python数据结构之图论

本篇学习笔记内容为图的各项性质、图的表示方法、图ADT的python实现

图（Graph）

是数据结构和算法学中最强大的框架之一（或许没有之一）。图几乎可以用来表现所有类型的结构或系统，从交通网络到通信网络，从下棋游戏到最优流程，从任务分配到人际交互网络，图都有广阔的用武之地。

我们会把图视为一种由“顶点”组成的抽象网络，网络中的各顶点可以通过“边”实现彼此的连接，表示两顶点有关联。我们要知道最基础最基本的2个概念，顶点（vertex）和边（edge）。

图可以分为有向图和无向图，一般用G=(V,E)来表示图。经常用邻接矩阵或者邻接表来描述一副图。

首先是链表、树与图的对比图：

圆为顶点、线为边

图的术语

图 G 是顶点V 和边 E的集合 

两个顶点之间：边 

如果顶点 x 和 y 共享边,则它们相邻,或者它们是相邻的 

无向图 ：无向图中的一个边可以在任一方向上遍历 

路径:：通过边连接的顶点序列 

周期：第一个和最后一个顶点相同的路径 

入度:：顶点的度数V是以V为端点的边数 

出度： 顶点的出度v是以v为起点的边的数量 

度：顶点的度数是其入度和出度的总和

图的ADT

数据成员 :

顶点 （vertex）

边缘 (edge)

操作 :

有多少顶点？

有多少个边缘？

添加一个新的顶点 

添加一个新的边缘 

获取所有邻居？ （进出）

U，V连接吗？

反转所有边缘？

获取2跳邻居

图表示法:邻接矩阵

class Vertex:
    def __init__(self, node):
        self.id = node
        # Mark all nodes unvisited        
        self.visited = False  

    def addNeighbor(self, neighbor, G):
        G.addEdge(self.id, neighbor)

    def getConnections(self, G):
        return G.adjMatrix[self.id]

    def getVertexID(self):
        return self.id

    def setVertexID(self, id):
        self.id = id

    def setVisited(self):
        self.visited = True

    def __str__(self):
        return str(self.id)

class Graph:
    def __init__(self, numVertices=10, directed=False):
        self.adjMatrix = [[None] * numVertices for _ in range(numVertices)]
        self.numVertices = numVertices
        self.vertices = []
        self.directed = directed
        for i in range(0, numVertices):
            newVertex = Vertex(i)
            self.vertices.append(newVertex)

    def addVertex(self, vtx, id):  #增加点，这个function没有扩展功能
        if 0 <= vtx < self.numVertices:
            self.vertices[vtx].setVertexID(id)

    def getVertex(self, n):
        for vertxin in range(0, self.numVertices):
            if n == self.vertices[vertxin].getVertexID():
                return vertxin
        return None

    def addEdge(self, frm, to, cost=0): #返回全部连线/航线
        #print("from",frm, self.getVertex(frm))
        #print("to",to, self.getVertex(to))
        if self.getVertex(frm) is not None and self.getVertex(to) is not None:
            self.adjMatrix[self.getVertex(frm)][self.getVertex(to)] = cost
            if not self.directed:
                # For directed graph do not add this
                self.adjMatrix[self.getVertex(to)][self.getVertex(frm)] = cost  

    def getVertices(self):
        vertices = []
        for vertxin in range(0, self.numVertices):
            vertices.append(self.vertices[vertxin].getVertexID())
        return vertices

    def printMatrix(self):
        for u in range(0, self.numVertices):
            row = []
            for v in range(0, self.numVertices):
                row.append(str(self.adjMatrix[u][v]) if self.adjMatrix[u][v] is not None else '/')
            print(row)

    def getEdges(self):
        edges = []
        for v in range(0, self.numVertices):
            for u in range(0, self.numVertices):
                if self.adjMatrix[u][v] is not None:
                    vid = self.vertices[v].getVertexID()
                    wid = self.vertices[u].getVertexID()
                    edges.append((vid, wid, self.adjMatrix[u][v]))
        return edges
    
    def getNeighbors(self, n):
        neighbors = []
        for vertxin in range(0, self.numVertices):
            if n == self.vertices[vertxin].getVertexID():
                for neighbor in range(0, self.numVertices):
                    if (self.adjMatrix[vertxin][neighbor] is not None):
                        neighbors.append(self.vertices[neighbor].getVertexID())
        return neighbors
    
    def isConnected(self, u, v):
        uidx = self.getVertex(u) 
        vidx = self.getVertex(v)
        return self.adjMatrix[uidx][vidx] is not None
    
    def get2Hops(self, u): #转一次机可以到达哪里
        neighbors = self.getNeighbors(u)
        print(neighbors)
        hopset = set()
        for v in neighbors:
            hops = self.getNeighbors(v)
            hopset |= set(hops)
        return list(hopset)

图表示法:邻接表

用邻接矩阵来表示，每一行表示一个节点与其他所有节点是否相连，但对于邻接表来说，一行只代表和他相连的节点：

可见邻接表在空间上是更省资源的。 
邻接表适合表示稀疏图，邻接矩阵适合表示稠密图。

import sys
class Vertex:
    def __init__(self, node):
        self.id = node
        self.adjacent = {}
        #为所有节点设置距离无穷大
        self.distance = sys.maxsize
        # 标记未访问的所有节点     
        self.visited = False  
        # Predecessor
        self.previous = None

    def addNeighbor(self, neighbor, weight=0):
        self.adjacent[neighbor] = weight

    # returns a list 
    def getConnections(self): # neighbor keys
        return self.adjacent.keys()  

    def getVertexID(self):
        return self.id

    def getWeight(self, neighbor):
        return self.adjacent[neighbor]

    def setDistance(self, dist):
        self.distance = dist

    def getDistance(self):
        return self.distance

    def setPrevious(self, prev):
        self.previous = prev

    def setVisited(self):
        self.visited = True

    def __str__(self):
        return str(self.id) + ' adjacent: ' + str([x.id for x in self.adjacent])
    
    def __lt__(self, other):
        return self.distance < other.distance and self.id < other.id    

class Graph:
    def __init__(self, directed=False):
        # key is string, vertex id
        # value is Vertex
        self.vertDictionary = {}
        self.numVertices = 0
        self.directed = directed
        
    def __iter__(self):
        return iter(self.vertDictionary.values())

    def isDirected(self):
        return self.directed
    
    def vectexCount(self):
        return self.numVertices

    def addVertex(self, node):
        self.numVertices = self.numVertices + 1
        newVertex = Vertex(node)
        self.vertDictionary[node] = newVertex
        return newVertex

    def getVertex(self, n):
        if n in self.vertDictionary:
            return self.vertDictionary[n]
        else:
            return None

    def addEdge(self, frm, to, cost=0):
        if frm not in self.vertDictionary:
            self.addVertex(frm)
        if to not in self.vertDictionary:
            self.addVertex(to)

        self.vertDictionary[frm].addNeighbor(self.vertDictionary[to], cost)
        if not self.directed:
            # For directed graph do not add this
            self.vertDictionary[to].addNeighbor(self.vertDictionary[frm], cost)

    def getVertices(self):
        return self.vertDictionary.keys()

    def setPrevious(self, current):
        self.previous = current

    def getPrevious(self, current):
        return self.previous

    def getEdges(self):
        edges = []
        for key, currentVert in self.vertDictionary.items():
            for nbr in currentVert.getConnections():
                currentVertID = currentVert.getVertexID()
                nbrID = nbr.getVertexID()
                edges.append((currentVertID, nbrID, currentVert.getWeight(nbr))) # tuple
        return edges
    
    def getNeighbors(self, v):
        vertex = self.vertDictionary[v]
        return vertex.getConnections()

学习资料参考：图论算法初步、python算法图论