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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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bridges of koenigsberg
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?

slide3
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

slide5
电路模拟

例:Pspice、Cadence、ADS…..

Pspice

Cadence

slide6
交通网络

航空网络!

捷運路線图!

slide7

计算机网络

某学校网络架构图

slide8
有向图

有单行道的街道!

行程表!

social network
Social Network

High School Dating

corporate e-mail

Reference: Bearman, Moody and Stovel, 2004

image by Mark Newman

Reference: Adamic and Adar, 2004

protein interaction network
Protein interaction network

Reference: Jeong et al, Nature Review | Genetics

the internet
The Internet

The Internet as mapped by The Opte Project

http://www.opte.org

more applications
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。
hamilton
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!

shortest path problem
最短路径问題 (Shortest Path Problem)

最快的routing

最快航線

dijkstra
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
depth first search
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

some graph processing problems
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?

slide22
什么是图?

例一 例二

一堆顶点和边的组合!

Set of vertices connected pairwise by edges.

slide23
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)}

slide24
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

digraph
有向图(Digraph)

有向图 (Digraph)

有向且无简单回路的图

(directed acyclic graph)

weighted graph
加权图(Weighted Graph)

图片來源:雷欽隆老師“資料結構”課投影片

spanning tree
生成树(Spanning Tree)

包括图中所有的顶点,并且是一棵树

生成树

complete graphs
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
bipartite graphs
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

p lanar graphs
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:

slide35
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
trees
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