第07章加权图WeightedGraphs学习资料
Classification of weighted graphs up to blowing-up and blowing-down
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xqrBiblioteka is denoted [x1 , . . . , xq ]. So we distinguish between the graph [x1 , . . . , xq ] and the sequence (x1 , . . . , xq ) and we note that [x1 , . . . , xq ] = [xq , . . . , x1 ]. An admissible chain is a linear chain in which every weight is strictly less than −1. The empty graph ø is an admissible chain. 1.4. Let G be a weighted graph. We define three types of “blowing-up of G”:
n 1.1. If E is a set then E ∗ = ∪∞ n=0 E denotes the set of finite sequences in E , including the empty sequence ∅ ∈ E ∗ . We write A− for the reversal of A ∈ E ∗ , i.e., if A = (a1 , . . . , an ) then A− = (an , . . . , a1 ).
1
CLASSIFICATION OF WEIGHTED GRAPHS
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Note, however, that if n is a positive integer then the linear chains −2 0 0 n r r r r . . . −r2 and
graph的复数形式
graph的复数形式什么是graph在计算机科学中，图（graph）是一种非常重要的数据结构。
图由节点（vertex）和边（edge）组成，节点表示对象，边表示对象之间的关系。
图可以用来解决许多实际问题，比如社交网络分析、路线规划、电子商务推荐等。
graph的复数形式“graph”一词是一个可数名词，它的复数形式是”graphs”。
当我们需要描述多个图时，就可以使用”graphs”这个词。
例如，在一个大型社交网络中，有许多不同的图。
每个图代表一个用户或者一组用户之间的关系。
我们可以说：“这个社交网络中有很多graphs”。
图的种类在计算机科学中，有许多不同类型的图。
下面介绍几种常见的图类型：1. 无向图（Undirected Graph）无向图是最简单的图类型之一。
它的边没有方向性，意味着从一个节点到另一个节点可以沿着任意方向移动。
2. 有向图（Directed Graph）有向图也被称为有向网络或有向关系。
它的边具有方向性，意味着从一个节点到另一个节点只能沿着特定方向移动。
3. 加权图（Weighted Graph）加权图是指每条边都有一个与之相关联的权重或者成本。
这些权重可以表示节点之间的距离、时间、成本等。
4. 无环图（Acyclic Graph）无环图是指不存在回路或环的图。
换句话说，从一个节点出发，沿着任意路径都不会回到起点。
5. 有环图（Cyclic Graph）有环图是指存在至少一个回路或环的图。
从一个节点出发，沿着某些路径可以回到起点。
图的应用图在许多领域中被广泛应用，包括计算机科学、网络分析、运输规划等。
下面介绍一些常见的应用场景：1. 社交网络分析社交网络分析使用图来研究人际关系和社交网络结构。
通过构建用户之间的关系图，可以分析用户之间的联系强度、社群结构以及信息传播路径。
2. 路线规划在地理信息系统中，使用图来进行路线规划是一种常见方法。
通过将道路和交叉口表示为节点，并将道路连接表示为边，可以使用各种算法找到最短路径或最快路径。
离散数学 图论基础
Graphs
第一节 图的基本概念
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一个图G定义为一个三元组：G=<V, E, Φ>
V —— 非空有限集合，V中的元素称为结点 (node)或 顶点(vertex)
E —— 有限集合（可以为空），E中的元素称为边(edge)
Φ —— 从E到V的有序对或无序对的关联映射 (associative mapping)
（va, vb V）则称e是有向边（或弧）
va是e的起始结点， vb是e的终结点
v3
v3
v3
v1 (a) v2
v1
v2
(b)
v1
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图的基本概念
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若va和vb与边（弧）e相联结，则称va和vb是e的端结点 va和vb是邻接结点，记作：va adj vb (adjoin) 也称e关联va和vb，或称va和vb关联e 若va和vb不与任何边（弧）相联结，则称va和vb是非邻接结点， 记作：va nadj vb 关联同一个结点的两条边（弧），称为邻接边（弧）
v1
构造双射函数f : V1 V2 ，f(v1)=a ，f(v2)=b
f(v3)=c ，f(v4)=d
v2
v4
可知，边[v1, v2], [v2, v3], [v3, v4], [v4, v1]被分别映射成[a, b], [b, c], [c, d], [d, a]，故G1 G2
a
b v3
G1
c
0 ≤ d(v1)<d(v2)<… <d(vn)≤n-1。
结点的次数
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问题2：是否存在这种情况：两个人或以上的人群中，至少
模糊数学方法(第七章权重)
如果u1，u2，u3不是三个旅游点而是三个元素， 则最后的结果：
(0.3617, 0.2538, 0.3845) 就是三个元素的权重：
u1 0.3617,u2 0.2538,u3 0.3845
W（2）
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n2
第三层n3个元素对第二层n2个元素的权重（排序）向量为
W1 ,W2 , ,Wn2
将它们构成分块矩阵：
W = (W1 ,W2 , ,Wn2 ) 则第三层元素对第一层目标的权重（排序）向量为
W（3） WW（2） (W1 ,W2 ,
,Wn2
)
p
a j wi xi i 1
得到权重集：
( j 1, 2, , n)
A (a1, a2, , an )
§7.2 层次分析法 (The Analytic Hierarchy process,简称AHP)
层次分析是一种决策分析的方法。它结合了 定性分析和定量分析，并把定性分析的结果量化。
特征向量归一化得第三层3个元素对第二层4个元素的权 重（排序）向量为：
0.6028 0.07023 0.09888 0.2791
W1
0.08236 源自,W2 0.3706
,W3
0.3643
,
W4
0.6494
0.3151
得到权重（排序）向量：
W (w1 , w2 , , wn )
3. 特征向量法
(1)计算判断矩阵A的最大特征值max ; (2)求A属于特征值max的正特征向量
Spectral Graph Theory
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1. EIGENVALUES AND THE LAPLACIAN OF A GRAPH
From the start, spectral graph theory has had applications to chemistry [27]. Eigenvalues were associated with the stability of molecules. Also, graph spectra arise naturally in various problems of theoretical physics and quantum mechanics, for example, in minimizing energies of Hamiltonian systems. The recent progress on expander graphs and eigenvalues was initiated by problems in communication networks. The development of rapidly mixing Markov chains has intertwined with advances in randomized approximation algorithms. Applications of graph eigenvalues occur in numerous areas and in different guises. However, the underlying mathematics of spectral graph theory through all its connections to the pure and applied, the continuous and discrete, can be viewed as a single unified subject. It is this aspect that we intend to cover in this book.
Multiway cuts in directed and node weighted graphs
Naveen Garg
Vijay V. Vazirani
y Mihalis Yannakakais
1 Introduction
Given an undirected graph G = (V; E) with weights on the edges and a set of k terminals, s1; s2 : : :sk, a multiway cut is a set of edges whose removal disconnects every pair of terminals. This generalizes the fundamental notion of an s-t cut. Whereas the problem of computing a minimum s-t cut can be solved in polynomial time using a maximum ow algorithm, the problem
is
at
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times
the
maximum
ow.
Closely related to the notion of a multiway cut is that of a k-cut. A k-cut is a set of edges
whose deletion disconnects the graph into k components. Goldschmidt and Hochbaum 3] show
paper. We cut problem
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例
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例
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邻接链表-时间需求
假设每个指针和整数均为2字节长，则 一个n顶点图的邻接链表所需要的空间 为2(n+m+1)，其中对于无向图，m=2e； 而对于有向图，m=e。 邻接链表便于进行边的插入和删除操 作。确定邻接表中顶点的数目所需要 的时间与表中顶点的数目成正比。
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关联/邻接
当且仅当(i, j) 是图中的边时，顶点 i 和j 是邻接的(adjacent)。边(i, j) 关联(incident)于顶点i 和j。
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例-邻接，关联
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关联/邻接
在有向图中，有时候对邻接和关联的 概念作更精确的定义非常有用。 有向边(i, j) 是关联至（incident to）顶点j 而关联于（incident from） 顶点i。顶点i 邻接至（adjacent to） 顶点j，顶点j邻接于（adjacent from） 顶点i。 对于无向图来说，“至”和“于”的 含义是相同的。
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生成树
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例
会议，此次大会上的所有发言人都只 会说英语，而参加会议的其他人说的 语言是{L1,L2,…,Ln}之一。翻译小组 能够将英语与其他语言互译。
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二分图
可以将顶点集合分成两个子集A和B， 这样每条边在A中有一个端点，在B中 有一个端点，具有这种特征的图叫作 二分图(bipartite graph)
第12讲 加权图
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普里姆算法构造最小生成树的过程
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Displays all edges and weights.
Returns a minimum spanning tree starting from vertex 0.
Returns a minimum spanning tree starting from vertex v.
Returns all single-source shortest paths.
New York
888
Atlanta
661 11 Miami
TestMinimumSpanningTree
TestMinimumSpanningTree
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普里姆算法实现采用邻接矩阵方法
lowCost 0 -1 1 50 2 60 3∞ 4∞ 5∞ 6∞
python下的复杂网络编程包networkx的使用（摘抄）
python下的复杂⽹络编程包networkx的使⽤（摘抄）NetworkX是⼀个⽤Python语⾔开发的图论与复杂⽹络建模⼯具，内置了常⽤的图与复杂⽹络分析算法，可以⽅便的进⾏复杂⽹络数据分析、仿真建模等⼯作。
我已经⽤了它⼀段时间了，感觉还不错（除了速度有点慢），下⾯介绍我的⼀些使⽤经验，与⼤家分享。
⼀、NetworkX及Python开发环境的安装⾸先到下载networkx-1.1-py2.6.egg，到下载pywin32-214.win32-py2.6.exe。
如果要⽤Networkx的制图功能，还要去下载matplotlib和numpy，地址分别在和。
注意都要⽤Python 2.6版本的。
上边四个包中，pywin32、matplotlib和numpy是exe⽂件，按提⽰⼀路next，⽐较容易安装。
⽽NetworkX是个egg⽂件，安装稍微⿇烦，需要⽤easyinstall安装。
具体⽅法：启动DOS控制台（在“运⾏”⾥输⼊cmd），输⼊C:\Python26\Lib\site-packages\easy_install.py C:\networkx-1.1-py2.6.egg，回车后会⾃动执⾏安装。
注意我是把networkx-1.1-py2.6.egg放到了C盘根⽬录，读者在安装时应该具体根据情况修改路径。
安装完成后，启动 “开始 - 程序 - ActiveState ActivePython 2.6 (32-bit) - PythonWin Editor”，在shell中输⼊：import networkx as nxprint nx如果能输出：说明Networkx已经安装好了，可以正常调⽤。
关于Python语⾔，如果没有接触过可以找⼀本Python的语法书来看看（推荐《Python 精要参考(第⼆版)》，⽹上有电⼦版）。
这个语⾔很简单易学，只要有点编程基础，⼏天就可以学会它，然后就可以⾃如的运⽤它调⽤NetworkX了。
数据结构与算法常用英语词汇
数据结构与算法常用英语词汇.txt 女人谨记：一定要吃好玩好睡好喝好。
一旦累死了，就别的女人花咱的钱，住咱的房，睡咱的老公，泡咱的男朋友，还打咱的娃。
第一部份计算机算法常用术语中英对照Data Structures 基本数据结构Dictionaries 字典Priority Queues 堆Graph Data Structures 图Set Data Structures 集合Kd-Trees 线段树Numerical Problems 数值问题Solving Linear Equations 线性方程组Bandwidth Reduction 带宽压缩Matrix Multiplication 矩阵乘法Determinants and Permanents 行列式Constrained and Unconstrained Optimization 最值问题Linear Programming 线性规划Random Number Generation 随机数生成Factoring and Primality Testing 因子分解/质数判定Arbitrary Precision Arithmetic 高精度计算Knapsack Problem 背包问题Discrete Fourier Transform 离散 Fourier 变换Combinatorial Problems 组合问题Sorting 排序Searching 查找Median and Selection 中位数Generating Permutations 罗列生成Generating Subsets 子集生成Generating Partitions 划分生成Generating Graphs 图的生成Calendrical Calculations 日期Job Scheduling 工程安排Satisfiability 可满足性Graph Problems -- polynomial 图论-多项式算法Connected Components 连通分支Topological Sorting 拓扑排序Minimum Spanning Tree 最小生成树Shortest Path 最短路径Transitive Closure and Reduction 传递闭包Matching 匹配Eulerian Cycle / Chinese Postman Euler 回路/中国邮路Edge and Vertex Connectivity 割边/割点Network Flow 网络流Drawing Graphs Nicely 图的描绘Drawing Trees 树的描绘Planarity Detection and Embedding 平面性检测和嵌入Graph Problems -- hard 图论-NP 问题Clique 最大团Independent Set 独立集Vertex Cover 点覆盖Traveling Salesman Problem 旅行商问题Hamiltonian Cycle Hamilton 回路Graph Partition 图的划分Vertex Coloring 点染色Edge Coloring 边染色Graph Isomorphism 同构Steiner Tree Steiner 树Feedback Edge/Vertex Set 最大无环子图Computational Geometry 计算几何Convex Hull 凸包Triangulation 三角剖分Voronoi Diagrams Voronoi 图Nearest Neighbor Search 最近点对查询Range Search 范围查询Point Location 位置查询Intersection Detection 碰撞测试Bin Packing 装箱问题Medial-Axis Transformation 中轴变换Polygon Partitioning 多边形分割Simplifying Polygons 多边形化简Shape Similarity 相似多边形Motion Planning 运动规划Maintaining Line Arrangements 平面分割Minkowski Sum Minkowski 和Set and String Problems 集合与串的问题Set Cover 集合覆盖Set Packing 集合配置String Matching 模式匹配Approximate String Matching 含糊匹配Text Compression 压缩Cryptography 密码Finite State Machine Minimization 有穷自动机简化Longest Common Substring 最长公共子串Shortest Common Superstring 最短公共父串DP——Dynamic Programming——动态规划recursion ——递归第二部份数据结构英语词汇数据抽象 data abstraction数据元素 data element数据对象 data object数据项 data item数据类型 data type抽象数据类型 abstract data type逻辑结构 logical structure物理结构 phyical structure线性结构 linear structure非线性结构 nonlinear structure基本数据类型 atomic data type固定聚合数据类型 fixed-aggregate data type可变聚合数据类型 variable-aggregate data type 线性表 linear list栈 stack队列 queue串 string数组 array树 tree图 grabh查找，线索 searching更新 updating排序(分类) sorting插入 insertion删除 deletion前趋 predecessor后继 successor直接前趋直接后继双端列表循环队列immediate predecessor immediate successor deque(double-ended queue) cirular queue指针 pointer先进先出表(队列) first-in first-out list 后进先出表(队列) last-in first-out list栈底栈定压入弹出队头bottom top push pop front队尾 rear上溢 overflow下溢 underflow数组 array矩阵 matrix多维数组 multi-dimentional array以行为主的顺序分配 row major order以列为主的顺序分配 column major order 三角矩阵 truangular matrix对称矩阵 symmetric matrix稀疏矩阵 sparse matrix转置矩阵 transposed matrix链表 linked list线性链表 linear linked list单链表 single linked list多重链表 multilinked list循环链表 circular linked list双向链表 doubly linked list十字链表 orthogonal list广义表 generalized list链 link指针域 pointer field链域 link field头结点 head 头指针 head 尾指针 tail 串 string node pointer pointer空白(空格)串blank string 空串(零串) null string子串 substring树 tree子树 subtree森林 forest根 root叶子结点深度层次双亲孩子leaf node depth level parents children兄弟 brother祖先 ancestor子孙 descentdant二叉树 binary tree平衡二叉树 banlanced binary tree 满二叉树 full binary tree彻底二叉树 complete binary tree遍历二叉树 traversing binary tree 二叉排序树 binary sort tree二叉查找树 binary search tree线索二叉树 threaded binary tree 哈夫曼树 Huffman tree有序数 ordered tree无序数 unordered tree判定树 decision tree双链树 doubly linked tree数字查找树 digital search tree树的遍历 traversal of tree先序遍历 preorder traversal中序遍历 inorder traversal后序遍历 postorder traversal图 graph子图 subgraph有向图无向图彻底图连通图digraph(directed graph) undigraph(undirected graph) complete graphconnected graph非连通图 unconnected graph强连通图 strongly connected graph 弱连通图 weakly connected graph 加权图 weighted graph有向无环图 directed acyclic graph 稀疏图 spares graph稠密图 dense graph重连通图 biconnected graph二部图 bipartite graph边 edge顶点 vertex弧 arc路径 path回路(环) cycle弧头弧尾源点终点汇点headtailsource destination sink权 weight连接点 articulation point 初始结点 initial node终端结点 terminal node相邻边 adjacent edge相邻顶点 adjacent vertex 关联边 incident edge入度 indegree出度 outdegree最短路径 shortest path有序对 ordered pair无序对 unordered pair简单路径简单回路连通分量邻接矩阵simple pathsimple cycle connected component adjacency matrix邻接表 adjacency list邻接多重表 adjacency multilist遍历图 traversing graph生成树 spanning tree最小(代价)生成树 minimum(cost)spanning tree生成森林 spanning forest拓扑排序 topological sort偏序 partical order拓扑有序 topological orderAOV 网 activity on vertex networkAOE 网 activity on edge network关键路径 critical path匹配 matching最大匹配 maximum matching增广路径 augmenting path增广路径图 augmenting path graph查找 searching线性查找(顺序查找) linear search (sequential search)二分查找 binary search分块查找 block search散列查找 hash search平均查找长度 average search length散列表 hash table散列函数 hash funticion直接定址法 immediately allocating method 数字分析法 digital analysis method平方取中法 mid-square method折叠法 folding method除法 division method随机数法 random number method排序 sort内部排序 internal sort外部排序 external sort插入排序 insertion sort随小增量排序 diminishing increment sort 选择排序 selection sort堆排序 heap sort快速排序归并排序基数排序外部排序quick sort merge sortradix sort external sort平衡归并排序 balance merging sort二路平衡归并排序 balance two-way merging sort 多步归并排序 ployphase merging sort置换选择排序 replacement selection sort文件 file主文件 master file顺叙文件 sequential file索引文件 indexed file索引顺叙文件 indexed sequential file索引非顺叙文件 indexed non-sequential file直接存取文件 direct access file多重链表文件 multilist file倒排文件 inverted file目录结构 directory structure树型索引 tree index。
图
无向图(树)
有向图
有向 完全图 n(n-1) 条边
G1的顶点集合为V(G1)={0,1,2,3} 边集合为E(G1)={(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)}
Data Structure – Graph
稀疏图(sparse graph) 稠密图(dense graph) 边较少的图。通常边数远少于nlogn。 无向图中，边数接近 n(n-1)/2。 有向图中，边数接近 n(n-1)。
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Data Structure – Graph
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在有向图中, 若对于每一对顶点vi 强连通图： (strongly connected graph) 和vj, 都存在一条从vi到vj和从vj到 vi的路径, 则称此图是强连通图。
① 建立一个顶点表和一个邻接矩阵。
记录各个顶点信息
表示各个顶点之间关系
② 设图 A = (V, E) 有 n 个顶点，则图的邻接矩阵是一个二维数 组 A.Edge[n][n]，定义为：
1, A. Edge[ i ][ j ] 0, 如果 < i , j > E 或者 ( i, j ) E 否则
运筹学第7章图与网络优化
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链，圈，初等链，初等圈，简单链（圈）
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相邻节点的序列 {v1 ,v2 ,…, vn} 构成一条链(link)p178；
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在无向图中，节点不重复出现的链称为初等链；
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首尾相连的链称为圈(loop) ；首尾相连的初等链称为初等圈；
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边不重复出现的链（圈）称为简单链（圈）
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子图，部分图；连通图，成分
(1).与v3相连的临时标号有v5
第五步：
T(v5)=min{T(v5),P(v3)+d35}=min{9,7+3}=9
(2).P(v5)=9
最短路线：
vs→v1→v4→ v5 vs→v2→v4→ v5
vS
v2
v3
v4
v5
1
2
2
2
3
3
3
4
4
0
4
5
3
7
9
*
也可以用表格的形式求解。p190
斯坦纳树问题
假设我们在北京、上海、西安三城市之间架设电话线，一种办法是分别联通北京--上海和北京--西安。另一种办法是选第四个点，假设郑州。由此分别向三城市架线，可能你不会想到第二种办法所用的电话线只是第一种办法的86.6%，即可取得比第一种办法节约13%的显著经济效益。这就是离散数学界30年代提出的著名的斯坦纳树问题，但一直未能得到证明。
平面图(planar graph)，若在平面上可以画出该图而没有任何边相交
*
7基础图，路，回路，欧拉回路
在有向图D(V,A)中去掉箭头，称为D的基础图，G（D）
01
在有向图中，链 路
02
圈 回路
03
第06章图Graph
Internet Web
John
Databases
Paul
David
Entity-relationship diagram
Graphs
5
Terminology
End vertices (or endpoints) of
an edge
U and V are the endpoints of a
ends with a vertex
each edge is preceded and followed by its endpoints
Simple path
path such that all its vertices and edges are distinct
Examples
P1=(V,b,X,h,Z) is a simple path P2=(U,c,W,e,X,g,Y,f,W,d,V) is a
Graphs
2
6.1 Graph
A graph is a pair (V, E), where
V is a set of nodes, called vertices
E is a collection of pairs of vertices, called edges
Vertices and edges are positions and store elements
Proof: each vertex has degree at most (n - 1)
Graphs
Notation
n m deg(v)
number of vertices number of edges degree of vertex v
第07章加权图习题答案
第07章加权图习题答案Homework 7 Solution Exercise R-7.7Answers vary. For example, the following is correct.Exercise C-7.3The greedy algorithm presented in this exercise is not guaranteed to find the shortest path between vertices in graph. This can be seen by a counterexample. Consider the following weighted graph:Vertices A, B, C, D. Edges (A: B), (A: C), (D: B), (D: C) and weights of 1, 2, 3, and 1 respectively.Suppose we wish to find the shortest path from start = A to goal = D and we make use of the proposed greedy strategy. Starting at A and with path initialized to A, the algorithm will next place B in path (because (A;B) is the minimum cost edge incident on A) and set start equal to B. The lightest edge incident on start = B which has not yet been traversed is (B;D). As a result, D will be appended to path and start will be assigned D. At this point the algorithm will terminate (since start = goal = D). In the end we have that path = A;B;D, which clearly is not the shortest path from A to D. (Note: in fact the algorithm is not guaranteed to find any path between two given nodes since it makes no provisions for backing up. In an appropriate situation, itwill fail to halt.)Exercise C-7.7We can model this problem using a graph. We associate a vertex of the graph with each switching center and an edge of the graph with each line between two switching centers. We assign the weight of each edge to be its bandwidth. Vertices that represent switching centers that are not connected by a line do not have an edge between them.We use the same basic idea as in Dijkstra’s algorithm. We keep a va riable d[v] associated with each vertex v that is the bandwidth on any path from a to this vertex. We initialize the d values of all vertices to 0, except for the value of the source (the vertex corresponding to a) that is initialized to infinity. We also keep a value associated with each vertex (that contains the predecessor vertex).The basic subroutine will be very similar to the subroutine Relax in Dijkstra. Assume that we have an edge (u; v). If min {d[u];w(u; v)} > d[v] then we should update d[v] to min{d[u];w(u; v)} (because the path from a to u and then to v has bandwidth min{d[u];w(u; v)}, which is more than the one we have currently). Algorithm Max-Bandwidth(G,a,b):for all vertices v in V dod[v]←0d[a]←∞Known←ǿUnknown← Vwhile Unknown ≠ ǿ dou←ExtractMax(Unknown)Known ←Known u U {u}for all vertices v in Ad j[u] doif (min(d[u];w(u; v))> d[v])d[v] min(d[u];w(u; v))return d[b]Here, the function ExtractMax(Unknown) finds the node in Unknown with the maximum value of d.Exercise C-7.8Consider the weighted graph G = (V;E), where V is the set of stations and E is the set of channels between the stations. Define the weight w(e) of an edge e ε E as the bandwidth of the corresponding channel.Given below are two ways of solving the problem:1. At every step of the greedy algorithm for constructing the minimum spanning tree, instead of picking the edge having the least weight, pick the edge having the greatest weight.2. Negate all the edge-weights. Run the usual minimum spanning tree algorithm on this graph. The algorithm gives us the desired solution.。
运筹学05.1图的基本概念
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运筹学
Operations Research
§5.1
over
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空图（empty graph）： 1 ≤ ν < +∞, ε = 0 平凡图（trivial graph）：ν = 1, ε = 0 无向图（graph），有向图（digraph）
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运筹学
Operations Research
连杆（link），环（loop），重边（multiedge）：
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运筹学
Operations Research
例5 图的各顶点的度按不增顺序排成的序列称为图的 度序列.问以下数列能否为某简单图的各顶点的度序列？ （1）3,2,2,2,1,1； （2）7，6，5，4，3，3，2； （3）10，6，3，2，2，1，1，1； （4）5，4，2，2，2，1，1；
迹（trail），路（path），圈（cycle）
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例8求证：若δ (G ) = 2，则图G中必含有圈.
证：
▍
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连通图（connected graph）
若顶点u, v之间有一条路相连，则称u与v是连通的. 若图G的任两顶点都是连通的，则称G是连通图.
简单图（simple graph）：不含有重边和环的图.
性质 若G是简单图，则(1)ε ≤ Cν2 ; ( 2)何时取 = ?
《图算法基础知识》PPT课件
精选ppt3
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图
● 通常用边数 |E|和顶点数|V|描述运行时间。
■ 无向图中 0≤|E|≤|V|(|V|-1)/2 ■ 有向图中 0≤|E|≤|V|(|V|-1) ■ 若|E| |V|2 ，图是稠密的 dense ■ 若 |E| |V| ，图是稀疏的 sparse
● 对稠密图和稀疏图使用不同的数据结构
● 完全图(Complete Grapth) 若有n个顶点的无向图n(n-1)/2条边，则此图为完 全图。
精选ppt2
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图的扩展
● 扩展:
■ 连通图（ connected graph） 从图中每一顶点都有到其
它顶点的路径 。
■ 无向图（ undirected graph）:
○ 边(u,v) = 边 (v,u)
精选ppt24
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//入度为0的顶点数小于n时，存在环；
否则图不存在环，且可进行拓扑排序，ord数组存储的就是排序后 的序号
If count<n then begin writeln(‘graph is cyclic’); topsort:=false; end;
End;
精选ppt25
图: 邻接矩阵
● Example: 1
2
4
3
A 12 1 01 2 00 3 00 4 00
34 10 10 00 10
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有向图
非对称矩阵
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图: 邻接矩阵
● Example:
1
a
2
d
4
b
c
3
A1 2 3 4
1
2
3
第07章加权图WeightedGraphs-PPT精品
We add to the cloud the vertex u outside the cloud with the smallest distance label, d(u)
We update the labels of the vertices adjacent to u
Recall that Sv deg(v) = 2m
The running time can also be expressed as O(m log n) since the graph is connected
Weighted Graphs
11
Extension
Using the template method pattern, we can extend Dijkstra’s algorithm to return a tree of shortest paths from the start vertex to all other vertices
Example:
Shortest path between Providence and Honolulu
Applications
Internet packet routing
Flight reservations
Driving directions
SFO
ORD
PVD
LGA
HNL
LAX
When the previous node, D, on the true shortest path was considered, its distance was correct.
But the edge (D,F) was relaxed at that time!
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8
8 B
7
0 A
2 2
C
53 9 2E
4
1
3 D
8
7 B
7
0 A
2 2
C
11 F5
53 9 2E
Weighted Graphs
4
1
3 D
8 F5
8
Example (cont.)
8
7 B
7
5 2E
0 A
2 2
C
39
4
1
3 D
8 F5
8
7 B
7
5 2E
0 A
2 2
C
39
4
1
3 D
8 F5
Weighted Graphs
6
Edge Relaxation
Consider an edge e = (u,z)
such that
u is the vertex most recently
added to the cloud
s
z is not in the cloud
The relaxation of edge e updates distance d(z) as follows:
Weighted Graphs
2
Weighted Graphs
In a weighted graph, each edge has an associated numerical value, called the weight of the edge
Edge weights may represent, distances, costs, etc.
Assumptions:
the graph is connected
the edges are undirected
the edge weights are nonnegative
We grow a “cloud” of vertices, beginning with s and eventually covering all the vertices
We store with each vertex v a label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices
At each step
MIA
3
Shortest Path Problem
Given a weighted graph and two vertices u and v, we want to find a path of minimum total weight between u and v.
Length of a path is the sum of the weights of its edges.
We add to the cloud the vertex u outside the cloud with the smallest distance label, d(u)
We update the labels of the vertices adjacent to u
Weighted Graphs
Example:
In a flight route graph, the weight of an edge represents the distance in miles between the endpoint airports
HNL
ORD SFO
LAX
DFW
Weighted Graphs
PVD LGA
Weighted Graphs
8
8 B
7
5 2E
0 A
2 2
C
39
4
1
3 D
8 F5
Weighted Graphs
1
Outline and Reading
Weighted graphs (§7.1) Dijkstra’s algorithm (§7.1.1) The Bellman-Ford algorithm (§7.1.2) Shortest paths in dags (§7.1.3) All-pairs shortest paths (§7.2.1) Shortest Path via Matrix Multiplication (§7.2.2) – No slide on this section- too mathematical for slides! Minimum Spanning Trees (§7.3) The Prim-Jarnik Algorithm (§7.3.2) Kruskal's Algorithm (§7.3.1) Baruvka's Algorithm (§7.3.3)
9
Dijkstra’s Algorithm
A priority queue stores the vertices outside the cloud
Key: distance
Element: vertex
Locator-based methods
insert(k,e) returns a locator
d(z) min{d(z),d(u) + weight(e)}
s
d(u) = 50
u
Байду номын сангаас
e
d(z) = 75
z
d(u) = 50
u
e
d(z) = 60
z
Weighted Graphs
7
Example
8
8 B
7
0 A
2 2
C
4
1
4 D
3 2E
9 F5
8
8 B
7
5 2E
0 A
2 2
C
39
4
1
3 D
8 F5
Weighted Graphs
MIA
4
Dijkstra’s Algorithm
The distance of a vertex v from a vertex s is the length of a shortest path between s and v
Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex s
Example:
Shortest path between Providence and Honolulu
Applications
Internet packet routing
Flight reservations
Driving directions
SFO
ORD
PVD
LGA
HNL
LAX
DFW