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研究生:梁閎鈞
研究生(外文):Hung-Chun Liang
論文名稱:有向平面圖上的最小切割與最短迴圈
論文名稱(外文):Minimum Cuts and Shortest Cycles in Directed Planar Graphs via Shortest Non-Crossing Paths
指導教授:呂學一
口試委員:顏嗣鈞、王大為、陳和麟
口試日期:2015-07-07
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:資訊工程學研究所
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2015
畢業學年度:103
語文別:英文
論文頁數:37
中文關鍵詞:演算法、平面圖、最小切割、最短路徑
外文關鍵詞:algorithm、planar graph、minimum cut、shortest path
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Let G be an n-node simple directed planar graph with nonnegative edge weights. We study the fundamental problems of computing (1) a global cut of G with minimum weight and (2) a cycle of G with minimum weight. The best
previously known algorithm for the former problem, running in O(n log3 n) time, can be obtained from the algorithm of Lacki, Nussbaum, Sankowski, and Wulff-Nilsen for single-source all-sinks maximum flows. The best previously known result for the latter problem is the O(n log3 n)-time algorithm of Wulff-Nilsen. By exploiting duality between the two problems in planar graphs, we solve both problems in O(n log n log log n) time via a divide-and-conquer algorithm that finds a shortest non-degenerate cycle. The kernel of our result is an O(n log log n)-time algorithm for computing shortest noncrossing paths among nodes well ordered on a common face of a directed plane graph, which is extended from the algorithm of Italiano, Nussbaum, Sankowski, and Wulff-Nilsen for an undirected plane graph.

摘要 iii
Abstract v
1 Introduction 1
1.1 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Technical overview and outline . . . . . . . . . . . . . . . . . . . . . . . 5
2 Reduction to finding shortest non-degenerate cycles 7
3 Divide-and-conquer via balanced separating cycles 11
4 Non-degenerate cycles that cross the separating cycle 15
5 Non-crossing shortest paths 19
6 Shortest paths via Monge units 27
7 Concluding remarks 29
Bibliography 31

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