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图论的介绍

图论的介绍. 哥尼斯 堡 七桥问題 ( Bridges of Koenigsberg ). 能不能走过每一个桥 刚 好一次并且回到原來的地方？. 欧拉路径 解決哥尼斯保七桥问題. 原來是一 笔画问题 啊！. 数学 家 欧拉 (Euler, 1707-1783) 于 1736 年 严格的证明了上述 哥尼斯堡 七桥问题无解 ， 并且 由此 开创 了 图论 的典型 思维 方式及 论证 方式. 实际生活中的图论 G raph Model. 电路模拟. 例： Pspice 、 Cadence 、 ADS…. Pspice. Cadence. 交通网络. 航空网络 !.

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图论的介绍

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  1. 图论的介绍

  2. 哥尼斯堡七桥问題(Bridges of Koenigsberg) 能不能走过每一个桥刚好一次并且回到原來的地方？

  3. 欧拉路径解決哥尼斯保七桥问題 原來是一笔画问题啊！ 数学家欧拉(Euler, 1707-1783) 于1736年严格的证明了上述哥尼斯堡七桥问题无解，并且由此开创了图论的典型思维方式及论证方式

  4. 实际生活中的图论Graph Model

  5. 电路模拟 例：Pspice、Cadence、ADS….. Pspice Cadence

  6. 交通网络 航空网络! 捷運路線图！

  7. 计算机网络 某学校网络架构图

  8. 有向图 有单行道的街道！ 行程表！

  9. Social Network High School Dating corporate e-mail Reference: Bearman, Moody and Stovel, 2004 image by Mark Newman Reference: Adamic and Adar, 2004

  10. Protein interaction network Reference: Jeong et al, Nature Review | Genetics

  11. Power transmission grid of Western US

  12. The Internet The Internet as mapped by The Opte Project http://www.opte.org

  13. More Applications • Hypertexts • 网页包含到其他网页的超链接。整个Web是一个图. 搜索引擎需要图处理算法。 • Matching • 职位招聘，如何有效将职位与应聘者匹配？ • Schedules • 工程项目的任务安排，如何满足限制条件，并在最短时间内完成？ • Program structure • 大型软件系统，函数（模块）之间调用关系。编译器分析调用关系图确定如何最好分配资源才能使程序更有效率。

  14. Graph Applications

  15. Graph Problems and Algorithms

  16. 哈密頓（Hamilton）周遊世界问題 正十二面体有二十个顶点 表示世界上20个城市 各经每个城市一次 最后返回原地 投影至平面 哈密頓路径至今尚无有效方法來解決！

  17. 最短路径问題 (Shortest Path Problem) 最快的routing 最快航線

  18. 最短路径算法Dijkstra算法 • 可以求出從某一点到图上其他任一点的最短路径

  19. 走迷宫与深度优先搜索(Depth-First Search) 一直往前走 碰壁就回头換条路找 沿途要记录下走过的路线 老鼠走迷宮深度优先搜索

  20. Some graph-processing problems • Path. Is there a path between s to t? • Shortest path. What is the shortest path between s and t? • Longest path. What is the longest simple path between s and t? • Cycle. Is there a cycle in the graph? • Euler tour. Is there a cycle that uses each edge exactly once? • Hamilton tour. Is there a cycle that uses each vertex exactly once? • Connectivity. Is there a way to connect all of the vertices? • MST. What is the best way to connect all of the vertices? • Biconnectivity. Is there a vertex whose removal disconnects the graph? • Planarity. Can you draw the graph in the plane with no crossing edges? First challenge: Which of these problems is easy? difficult? intractable?

  21. 图论的术语

  22. 什么是图？ 例一 例二 一堆顶点和边的组合！ Set of vertices connected pairwise by edges.

  23. 图论的术语 顶点 (Vertex) 边 (Edge) 一个图G = (V,E) V: 顶点的集合 E: 边的集合 例：如右图 V= {a,b,c,d,e} E= {(a,b),(a,c),(a,d), (b,e),(c,d),(c,e), (d,e)}

  24. 再來一些术语 连通图(connected graph) 子图(subgraph) 树(tree)沒有回路的连通图 森林(forest) 一堆树的集合

  25. 树的实例 ─ 行政组织图

  26. 有向图(Digraph) 有向图 (Digraph) 有向且无简单回路的图 (directed acyclic graph)

  27. 加权图(Weighted Graph) 图片來源：雷欽隆老師“資料結構”課投影片

  28. 生成树(Spanning Tree) 包括图中所有的顶点，并且是一棵树 生成树

  29. 可运用生成树的实例

  30. Graph Terminology

  31. 一些特殊的图

  32. 完全图Complete graphs • 任意两点之间都有一条边与其相连的图称为完全图，以Kn來表示，n为顶点数

  33. 二分图（偶图）Bipartite graphs • A graph that can be decomposed into two partite sets but not fewer is bipartite • It is a complete bipartite if its vertices can be divided into two non-empty groups, A and B. Each vertex in A is connected to B, and vice-versa The graph is bipartite Complete bipartite graph K2,3

  34. 平面图 Planar graphs • A planar graph is a graph that can be embedded in a plane so that no edges intersect Planar: = NOT Planar:

  35. 平面图实例 • 8个顶点(V=8) • 12条边(E=12) • 6个面 (F=6) • 8-12+6=2

  36. 更多平面图实例

  37. 树Trees • 树(tree):连通无简单回路无向图 • 若有n个顶点，則有n-1条边 • 任两点之间仅有一条路径 • 生成树(spanning tree):包括图中所有的顶点，并且是一棵树 tree A connected graph G Spanning trees of G

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