令 G = (V, E) 為一個無向的平面圖 (undirected planar graph)，每一個edge都有一個非負整數的長度 (length)，(r1, s1) 和 (r2, s2)是兩組成對且相異的點．而這些點都跟無界限的區域 (unbounded face) 相鄰，最後再給另外兩個正整數b1, b2．令L = max{b1, b2}．在這篇論文中我們要探討的問題是如何在平面圖中尋找兩條點相異且長度有限制的路徑 (problem of finding two length-bounded vertex-disjoint paths in planar graphs)，這個問題常常被應用在設計網路的傳輸策略 (routing strategy)，一般來說，當我們傳輸兩個以上的封包時，我們不希望這些封包會同時傳給同一個路由器 (router)，並且我們也不希望這些封包花太多的時間到達目的地．能不能找到這樣的傳輸路徑對網路的可靠度（reliability）是一個重要的因素．
這個問題已經在2002年時被Holst 和 Pina證明了是NP-Hard，在同一篇論文中他們同時也對這個問題提出一個利用dynamic-programming技術來解決這個問題，這個演算法需要花 O(|V|^4*L^2) 時間．
我們這篇論文對同樣的一個問題提出了一個更有效率的演算法，這個新的演算法同樣也是利用dynamic-programming技術，這個改良過後的演算法只需要花O(|V|^3*L^2) 時間．跟Ｈolst和Pina的結果相比，我們的演算法有較好的時間複雜度．

Let G = (V, E) be an undirected planar graph embedded in R2 with vertex set V and edge set E. Each edge e□E has a non-negative integral length l(e). Let (r1, s1) and (r2, s2) be two distinct pairs of vertices of G adjacent to the unbounded face. Let b1 and b2 be two positive integers. Given G, (r1, s1), (r2, s2), b1 and b2, we consider the problem of finding two vertex-disjoint paths P1 and P2 such that Pi is a path from ri to si and the length of Pi is at most bi for i = 1, 2.

Previously, Holst and Pina [18] proposed a pseudo-polynomial time algorithm for this problem, which takes O(|V|^4*L^2) time, where L=max{b1, b2}. In this thesis, an improved algorithm is proposed. The proposed algorithm requires O(|V|^3*L^2) time.

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