorithms for the tree bisector problem and applications,” in Proceedings of the 1999 International Conference on Parallel Processing, 1999, pp. 192-199.