Fleury算法简介

Fleury算法是一种用于寻找欧拉路径或欧拉电路的算法。它是由欧拉于1736年所提出的，并且被证明是正确的。

Fleury算法的主要思想是通过不断删除桥边（会导致分离）或割边（会导致不连通），来求解欧拉路径或电路。

实现步骤

1. 选择一个起点
2. 从起点开始走，将所经过的边都标记为已访问
3. 每次可以选择任意一条未访问过的边，前提是这条边不是桥边或割边
4. 重复步骤3，直到无法再选择未访问过的边
5. 如果走过所有的边，则得到了一个欧拉电路或路径；否则无解。

代码实现

def fleury_algorithm(graph, start_node):
    """
    A function that implements the Fleury algorithm.

    Parameters:
    graph: A dictionary representing the graph. The keys are the nodes, and the values are lists of adjacent nodes.
    start_node: A string representing the starting node.

    Returns: A list of nodes, representing the eulerian path or circuit.
    """
    # Initialize variables
    path = []
    visited_edges = set()
    current_node = start_node

    # While there are still edges to visit
    while bool(graph[current_node]):
        # Select an unvisited edge
        unvisited_edges = [edge for edge in graph[current_node] if edge not in visited_edges]

        # If all edges have been visited, end the algorithm
        if not unvisited_edges:
            break

        # If there is only one unvisited edge, take it
        elif len(unvisited_edges) == 1:
            next_node = unvisited_edges[0]

        # Find a non-bridge/non-cut edge
        else:
            for edge in unvisited_edges:
                graph_copy = graph.copy()
                graph_copy[current_node].remove(edge)
                graph_copy[edge].remove(current_node)
                if not is_bridge(graph_copy, current_node, edge):
                    next_node = edge
                    break

        # Add the edge to the visited edges set
        visited_edges.add(next_node)

        # Add the new node to the path
        if next_node[0] == current_node:
            path.append(next_node[1])
            current_node = next_node[1]
        else:
            path.append(next_node[0])
            current_node = next_node[0]

    # If there are no more edges to visit, end the algorithm
    return path

def is_bridge(graph, node1, node2):
    """
    A helper function that determines whether an edge is a bridge.

    Parameters:
    graph: A dictionary representing the graph. The keys are the nodes, and the values are lists of adjacent nodes.
    node1: A string representing node1.
    node2: A string representing node2.

    Returns: True if the edge is a bridge, False otherwise.
    """
    # Get the number of connected components before removing the edge
    num_components_before = count_connected_components(graph)

    # Remove the edge
    graph_copy = graph.copy()
    graph_copy[node1].remove(node2)
    graph_copy[node2].remove(node1)

    # Get the number of connected components after removing the edge
    num_components_after = count_connected_components(graph_copy)

    # If the number of connected components increased, the edge is a bridge
    return num_components_after > num_components_before

def count_connected_components(graph):
    """
    A helper function that counts the number of connected components in a graph.

    Parameters:
    graph: A dictionary representing the graph. The keys are the nodes, and the values are lists of adjacent nodes.

    Returns: The number of connected components in the graph.
    """
    visited = set()
    num_components = 0

    for node in graph:
        if node not in visited:
            num_components += 1
            dfs(graph, node, visited)

    return num_components

def dfs(graph, node, visited):
    """
    A helper function that performs a depth-first search of a graph.

    Parameters:
    graph: A dictionary representing the graph. The keys are the nodes, and the values are lists of adjacent nodes.
    node: A string representing the starting node.
    visited: A set representing the nodes that have already been visited.

    Returns: None.
    """
    visited.add(node)
    for neighbor in graph[node]:
        if neighbor not in visited:
            dfs(graph, neighbor, visited)

总结

Fleury算法是一种简单而有效的欧拉路径/电路算法。它的核心思想是在未访问过的边中选择一个非割边/非桥边来行走，通过不断删除桥边或割边，最终得到欧拉路径/电路。