使用 Tarjan 算法枚举图中的循环

Question

我正在尝试使用 Tarjan 的算法确定有向图中的循环，该算法在他 1972 年 9 月的研究论文“有向图的基本电路的枚举”中提出。

我使用 Python 来编写算法，并使用邻接表来跟踪节点之间的关系。

所以在下面的“G”中，节点0指向节点1，节点1指向节点4,6,7...等

G = [[1], [4, 6, 7], [4, 6, 7], [4, 6, 7], [2, 3], [2, 3], [5, 8], [5, 8], [], []]
N = len(G)

points = []
marked_stack = []
marked = [False for x in xrange(0,N)]

g = None
def tarjan(s, v, f):
    global g
    points.append(v)
    marked_stack.append(v)
    marked[v] = True

    for w in G[v]:
        if w < s:            
            G[v].pop(G[v].index(w))
        elif w == s:
            print points
            f = True
        elif marked[w] == False:
            if f == g and f == False:
                f = False
            else:
                f = True
            tarjan(s, w, g)

    g = f
    
    if f == True:
        u = marked_stack.pop()
        while (u != v):
            marked[u] = False
            u = marked_stack.pop()
        #v is now deleted from mark stacked, and has been called u
        #unmark v
        marked[u] = False
    points.pop(points.index(v))

for i in xrange(0,N):
    marked[i] = False

for i in xrange(0,N):
    points = []
    tarjan(i,i, False)
    while (len(marked_stack) > 0):
        u = marked_stack.pop()
        marked[u] = False

Tarjan 的算法检测以下循环：

[2, 4]

[2, 4, 3, 6, 5]

[2, 4, 3, 7, 5]

[2, 6, 5]

[2, 6, 5, 3, 4]

[2, 7, 5]

[2, 7, 5, 3, 4]

[3,7,5]

我也完成了 Tiernan 的算法，它（正确地）找到了 2 个 extra 循环：

[3,4]

[3,6,5]

如果能帮助我找出 Tarjan 跳过这两个周期的原因，我将不胜感激。可能是我的代码有问题？

Answer 1

在这些行中：

for w in G[v]:
    if w < s:            
        G[v].pop(G[v].index(w))

您正在遍历一个列表并从中弹出元素。这会阻止迭代按预期工作。

如果您更改代码以制作列表的副本，则会产生额外的循环：

for w in G[v][:]:

Answer 2

我有点好奇，因为我没有看到使用低链接值或动态分配索引，因为我认为 Tarjan 的查找 SCC 的算法就是这种情况。这是不同的算法还是修改？

我重命名了一些变量以使这更容易理解：

G = [
    [1],
    [4, 6, 7],
    [4, 6, 7],
    [4, 6, 7],
    [2, 3],
    [2, 3],
    [5, 8],
    [5, 8],
    [],
    [],
]
# %%


def entry_tarjan(G_):

    G = G_.copy()

    marked = [False] * len(G_)
    cycles = []
    point_stack = []
    marked_stack = []

    def tarjan(src, v):
        nonlocal cycles, marked, G
        cycle_found = False
        point_stack.append(v)
        marked_stack.append(v)
        marked[v] = True

        for nei in G[v]:
            if nei < src:
                # prevent prior traversals
                G[nei] = []

            elif nei == src:
                # neighbor is source => cycle
                cycles.append(point_stack.copy())
                cycle_found = True

            elif marked[nei] == False:
                # dfs continues
                cycle_in_nei = tarjan(src, nei)
                cycle_found = cycle_found or cycle_in_nei

        if cycle_found == True:
            # adjust marked to current vertex
            while marked_stack[len(marked_stack) - 1] != v:
                u = marked_stack.pop()
                marked[u] = False
            marked_stack.pop()
            marked[v] = False

        point_stack.pop()
        return cycle_found

    for i in range(len(G)):
        # start at each vertex
        tarjan(i, i)

        while marked_stack:
            u = marked_stack.pop()
            marked[u] = False

    return cycles


print(entry_tarjan(G))

Answer 3

下面的代码运行没有错误，并且有预期的输出。

G = [[1], [4, 6, 7], [4, 6, 7], [4, 6, 7], [2, 3], [2, 3], [5, 8], [5, 8], [], []]
N = len(G)

points = []
marked_stack = []
marked = [False for x in xrange(0,N)]

g = None
def tarjan(s, v, f):
    global g
    points.append(v)
    marked_stack.append(v)
    marked[v] = True

    for w in G[v][:]:
        if w < s:            
            G[v].pop(G[v].index(w))
        elif w == s:
            print points
            f = True
        elif marked[w] == False:
            tarjan(s, w, g)

    g = f
    
    if f == True:
        u = marked_stack.pop()
        while (u != v):
            marked[u] = False
            u = marked_stack.pop()
        #v is now deleted from mark stacked, and has been called u
        #unmark v
        marked[u] = False
    points.pop(points.index(v))

for i in xrange(0,N):
    marked[i] = False

for i in xrange(0,N):
    points = []
    tarjan(i,i, False)
    while (len(marked_stack) > 0):
        u = marked_stack.pop()
        marked[u] = False

输出：

[2, 4]
[2, 4, 3, 6, 5]
[2, 4, 3, 7, 5]
[2, 6, 5]
[2, 6, 5, 3, 4]
[2, 7, 5]
[2, 7, 5, 3, 4]
[3, 4]
[3, 6, 5]
[3, 7, 5]