【算法——Python实现】有权图求单源最短路径Dijkstra算法

class Edge(object):
    """边"""
    def __init__(self, a, b, weight):
        self.a = a # 第一个顶点
        self.b = b # 第二个顶点
        self.weight = weight # 权值

    def v(self):
        return self.a

    def w(self):
        return self.b

    def wt(self):
        return self.weight

    def other(self, x):
        # 返回x顶点连接的另一个顶点
        if x == self.a or x == self.b:
            if x == self.a:
                return self.b
            else:
                return self.a

    def __lt__(self, other):
        # 小于号重载
        return self.weight < other.wt()

    def __le__(self, other):
        # 小于等于号重载
        return self.weight <= other.wt()

    def __gt__(self, other):
        # 大于号重载
        return self.weight > other.wt()

    def __ge__(self, other):
        # 大于等于号重载
        return self.weight >= other.wt()

    def __eq__(self, other):
        # ==号重载
        return self.weight == other.wt()


class DenseGraph(object):
    """有权稠密图 - 邻接矩阵"""
    def __init__(self, n, directed):
        self.n = n  # 图中的点数
        self.m = 0  # 图中的边数
        self.directed = directed  # bool值，表示是否为有向图
        self.g = [[None for _ in range(n)] for _ in range(n)]  # 矩阵初始化都为None的二维矩阵

    def V(self):
        # 返回图中点数
        return self.n

    def E(self):
        # 返回图中边数
        return self.m

    def addEdge(self, v, w, weight):
        # v和w中增加一条边，v和w都是[0,n-1]区间
        if v >= 0 and v < n and w >= 0 and w < n:
            if self.hasEdge(v, w):
                self.m -= 1
            self.g[v][w] = Edge(v, w, weight)
            if not self.directed:
                self.g[w][v] = Edge(w, v, weight)
            self.m += 1

    def hasEdge(self, v, w):
        # v和w之间是否有边，v和w都是[0,n-1]区间
        if v >= 0 and v < n and w >= 0 and w < n:
            return self.g[v][w] != None

    class adjIterator(object):
        """相邻节点迭代器"""
        def __init__(self, graph, v):
            self.G = graph  # 需要遍历的图
            self.v = v  # 遍历v节点相邻的边
            self.index = 0  # 遍历的索引

        def __iter__(self):
            return self

        def next(self):
            while self.index < self.G.V():
                # 当索引小于节点数量时遍历，否则为遍历完成，停止迭代
                if self.G.g[self.v][self.index]:
                    r = self.G.g[self.v][self.index]
                    self.index += 1
                    return r
                self.index += 1
            raise StopIteration()


class SparseGraph(object):
    """有权稀疏图- 邻接表"""
    def __init__(self, n, directed):
        self.n = n  # 图中的点数
        self.m = 0  # 图中的边数
        self.directed = directed  # bool值，表示是否为有向图
        self.g = [[] for _ in range(n)]  # 矩阵初始化都为空的二维矩阵

    def V(self):
        # 返回图中点数
        return self.n

    def E(self):
        # 返回图中边数
        return self.m

    def addEdge(self, v, w, weight):
        # v和w中增加一条边，v和w都是[0,n-1]区间
        if v >= 0 and v < n and w >= 0 and w < n:
            # 考虑到平行边会让时间复杂度变为最差为O(n)
            # if self.hasEdge(v, w):
            #   return None
            self.g[v].append(Edge(v, w, weight))
            if v != w and not self.directed:
                self.g[w].append(Edge(w, v, weight))
            self.m += 1

    def hasEdge(self, v, w):
        # v和w之间是否有边，v和w都是[0,n-1]区间
        # 时间复杂度最差为O(n)
        if v >= 0 and v < n and w >= 0 and w < n:
            for i in self.g[v]:
                if i.other(v) == w:
                    return True
            return False

    class adjIterator(object):
        """相邻节点迭代器"""
        def __init__(self, graph, v):
            self.G = graph  # 需要遍历的图
            self.v = v  # 遍历v节点相邻的边
            self.index = 0  # 遍历的索引

        def __iter__(self):
            return self

        def next(self):
            if len(self.G.g[self.v]):
                # v有相邻节点才遍历
                if self.index < len(self.G.g[self.v]):
                    r = self.G.g[self.v][self.index]
                    self.index += 1
                    return r
                else:
                    raise StopIteration()
            else:
                raise StopIteration()


class ReadGraph(object):
    """读取文件中的图"""
    def __init__(self, graph, filename):
        with open(filename, 'r') as f:
            line = f.readline()
            line = line.strip('\n')
            line = line.split()
            v = int(line[0])
            e = int(line[1])
            if v == graph.V():
                lines = f.readlines()
                for i in lines:
                    a, b, w = self.stringstream(i)
                    if a >= 0 and a < v and b >=0 and b < v:
                        graph.addEdge(a, b, w)

    def stringstream(self, text):
        result = text.strip('\n')
        result = result.split()
        a, b, w = result
        return int(a), int(b), int(w)


class IndexMinHeap(object):
    """
    最小反向索引堆，出堆不删除data，只删除indexs
    """
    def __init__(self, n):
        self.capacity = n # 堆的最大容量
        self.data = [-1 for _ in range(n)]  # 创建堆
        self.indexs = [] # 创建索引堆
        self.reverse = [-1 for _ in range(n)] # 创建反向索引
        self.count = 0  # 元素数量

    def size(self):
        return self.count

    def isEmpty(self):
        return self.count == 0

    def contain(self, i):
        # 最小堆中是否包含i索引的元素
        return self.reverse[i] != -1

    def insert(self, i, item):
        # 插入元素入堆
        self.data[i] = item
        self.indexs.append(i)
        self.reverse[i] = self.count
        self.count += 1
        self.shiftup(self.count)

    def shiftup(self, count):
        # 将插入的元素放到合适位置，保持最小堆
        while count > 1 and self.data[self.indexs[(count/2)-1]] > self.data[self.indexs[count-1]]:
            self.indexs[(count/2)-1], self.indexs[count-1] = self.indexs[count-1], self.indexs[(count/2)-1]
            self.reverse[self.indexs[(count/2)-1]] = (count/2)-1
            self.reverse[self.indexs[count-1]] = count-1
            count /= 2

    def extractMin(self):
        # 出堆
        if self.count > 0:
            ret = self.data[self.indexs[0]]
            self.indexs[0], self.indexs[self.count-1] = self.indexs[self.count-1], self.indexs[0]
            self.reverse[self.indexs[0]] = 0
            self.reverse[self.indexs[self.count-1]] = self.count-1
            # for i in range(self.indexs[self.count-1]+1, self.count):
            #   self.indexs[self.reverse[i]] -= 1
            self.reverse[self.indexs[self.count-1]] = -1
            # self.data[self.indexs[self.count-1]] = -1
            self.indexs.pop(self.count-1)
            self.count -= 1
            self.shiftDown(1)
            return ret

    def extractMinIndex(self):
        # 出堆返回索引
        if self.count > 0:
            ret = self.indexs[0]
            self.indexs[0], self.indexs[self.count-1] = self.indexs[self.count-1], self.indexs[0]
            self.reverse[self.indexs[0]] = 0
            self.reverse[self.indexs[self.count-1]] = self.count-1
            # for i in range(self.indexs[self.count-1]+1, self.count):
            #   self.indexs[self.reverse[i]] -= 1
            self.reverse[self.indexs[self.count-1]] = -1
            # self.data[self.indexs[self.count-1]] = -1
            self.indexs.pop(self.count-1)
            self.count -= 1
            self.shiftDown(1)
            return ret

    def shiftDown(self, count):
        # 将堆的索引位置元素向下移动到合适位置，保持最小堆
        while 2 * count <= self.count :
            # 证明有孩子
            j = 2 * count
            if j + 1 <= self.count:
                # 证明有右孩子
                if self.data[self.indexs[j]] < self.data[self.indexs[j-1]]:
                    # 右孩子数值比左孩子数值小
                    j += 1
            if self.data[self.indexs[count-1]] <= self.data[self.indexs[j-1]]:
                # 堆的索引位置已经小于两个孩子节点，不需要交换了
                break
            self.indexs[count-1], self.indexs[j-1] = self.indexs[j-1], self.indexs[count-1]
            self.reverse[self.indexs[count-1]] = count-1
            self.reverse[self.indexs[j-1]] =j-1
            count = j

    def getItem(self, i):
        # 根据索引获取数值
        if i >=0 and i <= self.count-1:
            return self.data[i]
        else:
            return None

    def change(self, i, newItem):
        # 改变i索引位置的数值
        if i >=0:
            self.data[i] = newItem
            j = self.reverse[i]
            self.shiftup(j+1)
            self.shiftDown(j+1)


class Dijkstra(object):
    """Dijkstra算法(不能有负权边)求从s点到所有节点最短路径，从原点s开始遍历相邻节点，选取权值最小的边连接的顶点，依次遍历选取最小边并做松弛操作"""
    def __init__(self, graph, s):
        self.G = graph # 图
        self.ipq = IndexMinHeap(self.G.V()) # 最小索引堆，存权值
        self.s = s # 原点，表示从此节点到所有节点求最短路径
        self.marked = [False for _ in range(self.G.V())]  # 用于标记已经找到到节点的最短路径，初始都为False
        self.distTo = [0 for _ in range(self.G.V())]  # 从原点到每个节点的权值，初始都为0
        self.fromed = [None for _ in range(self.G.V())] # 记录此节点来自哪个节点的连接，存储边。None表示未被访问过

        # Dijkstra
        self.marked[s] = True
        self.ipq.insert(s, self.distTo[s])
        while not self.ipq.isEmpty():
            v = self.ipq.extractMinIndex()
            self.marked[v] = True  # distTo[v]就是s到v的最短距离，v确认找到最短路径
            # 松弛操作
            adj = self.G.adjIterator(self.G, v)
            for e in adj:
                w = e.other(v)
                # 如果w还没有确定找到最短路径
                if not self.marked[w]:
                    # 如果w还未被访问过，或经过v的最短路径再到w节点的权值小于w之前被访问时记录的路径，则用经过v再到w的路径替代
                    if not self.fromed[w] or self.distTo[v] + e.wt() <  self.distTo[w]:
                        self.distTo[w] = self.distTo[v] + e.wt()
                        self.fromed[w] = e
                        if self.ipq.contain(w):
                            self.ipq.change(w, self.distTo[w])
                        else:
                            self.ipq.insert(w, self.distTo[w])

    def shortestPathTo(self, w):
        # 从原点到w节点的权值
        return self.distTo[w]

    def hasPathTo(self, w):
        # 从原点到w是否有路径
        return self.marked[w]

    def shortestPath(self, w):
        # 从原点到w的最短路径
        s = []
        e = self.fromed[w]
        while e:
            s.append(e)
            e = self.fromed[e.v()]
        s.reverse()
        return s

    def showPath(self, w):
        # 输出路径
        s = self.shortestPath(w)
        i = 0
        while i < len(s):
            print s[i].v()
            if i == len(s) - 1:
                print s[i].w()
            i += 1