在廣播式的行動無線網路環境下，為節省頻寬的需求，通常我們會希望將相同的訊息需求排程後以廣播的方式一次傳送給多位使用者，以節省頻寬的需求(不需重複傳送同樣的訊息），然而，在這樣的目標之下，使用者必須等候訊息，以服務供應商的角度出發，我們必須運算出一個訊息的排列次序，以使得使用者的總等候時間最短，而本論文即嚐試將訊息間的相依性一般化至圖形(Graph)上，並以各節點的權重w及延遲時間l來分別表示使用者的數量及訊息長度，這樣的問題，已知在一般的圖上是一個NP-Hard的問題，因此在本文中，我們嚐試在具某些特殊性質的圖形上，尋求最小延遲問題的最佳演算法。
在將上述問題一般化至G(v,e)後，假設共計有n個訊息(v1…vn)要傳送，訊息的相依存取關係可以圖G(v,e)表示，邊的關係表示vi間的相依性(如Hyperlink)，亦即傳送某一訊息vi前，至少該訊息的其中一個祖先節點必須要已經被傳送，令每一個訊息的傳送時間為d(vi)，每個訊息等候的人數為w(vi)，總等候時間 ，其中l(vi)為到達vi前所需經的路徑延遲總和。
在這樣的問題假設下，我們分別地針對了Tree、k-ary Tree、Path、及K-Path等四種圖形上提出排程的演算法，我們也分別獲得了O(n log n)、O(n log k)、O(n)、O(n)的時間複雜度，並且證明了各個演算法的正確性，除此之外，我們亦提出了MLP問題在Tree的圖形下最佳的時間複雜度必然不可能小於O(n log n)。

In mobile environment, users retrieve information by portable devices. Since the mobile devices usually have limited power, the issue of minimization the data access latency is important. Periodic broadcasts of frequently requested data can thus reduce the traffics in the air and save the powers of the mobile devices. However, users need to wait for the required data to appear on the broadcast channel. It follows the rule “the more time they wait then the more power devices have to consume. Finding the minimum latency tour can thus help us in solving this kind of problem.
In this paper we study the variation of the minimum latency problem (MLP) [2]. The MLP is to find a walk tour on the graph G(V,E) with a distance matrix di,j.Where di,j indicate the distance between vi and vj. Let l(vi) is the latency length of vi, defined to be the distance traveled before the first visiting vi. The minimum latency tour is to minimize the . In some message broadcast and scheduling problem [8] the vertex also has latency time and weight. Those problem need to extend the objective function of the minimum latency tour as . The definition is equivalent to the MLP with no edge distance but vertex latency time and vertex weight. We give a linear algorithm for the un-weighted full k-ary tree or k-path graphs, and O(n log n) time for general tree graphs. The time complexity in trees is the same as Adolphson's result; however, the algorithm given here is not only simpler, easier to understand, but also more flexible and thus can be easily extended to other classes of graphs.

[1] D. Adolphson and T. C. Hu. Optimal linear ordering. SIAM J. Appl. Math., 25:403-423, 1973.
[2] A. Blum, P. Chalasani, D. Coppersmith, B. Pulley blank, P. Raghavan, and M. Sudan. The minimum latency problem. Proc. 26th Annu. ACM Sympos.Theory Comput., pages 163-171, 1994.
[3] T. Corman, C. Leiserson, and R. Rivest. Introduction to Algorithms. MIT Press, 1990.
[4] T. Imielinski, S. Viswanathan, and B.R. Badrinath.Power efficient filtering of data on air. 4th International Conference on Extending Database Technoloy, pages 245-258, March 1994.
[5] T. Imielinski, S. Viswanathan, and B.R. Badrinath. Data on air: Organization and access. IEEE Trans. on Knowledge and Data Engineering, 9(3):353-372, May 1997.
[6] Shou-Chih Lo and Arbee L.P. Chen. Index and data allocation in multiple broadcast channels. IEEE Internation Conference on Data Engineering 2000, pages 293-302, 2000.
[7] N. Shivakumar and S. Venkatasubramanian. Energy-efficient indexing for information dissemination in wireless systems. ACM, Journal of Wireless and Nomadic Application, 1996.
[8] M. Chen P. Yu and K. Wu. Indexed sequential data broadcasting in wireless mobile computing. IEEE Interantional Conference on Distributed Computing Systems, pages 124-131, 1997.