Question

我以前习惯使用dicts来表示python中的图形，但是我遇到了大图和复杂计算的严重性能问题，所以我想我应该跨越使用邻接矩阵来绕过哈希表的开销。我的问题是，如果我有一个形式g：{node：{vertex：weight。。。 }。。。什么是将其转换为基于列表的邻接矩阵的最有效方法？

Answer 1

可能不是最有效的，但是在列表基础上将格式转换为邻接矩阵的简单方法可能如下所示：

g = {1:{2:.5, 3:.2}, 2:{4:.7}, 4:{5:.6, 3:.3}}
hubs = g.items() # list of nodes and outgoing vertices
size=max(map(lambda hub: max(hub[0], max(hub[1].keys())), hubs))+1 # matrix dimension is highest known node index + 1
matrix=[[None]*size for row in range(size)] # set up a matrix of the appropriate size

for node, vertices in hubs: # loop through every node in dictionary
    for vertice, weight in vertices.items(): # loop through vertices
        matrix[vertice][node] = weight # define adjacency of both nodes by assigning the vertice's weight

这适用于有向图，假设节点仅由其从零开始的索引表示。这是一个可视化，以及在此示例中处理的图形的结果矩阵：

     0    1    2    3    4    5
   ------------------------------
0 |                              
1 |                              
2 |      0.5                     
3 |      0.2            0.3      
4 |           0.7                
5 |                     0.6

如果您的图表实际上是无向的，我可以想到一些优化的机会。如果字典将每个节点都包含为列出其所有顶点的键，例如{1:{2:.2, 3:.3}, 2:{1:.2}, 3:{1:.3}}，则可以在遍历之前对相应的列表进行排序并限制内部循环：

hubs = sorted(g.items())
for node, vertices in hubs:
    for vertice, weight in reversed(sorted(vertices.items())):
        if vertice >= node: 
            matrix[vertice][node] = weight
            matrix[node][vertice] = weight
        else: # do only care about vertices that haven't been saved before,
            break # continue with next node when the current one won't introduce any more vertices

使用binary search可以提高效率。由于生成的矩阵显然是镜像对称的，你也可以更进一步，只存储它的一半。最简单的方法是在垂直轴上翻转它：

# unlike the one before, this sample doesn't rely on the dictionary containing every vertice twice
matrix=[[None]*size for row in range(size)]  
for node, vertices in hubs:
    for vertice, weight in vertices.items():
        matrix[vertice][size-node-1] = weight

由于矩阵的一半被截断，因此不是每个节点(u,v)之间的顶点查找都能正常工作，因此必须确保列的索引大于行的索引。要查找的单元格：

u,v = sorted((u,v))
weight = matrix[v][u]

祝你好运！

Answer 2

要在邻接列表中实现，您可以创建两个类，一个用于存储有关顶点的信息。

# Vertex, which will represent each vertex in the graph.Each Vertex uses a dictionary
# to keep track of the vertices to which it is connected, and the weight of each edge. 
class Vertex:

# Initialze a object of this class
# we use double underscore 
def __init__(self, key):
    # we identify the vertex with its key
    self.id = key
    # this stores the info about the various connections any object 
    # (vertex) of this class has using a dictionary which is called connectedTo.
    # initially its not connected to any other node so,
    self.connectedTo={}

# Add the information about connection between vertexes into the dictionary connectedTo
def addNeighbor(self,neighbor,weight=0):
    # neighbor is another vertex we update the connectedTo dictionary ( Vertex:weight )
    # with the information of this new Edge, the key is the vertex and
    # the edge's weight is its value. This is the new element in the dictionary
    self.connectedTo[neighbor] = weight

# Return a string containing a nicely printable representation of an object.
def __str__(self):
    return str(self.id) + ' connectedTo: ' + str([x.id for x in self.connectedTo])

# Return the vertex's self is connected to in a List
def getConnections(self):
    return self.connectedTo.keys()

# Return the id with which we identify the vertex, its name you could say
def getId(self):
    return self.id

# Return the value (weight) of the edge (or arc) between self and nbr (two vertices)
def getWeight(self,nbr):
    return self.connectedTo[nbr]

从评论中可以看出，每个顶点都存储一个'key'，用于识别它，它有一个字典'connectedTo'，它保存来自这个顶点的所有连接的密钥 - 权重对。连接顶点的键和边缘的重量。

现在我们可以使用Graph类来存储这样一个顶点的集合，这可以像这样实现，

# The Graph class contains a dictionary that maps vertex keys to vertex objects (vertlist) and a    count of the number of vertices in the graph
class Graph:

def __init__(self):
    self.vertList = {}
    self.numVertices = 0


# Returns a vertex which was added to the graph with given key
def addVertex(self,key):
    self.numVertices = self.numVertices + 1
    # create a vertex object
    newVertex = Vertex(key)
    # set its key
    self.vertList[key] = newVertex
    return newVertex

# Return the vertex object corresponding to the key - n
def getVertex(self,n):
    if n in self.vertList:
        return self.vertList[n]
    else:
        return None

# Returns boolean - checks if graph contains a vertex with key n
def __contains__(self,n):
    return n in self.vertList

# Add's an edge to the graph using addNeighbor method of Vertex
def addEdge(self,f,t,cost=0):
    # check if the 2 vertices involved in this edge exists inside
    # the graph if not they are added to the graph
    # nv is the Vertex object which is part of the graph
    # and has key of 'f' and 't' respectively, cost is the edge weight
    if f not in self.vertList:
        nv = self.addVertex(f)
    if t not in self.vertList:
        nv = self.addVertex(t)
    # self.vertList[f] gets the vertex with f as key, we call this Vertex
    # object's addNeighbor with both the weight and self.vertList[t] (the vertice with t as key) 
    self.vertList[f].addNeighbor(self.vertList[t], cost)

# Return the list of all key's corresponding to the vertex's in the graph
def getVertices(self):
    return self.vertList.keys()

# Returns an iterator object, which contains all the Vertex's
def __iter__(self):
    return iter(self.vertList.values())

这里，我们有图表类，它保存'numVertices'中的顶点数并且具有字典'vertList'，其中包含键和Vertex（我们刚刚创建的类）对象图表。我们可以通过调用

来创建图表并进行设置

# Now lets make the graph
the_graph=Graph()

print "enter the number of nodes in the graph"
no_nodes=int(raw_input())

# setup the nodes
for i in range(no_nodes):
        print "enter the "+str(i+1)+" Node's key"
        the_graph.addVertex(raw_input())

print "enter the number of edges in the graph"
no_edges=int(raw_input())
print "enter the maximum weight possible for any of edges in the graph"
max_weight=int(raw_input())


# setup the edges
for i in range(no_edges):
    print "For the "+str(i+1)+" Edge, "
    print "of the 2 nodes involved in this edge \nenter the first Node's key"
    node1_key=raw_input()
    print "\nenter the second Node's key"
    node2_key=raw_input()
    print "\nenter the cost (or weight) of this edge (or arc) - an integer"
    cost=int(raw_input())
    # add the edge with this info
    the_graph.addEdge(node1_key,node2_key,cost)

如果您想要无向图，请添加此行the_graph.addEdge(node2_key,node1_key,cost) 因此，连接将不是作为连接到b而是连接到b和b连接到a。还要注意上面两个类实现的缩进，它可能是不正确的。

在Python中从Dicts创建邻接列表

2 个答案: