BFS查找所有固定长度的循环

Question

I have to find all cycles of length 3 in a given graph. I've implemented it using BFS, but so far it works only for relatively small inputs. It still works for bigger ones and gives correct answer, but the time it takes to find an answer is extremely high. Is there any way to improve the following code to make it more efficient?

num_res = 0
adj_list = []
cycles_list = []


def bfs_cycles(start):
    queue = [(start, [start])]
    depth = 0
    while queue and depth <= 3:
        (vertex, path) = queue.pop(0)
        current_set = set(adj_list[vertex]) - set(path)
        if start in set(adj_list[vertex]):
            current_set = current_set.union([start])
        depth = len(path)
        for node in current_set:
            if node == start:
                if depth == 3 and sorted(path) not in cycles_list:
                    cycles_list.append(sorted(path))
                    yield path + [node]
            else:
                queue.append((node, path + [node]))


if __name__ == "__main__":
    num_towns, num_pairs = [int(x) for x in input().split()]
    adj_list = [[] for x in range(num_towns)]
    adj_matrix = [[0 for x in range(num_towns)] for x in range(num_towns)]

    # EDGE LIST TO ADJACENCY LIST
    for i in range(num_pairs):
        cur_start, cur_end = [int(x) for x in input().split()]
        adj_list[cur_start].append(cur_end)
        adj_list[cur_end].append(cur_start)

    num_cycles = 0

    for i in range(num_towns):
        my_list = list(bfs_cycles(i))
        num_cycles += len(my_list)

    print(num_cycles)

Examples of inputs:

6 15
5 4
2 0
3 1
5 1
4 1
5 3
1 0
4 0
4 3
5 2
2 1
3 0
3 2
5 0
4 2

(output: 20; works ok)

52 1051
48 5
41 28
12 4
33 27
12 5
1 0
15 12
50 8
33 8
38 28
26 10
13 7
39 18
31 11
48 19
41 19
40 25
47 45
27 16
46 25
42 6
5 4
51 2
30 21
41 27
26 25
33 11
45 26
16 7
23 15
17 6
45 22
32 6
29 8
36 20
30 1
36 25
41 6
46 4
46 40
18 8
38 1
28 5
43 22
21 11
39 14
31 29
18 9
50 35
32 17
48 27
49 40
16 1
49 47
41 12
30 28
33 14
48 12
37 20
49 20
48 8
48 6
27 17
46 44
31 12
17 9
32 27
14 11
40 23
36 19
38 10
42 2
35 22
26 23
29 23
30 11
11 7
47 12
30 13
38 34
48 11
46 8
42 31
30 4
35 17
50 2
51 1
12 10
44 25
47 17
45 24
25 2
45 11
39 21
39 31
9 6
16 3
10 6
15 11
37 2
23 6
41 40
34 26
45 33
35 23
45 36
11 4
38 7
36 6
10 3
33 12
39 12
41 24
47 8
33 5
44 18
45 8
48 41
44 37
11 3
16 6
21 10
20 0
44 36
29 4
43 33
48 4
46 35
33 6
42 12
45 19
12 8
37 15
43 41
36 11
12 11
50 37
9 7
51 30
36 0
33 17
36 35
50 36
49 37
50 16
46 21
36 22
49 15
46 28
50 27
20 10
23 0
36 29
35 33
42 17
31 16
48 47
48 23
17 2
40 14
10 5
45 7
48 42
39 32
51 4
42 8
38 19
34 10
50 5
51 36
46 26
42 38
20 12
44 32
34 4
49 6
50 45
37 10
45 41
38 11
42 30
21 20
43 23
42 26
33 1
17 7
26 6
16 12
44 16
21 9
36 30
39 24
26 4
47 10
18 7
36 12
26 17
28 13
18 11
23 7
44 4
43 26
26 16
22 21
37 0
36 28
34 5
22 17
41 20
31 8
27 25
12 2
42 11
29 28
39 33
34 12
30 2
22 8
40 15
42 9
28 7
44 41
41 35
44 17
12 7
13 10
23 20
48 38
43 12
32 19
43 30
50 1
10 1
17 12
32 2
26 14
29 12
32 5
7 6
36 16
49 7
31 1
45 17
33 29
28 11
32 0
49 32
42 36
16 4
45 20
21 14
39 15
34 18
13 8
27 15
19 11
37 36
36 14
28 4
36 13
17 11
38 13
35 28
50 10
39 28
40 2
35 8
32 24
47 34
45 27
41 21
21 4
47 27
48 1
35 30
21 5
20 14
27 26
17 1
28 17
43 7
31 6
20 3
34 21
8 2
21 1
32 9
29 1
45 43
50 39
19 15
22 12
48 7
46 18
45 35
50 42
51 17
37 6
24 23
29 3
39 20
51 50
38 6
50 11
38 14
25 24
14 7
45 44
28 14
50 49
42 28
36 7
35 25
13 4
46 1
48 21
51 11
39 11
17 5
31 0
49 36
40 4
37 21
35 1
23 4
43 4
46 36
38 20
37 27
30 0
44 34
49 10
48 14
48 45
38 31
47 29
40 16
51 20
34 17
51 19
24 9
24 5
5 1
15 13
26 2
19 12
50 14
42 7
35 14
46 20
43 28
8 3
38 37
28 1
21 0
51 5
17 16
38 17
34 30
46 12
17 14
50 9
16 13
30 27
45 0
41 16
41 32
48 18
30 8
51 47
11 8
40 13
34 32
23 11
51 28
42 35
36 2
13 11
28 8
15 10
39 35
27 1
50 7
41 23
46 39
38 9
44 10
46 38
6 4
44 27
36 21
35 9
45 30
44 7
37 1
44 28
9 1
32 31
39 16
4 0
44 13
24 0
17 15
15 1
32 8
39 22
42 34
24 6
49 18
36 1
51 42
38 5
14 12
33 3
51 45
24 18
37 32
46 6
44 12
23 10
32 12
50 26
29 20
41 30
6 0
48 31
39 8
21 19
47 6
47 16
18 3
46 27
11 10
36 3
47 2
17 10
43 6
36 8
4 1
14 9
42 1
44 1
46 22
44 23
40 26
30 17
21 17
42 29
45 16
49 45
11 6
35 7
46 42
14 10
26 13
49 44
19 18
26 12
46 2
50 41
43 20
38 24
48 30
34 29
25 19
32 11
46 16
30 25
38 15
50 38
51 23
47 28
14 5
40 12
21 8
47 36
38 32
32 15
28 21
45 10
44 8
34 0
32 14
43 25
32 21
38 2
27 2
24 17
33 31
49 26
22 13
13 1
32 20
43 0
46 0
45 29
40 32
48 44
45 34
29 2
39 27
14 8
26 3
40 19
45 38
40 11
34 6
43 39
40 8
35 0
18 0
47 25
21 18
24 8
18 4
25 14
20 11
18 17
24 14
27 23
47 15
38 21
19 2
6 1
46 11
51 38
6 3
31 17
3 0
13 2
41 1
51 14
19 5
39 2
41 22
16 9
22 3
13 0
42 21
24 16
44 31
51 25
40 33
46 29
47 31
51 35
35 18
43 1
47 22
20 18
48 29
39 23
31 25
32 25
22 10
46 24
32 3
46 13
24 15
34 13
50 18
41 4
41 2
43 27
29 10
30 20
32 7
50 20
42 10
42 24
15 7
48 25
41 39
32 1
40 36
20 7
32 13
27 3
34 7
48 34
47 39
39 36
40 5
19 0
25 20
38 12
27 14
44 3
36 4
37 4
33 28
37 23
34 9
46 45
25 9
30 16
34 14
46 37
28 26
26 22
18 5
16 0
36 27
45 42
38 33
37 22
27 0
44 15
49 42
34 23
29 11
30 12
17 8
48 28
10 4
36 15
44 14
23 19
43 18
27 5
40 1
18 12
34 20
50 23
9 3
35 4
46 15
37 11
27 4
19 3
45 1
47 1
48 17
9 2
39 26
33 10
38 30
45 25
48 24
29 17
37 28
34 31
51 21
43 8
31 4
20 16
39 25
31 13
24 3
50 43
13 9
32 23
40 18
45 40
37 35
47 38
42 13
51 26
43 31
49 23
18 15
15 0
43 9
7 2
48 46
35 11
42 23
47 40
3 1
25 6
46 3
42 19
28 9
15 3
43 3
35 10
42 41
51 46
9 4
46 34
28 0
6 5
45 14
26 11
48 13
33 23
40 9
23 21
18 16
28 12
43 29
35 31
30 14
36 34
49 38
49 22
24 11
23 14
45 13
49 21
48 16
51 10
39 4
50 46
50 48
43 17
31 18
38 23
2 0
41 0
30 19
20 1
29 19
48 32
30 15
40 22
51 12
50 40
24 4
39 10
31 20
7 0
40 17
41 31
37 29
33 32
30 3
40 6
51 15
46 19
31 28
34 22
31 5
33 7
29 14
34 24
44 6
24 2
44 40
35 6
37 18
47 0
43 42
49 30
49 25
19 1
25 3
49 5
40 10
25 21
48 15
35 19
50 6
36 17
44 33
21 13
15 4
36 32
28 6
49 35
47 9
49 46
47 14
25 4
44 29
38 25
23 12
51 41
20 5
39 34
15 6
47 23
21 6
47 11
22 7
41 29
34 2
43 38
6 2
3 2
40 20
40 24
37 16
32 26
49 31
49 16
50 13
31 2
26 1
5 0
19 16
45 32
42 40
16 5
15 8
38 27
12 6
47 4
39 6
31 19
26 9
47 18
42 32
4 2
42 20
46 10
27 6
41 7
49 2
49 28
20 9
46 33
16 11
14 4
34 1
33 2
30 6
47 44
41 8
23 17
33 25
23 5
24 13
33 20
44 35
47 46
47 7
41 25
45 5
28 23
31 15
31 10
39 9
40 7
45 6
43 11
35 26
51 34
44 38
45 3
24 19
51 22
47 42
34 15
37 33
29 9
49 3
14 3
23 2
39 7
46 23
40 31
33 16
44 43
41 36
37 17
43 40
32 18
46 32
26 18
4 3
39 5
44 11
28 20
44 21
41 26
39 38
36 5
7 3
39 0
27 18
26 20
18 2
50 28
37 26
40 27
17 4
50 3
39 30
32 29
50 34
18 1
20 4
36 23
25 15
49 0
45 39
39 1
37 5
23 16
47 20
27 20
38 4
46 43
34 27
15 5
31 23
39 29
46 7
38 35
41 14
45 9
25 22
10 9
35 21
19 14
37 8
47 35
9 0
35 13
21 16
50 32
37 7
19 8
22 5
51 24
51 9
29 0
51 39
44 19
42 5
31 9
40 30
51 37
25 12
26 0
32 16
25 1
41 13
47 43
25 18
35 29
50 44
45 23
44 20
50 47
22 2
45 4
34 19
48 33
34 16
18 10
29 18
37 13
45 2
43 14
48 10
15 2
28 22
29 16
45 15
19 17
35 16
46 9
9 5
35 27
30 5
49 39
32 28
42 3
48 37
43 32
44 30
37 30
14 2
47 32
20 8
18 13
25 5
44 5
29 15
49 11
42 14
30 29
42 27
19 6
51 49
51 13
12 1
40 34
23 13
27 11
51 43
27 24
19 13
26 19
16 10
23 1
46 5
35 15
30 10
48 3
19 9
25 23
16 14
23 3
34 11
27 9
32 30
39 19
50 33
45 21
50 12
13 3
50 15
25 16
49 14
41 17
47 19
43 36
13 12
30 7
49 48
14 0
24 7
49 27
30 26
47 21
14 6
30 22
22 9
29 5
23 22
51 40
42 37
29 6
8 5
51 29
22 4
28 19
21 3
45 12
47 26
43 35
48 43
20 2
24 21
33 22
24 20
41 5
35 3
43 15
43 34
19 10
47 41
49 8
29 21
51 31
43 19
50 17
47 24

(output: 11061; takes around 10 seconds)

Answer 1

A few problems in your code:

• the operation sorted(path) not in cycles_list has O(n) complexity, where n is the size of cycles_list
• queue.pop(0) has O(n) complexity, where n is the size of the queue . You should use the collections.deque structure , not a list here.

As a general note, unless you really need to solve the question using specifically BFS (eg because some asked you to use this method), a simple combination of loops would do the job better. Pseudocode:

 num_loops = 0
 for a in nodes:
   for b in neighbors(a)
     if b > a:
       for c in neighbors(b):
         if c > b and a in neighbors(c):
           num_loops += 1

The b > a and c > b checks are added to count each loop only once.

Answer 2

For a small number of steps like 3, you can just check for each node if you can walk away from and back to the node within 3 steps.

This works reasonably fast:

import fileinput

graph = {}

# Recursive function to find a goal in a number of steps
def count_unique_walks(start, goal, length, visited=[]):
    if length == 0:
        # Out of steps
        return 1 if start == goal else 0
    if start in visited:
        # Already been here
        return 0
    result = 0
    for neighbor in graph[start]:
        if neighbor < start and neighbor != goal:
            # Count only unique cycles
            continue
        result += count_unique_walks(neighbor, goal, length-1, visited+[start])
    return result

# Read input
for line in fileinput.input():
    a, b = map(int, line.split())
    if a not in graph:
        graph[a] = set()
    graph[a].add(b)
    if b not in graph:
        graph[b] = set()
    graph[b].add(a)

# Sum up the cycles of each node
result = 0
for node in graph:
    result += count_unique_walks(node, node, 3)
print result