MITx - 6.00.2x 笔记(Unit1 Lecture 3 Graph Problems)

Lecture 3 Graph Problems

Graph Theory(图论）

图论是数学的一个分支，它以图为研究对象，研究顶点和边组成的图形的数学理论和方法。nodes, edges
现实世界中很多事物都是有联系的，因此图论特别适合于解决现实问题。
可应用于路径规划

AdjList
Adjacency List（邻接表）
在图论中，邻接表代表一个图中的所有边或弧。
如果是无向图，那么每条边由两个结点组成，分别代表边的两个端点；如果是有向图，那么每条边是一个结点对，分别代表边的始点和终点。一般来说，邻接表是无向的。

# 定义Node class而不直接使用string
# 因为可能会涉及拥有多个属性的对象

# 定义节点
class Node(object):
    def __init__(self, name):
        """Assume name is a string"""
        self.name = name
    def getName(self):
        return self.name
    def __str__(self):
        return self.name

# 定义路径（起点src, 终点dest)    
class Edge(object):
    def __init__(self, src, dest):
        """Assumes src and dest are nodes,
            src -> dest """
        self.src = src
        self.dest = dest
    def getSource(self):
        return self.src
    def getDestination(self):
        return self.dest
    def __str__(self):
        return self.src.getName() + '->' + self.dest.getName()

# 绘制模块
class Digraph(object):
    """edges is a dict mapping each node to a list of children"""

    def __init__(self):
        self.edges = []

    def addNode(self, node):
        if node in self.edges:
            raise ValueError('Duplicate node')
        else:
            self.edges[node] = []

    def addEdge(self, edge):
        src = edge.getSource()
        dest = edge.getDestination()
        if not (src in self.edges and dest in self.edges):
            raise ValueError('Node not in graph')
        self.edges[src].append(dest)

    def childrenOf(self, node):
        return self.edges[node]

    def hasNode(self, node):
        return node in self.edges

    def getNode(self, name):
        for n in self.edges:
            if n.getName() == name:
                return n
        raise NameError(name)

    def __str__(self):
        result = ''
        for src in self.edges:
            for dest in self.edges[src]:
                result = result + src.getName() + ' -> ' + dest.getName() + '\n'

            return result[:-1] # omit final newline

class Graph(Digraph):
    def addEdge(self, edge):
        Digraph.addEdge(self, edge)
        rev = Edge(edge.getDestination(), edge.getSource())
        Digraph.addEdge(self, rev)
        # Digraph addEdge add TWO edges, one for each direction

寻找节点n1到n2的最短路径的基本规则：
- 第一edge的起点为n1
- 最后一条edge的终点为n2
- 如果e2跟在e1之后，那么e2的起点是e1的终点
- 目标：尽可能减少edges的数量

GraphOptmProblem

def buildCityGraph(graphType):
    g = graphType()
    for name in ('Boston', 'Providence', 'New York', 'Chicago',
                 'Denver', 'Phoenix', 'Los Angeles'): #Create 7 nodes
        g.addNode(Node(name))
    g.addEdge(Edge(g.getNode('Boston'), g.getNode('Providence')))
    g.addEdge(Edge(g.getNode('Boston'), g.getNode('New York')))
    g.addEdge(Edge(g.getNode('Providence'), g.getNode('Boston')))
    g.addEdge(Edge(g.getNode('Providence'), g.getNode('New York')))
    g.addEdge(Edge(g.getNode('New York'), g.getNode('Chicago')))
    g.addEdge(Edge(g.getNode('Chicago'), g.getNode('Denver')))
    g.addEdge(Edge(g.getNode('Denver'), g.getNode('Phoenix')))
    g.addEdge(Edge(g.getNode('Denver'), g.getNode('New York')))
    g.addEdge(Edge(g.getNode('Los Angeles'), g.getNode('Boston')))
    return g

# In:
print('Digraph:')
print(buildCityGraph(Digraph))

# Out:
Digraph:
Boston -> Providence
Boston -> New York
Providence -> Boston
Providence -> New York
New York -> Chicago
Chicago -> Denver
Denver -> Phoenix
Denver -> New York
Los Angeles -> Boston

Depth-first Search (DFS，深度优先搜索)

先找准一个节点，搜索所有可能路径，遍历顺序图示：

def printPath(path):
    """Assumes path is a list of nodes"""
    result = ''
    for i in range(len(path)):
        result = result + str(path[i])
        if i != len(path) - 1:
            result = result + '->'
    return result 

def DFS(graph, start, end, path, shortest, toPrint = False):
    """Assumes graph is a Digraph; start and end are nodes;
          path and shortest are lists of nodes
       Returns a shortest path from start to end in graph"""
    path = path + [start]
    if toPrint:
        print('Current DFS path:', printPath(path))
    if start == end:
        return path
    for node in graph.childrenOf(start):
        if node not in path: #avoid cycles
            if shortest == None or len(path) < len(shortest):
                newPath = DFS(graph, node, end, path, shortest,
                              toPrint)
                if newPath != None:
                    shortest = newPath
        elif toPrint:
            print('Already visited', node)
    return shortest

def shortestPath(graph, start, end, toPrint = False):
    """Assumes graph is a Digraph; start and end are nodes
       Returns a shortest path from start to end in graph"""
    return DFS(graph, start, end, [], None, toPrint)

def testSP(source, destination):
    g = buildCityGraph(Digraph)
    sp = shortestPath(g, g.getNode(source), g.getNode(destination),
                      toPrint = True)
    if sp != None:
        print('Shortest path from', source, 'to',
              destination, 'is', printPath(sp))
    else:
        print('There is no path from', source, 'to', destination)

testSP('Chicago', 'Boston')

# Out：
Current DFS path: Chicago
Current DFS path: Chicago->Phoenix
Current DFS path: Chicago->Denver
Current DFS path: Chicago->Denver->Phoenix
Current DFS path: Chicago->Denver->New York
Already visited Chicago
There is no path from Chicago to Boston