python數據結構之圖論

本篇學習筆記內容為圖的各項性質、圖的表示方法、圖ADT的python實現

圖（Graph）

是數據結構和算法學中最強大的框架之一（或許沒有之一）。圖幾乎可以用來表現所有類型的結構或系統，從交通網絡到通信網絡，從下棋游戲到最優流程，從任務分配到人際交互網絡，圖都有廣闊的用武之地。

我們會把圖視為一種由“頂點”組成的抽象網絡，網絡中的各頂點可以通過“邊”實現彼此的連接，表示兩頂點有關聯。我們要知道最基礎最基本的2個概念，頂點（vertex）和邊（edge）。

圖可以分為有向圖和無向圖，一般用 G=(V,E)來表示圖。經常用鄰接矩陣或者鄰接表來描述一副圖。

首先是鏈表、樹與圖的對比圖：

圓為頂點、線為邊

圖的術語

圖 G 是頂點V 和邊 E的集合 

兩個頂點之間：邊 

如果頂點 x 和 y 共享邊,則它們相鄰,或者它們是相鄰的 

無向圖 ：無向圖中的一個邊可以在任一方向上遍歷 

路徑:：通過邊連接的頂點序列 

周期：第一個和最后一個頂點相同的路徑 

入度:：頂點的度數V是以V為端點的邊數 

出度： 頂點的出度v是以v為起點的邊的數量 

度：頂點的度數是其入度和出度的總和

圖的ADT

數據成員 :

頂點 （vertex）

邊緣 (edge)

操作 :

有多少頂點？

有多少個邊緣？

添加一個新的頂點 

添加一個新的邊緣 

獲取所有鄰居？ （進出）

U，V連接嗎？

反轉所有邊緣？

獲取2跳鄰居

圖表示法:鄰接矩陣

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):  #增加點，這個function沒有擴展功能
        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): #返回全部連線/航線
        #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): #轉一次機可以到達哪里
        neighbors = self.getNeighbors(u)
        print(neighbors)
        hopset = set()
        for v in neighbors:
            hops = self.getNeighbors(v)
            hopset |= set(hops)
        return list(hopset)

圖表示法:鄰接表

用鄰接矩陣來表示，每一行表示一個節點與其他所有節點是否相連，但對於鄰接表來說，一行只代表和他相連的節點：

可見鄰接表在空間上是更省資源的。 
鄰接表適合表示稀疏圖，鄰接矩陣適合表示稠密圖。

import sys
class Vertex:
    def __init__(self, node):
        self.id = node
        self.adjacent = {}
        #為所有節點設置距離無窮大
        self.distance = sys.maxsize
        # 標記未訪問的所有節點     
        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()

 

學習資料參考：圖論算法初步、python算法圖論