我找到的唯一一个接近我的问题的答案是这个:
似乎找到一组基本循环并对它们进行异或运算可以解决问题。找到一组基本循环很容易，但我不知道如何将它们组合起来以获取图中的所有循环...
对于一个无向图，标准的方法是寻找所谓的循环基：一组简单的循环，通过组合可以生成所有其他循环。这些不一定是图中的所有简单循环。例如考虑下面的图：
A
/ \
B ----- C
\ /
D
这里有三个简单的循环：A-B-C-A，B-C-D-B和A-B-D-C-A。然而，您可以将其中每2个作为基础，并通过这两个的组合获得第3个。这与有向图有实质区别，因为由于需要观察边缘方向，无法自由地组合循环。
找到无向图的循环基线的标准基线算法如下：建立一个生成树，然后对于不在该树中的每条边，从该边和一些树上的边构建一个循环。这样的循环必须存在，否则该边将成为树的一部分。
例如，上面示例图形的可能生成树之一是：
A
/ \
B C
\
D
不在树中的两个边是B-C和C-D。相应的简单循环是A-B-C-A和A-B-D-C-A。
你还可以构建以下生成树：
A
/
B ----- C
\
D
对于这个生成树而言，简单的循环路径包括A-B-C-A和B-C-D-B。
基本算法可以通过不同的方式进行优化。据我所知，最好的改进方法属于帕顿 (K. Paton, An algorithm for finding a fundamental set of cycles for an undirected linear graph, Comm. ACM 12 (1969), pp. 514-518)。Java的开源实现可在此处找到: http://code.google.com/p/niographs/。
我应该提到如何组合来自循环基的简单循环路径以形成新的简单循环路径。您首先按任意(但在此之后固定)顺序列出图的所有边缘。然后，通过将归属于该循环的边缘的位置放置为1，并将未涉及该循环的边缘的位置放置为0，将循环表示为0和1的序列。然后对序列进行按位异或(XOR)操作。进行XOR的原因是您要排除同时属于两个循环路径的边缘，从而使组合循环路径非简单。您还需要检查两个循环路径具有某些公共边缘，方法是通过检查序列的按位AND是否不全为0。否则，XOR的结果将是2个不相交的循环路径，而不是新的简单循环路径。
以下是上面样例图的示例:
我们首先列出边缘:((AB),(AC),(BC),(BD),(CD))。然后将简单循环路径A-B-C-A、B-D-C-B和A-B-D-C-A表示为(1,1,1,0,0)、(0,0,1,1,1)和(1,1,0,1,1)。例如，我们可以将A-B-C-A与B-D-C-B进行XOR操作，结果为(1, 1, 0, 1, 1)，这正好是A-B-D-C-A。或者我们可以将A-B-C-A和A-B-D-C-A进行XOR操作，结果为(0,0,1,1,1)，这正好是B-D-C-B。
通过检查2个或多个不同基本循环路径的所有可能组合，您可以发现所有简单循环路径。该过程在此处有更详细的描述：http://dspace.mit.edu/bitstream/handle/1721.1/68106/FTL_R_1982_07.pdf的第14页。
为了完整起见，我应该指出似乎可以(且效率低下地)使用用于查找有向图中所有简单循环路径的算法。每个无向图的边缘可以被替换为两条相反方向的有向边缘。然后应该使用针对有向图的算法。每个无向图的边都将有1个“错误”的2节点循环路径需要忽略，每个简单的无向图循环路径都将有一个顺时针和逆时针版本。在我已经引用的链接中，可以找到用于查找有向图中所有循环路径的Java开源实现。
XOR的循环有3个，我们应该如何进行AND位运算？ - Peter LeeAxel，我已经将您的代码翻译成Python。代码行数约为原来的四分之一，且更易读懂。
graph = [[1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [2, 6], [4, 6], [8, 7], [8, 9], [9, 7]]
cycles = []
def main():
global graph
global cycles
for edge in graph:
for node in edge:
findNewCycles([node])
for cy in cycles:
path = [str(node) for node in cy]
s = ",".join(path)
print(s)
def findNewCycles(path):
start_node = path[0]
next_node= None
sub = []
#visit each edge and each node of each edge
for edge in graph:
node1, node2 = edge
if start_node in edge:
if node1 == start_node:
next_node = node2
else:
next_node = node1
if not visited(next_node, path):
# neighbor node not on path yet
sub = [next_node]
sub.extend(path)
# explore extended path
findNewCycles(sub);
elif len(path) > 2 and next_node == path[-1]:
# cycle found
p = rotate_to_smallest(path);
inv = invert(p)
if isNew(p) and isNew(inv):
cycles.append(p)
def invert(path):
return rotate_to_smallest(path[::-1])
# rotate cycle path such that it begins with the smallest node
def rotate_to_smallest(path):
n = path.index(min(path))
return path[n:]+path[:n]
def isNew(path):
return not path in cycles
def visited(node, path):
return node in path
main()
using System;
using System.Collections.Generic;
namespace akCyclesInUndirectedGraphs
{
class Program
{
// Graph modelled as list of edges
static int[,] graph =
{
{1, 2}, {1, 3}, {1, 4}, {2, 3},
{3, 4}, {2, 6}, {4, 6}, {7, 8},
{8, 9}, {9, 7}
};
static List<int[]> cycles = new List<int[]>();
static void Main(string[] args)
{
for (int i = 0; i < graph.GetLength(0); i++)
for (int j = 0; j < graph.GetLength(1); j++)
{
findNewCycles(new int[] {graph[i, j]});
}
foreach (int[] cy in cycles)
{
string s = "" + cy[0];
for (int i = 1; i < cy.Length; i++)
s += "," + cy[i];
Console.WriteLine(s);
}
}
static void findNewCycles(int[] path)
{
int n = path[0];
int x;
int[] sub = new int[path.Length + 1];
for (int i = 0; i < graph.GetLength(0); i++)
for (int y = 0; y <= 1; y++)
if (graph[i, y] == n)
// edge referes to our current node
{
x = graph[i, (y + 1) % 2];
if (!visited(x, path))
// neighbor node not on path yet
{
sub[0] = x;
Array.Copy(path, 0, sub, 1, path.Length);
// explore extended path
findNewCycles(sub);
}
else if ((path.Length > 2) && (x == path[path.Length - 1]))
// cycle found
{
int[] p = normalize(path);
int[] inv = invert(p);
if (isNew(p) && isNew(inv))
cycles.Add(p);
}
}
}
static bool equals(int[] a, int[] b)
{
bool ret = (a[0] == b[0]) && (a.Length == b.Length);
for (int i = 1; ret && (i < a.Length); i++)
if (a[i] != b[i])
{
ret = false;
}
return ret;
}
static int[] invert(int[] path)
{
int[] p = new int[path.Length];
for (int i = 0; i < path.Length; i++)
p[i] = path[path.Length - 1 - i];
return normalize(p);
}
// rotate cycle path such that it begins with the smallest node
static int[] normalize(int[] path)
{
int[] p = new int[path.Length];
int x = smallest(path);
int n;
Array.Copy(path, 0, p, 0, path.Length);
while (p[0] != x)
{
n = p[0];
Array.Copy(p, 1, p, 0, p.Length - 1);
p[p.Length - 1] = n;
}
return p;
}
static bool isNew(int[] path)
{
bool ret = true;
foreach(int[] p in cycles)
if (equals(p, path))
{
ret = false;
break;
}
return ret;
}
static int smallest(int[] path)
{
int min = path[0];
foreach (int p in path)
if (p < min)
min = p;
return min;
}
static bool visited(int n, int[] path)
{
bool ret = false;
foreach (int p in path)
if (p == n)
{
ret = true;
break;
}
return ret;
}
}
}
演示图表的周期：
1,3,2
1,4,3,2
1,4,6,2
1,3,4,6,2
1,4,6,2,3
1,4,3
2,6,4,3
7,9,8
Java编写的算法：
import java.util.ArrayList;
import java.util.List;
public class GraphCycleFinder {
// Graph modeled as list of edges
static int[][] graph =
{
{1, 2}, {1, 3}, {1, 4}, {2, 3},
{3, 4}, {2, 6}, {4, 6}, {7, 8},
{8, 9}, {9, 7}
};
static List<int[]> cycles = new ArrayList<int[]>();
/**
* @param args
*/
public static void main(String[] args) {
for (int i = 0; i < graph.length; i++)
for (int j = 0; j < graph[i].length; j++)
{
findNewCycles(new int[] {graph[i][j]});
}
for (int[] cy : cycles)
{
String s = "" + cy[0];
for (int i = 1; i < cy.length; i++)
{
s += "," + cy[i];
}
o(s);
}
}
static void findNewCycles(int[] path)
{
int n = path[0];
int x;
int[] sub = new int[path.length + 1];
for (int i = 0; i < graph.length; i++)
for (int y = 0; y <= 1; y++)
if (graph[i][y] == n)
// edge refers to our current node
{
x = graph[i][(y + 1) % 2];
if (!visited(x, path))
// neighbor node not on path yet
{
sub[0] = x;
System.arraycopy(path, 0, sub, 1, path.length);
// explore extended path
findNewCycles(sub);
}
else if ((path.length > 2) && (x == path[path.length - 1]))
// cycle found
{
int[] p = normalize(path);
int[] inv = invert(p);
if (isNew(p) && isNew(inv))
{
cycles.add(p);
}
}
}
}
// check of both arrays have same lengths and contents
static Boolean equals(int[] a, int[] b)
{
Boolean ret = (a[0] == b[0]) && (a.length == b.length);
for (int i = 1; ret && (i < a.length); i++)
{
if (a[i] != b[i])
{
ret = false;
}
}
return ret;
}
// create a path array with reversed order
static int[] invert(int[] path)
{
int[] p = new int[path.length];
for (int i = 0; i < path.length; i++)
{
p[i] = path[path.length - 1 - i];
}
return normalize(p);
}
// rotate cycle path such that it begins with the smallest node
static int[] normalize(int[] path)
{
int[] p = new int[path.length];
int x = smallest(path);
int n;
System.arraycopy(path, 0, p, 0, path.length);
while (p[0] != x)
{
n = p[0];
System.arraycopy(p, 1, p, 0, p.length - 1);
p[p.length - 1] = n;
}
return p;
}
// compare path against known cycles
// return true, iff path is not a known cycle
static Boolean isNew(int[] path)
{
Boolean ret = true;
for(int[] p : cycles)
{
if (equals(p, path))
{
ret = false;
break;
}
}
return ret;
}
static void o(String s)
{
System.out.println(s);
}
// return the int of the array which is the smallest
static int smallest(int[] path)
{
int min = path[0];
for (int p : path)
{
if (p < min)
{
min = p;
}
}
return min;
}
// check if vertex n is contained in path
static Boolean visited(int n, int[] path)
{
Boolean ret = false;
for (int p : path)
{
if (p == n)
{
ret = true;
break;
}
}
return ret;
}
}
这是Python代码的C++版本：
std::vector< std::vector<vertex_t> > Graph::findAllCycles()
{
std::vector< std::vector<vertex_t> > cycles;
std::function<void(std::vector<vertex_t>)> findNewCycles = [&]( std::vector<vertex_t> sub_path )
{
auto visisted = []( vertex_t v, const std::vector<vertex_t> & path ){
return std::find(path.begin(),path.end(),v) != path.end();
};
auto rotate_to_smallest = []( std::vector<vertex_t> path ){
std::rotate(path.begin(), std::min_element(path.begin(), path.end()), path.end());
return path;
};
auto invert = [&]( std::vector<vertex_t> path ){
std::reverse(path.begin(),path.end());
return rotate_to_smallest(path);
};
auto isNew = [&cycles]( const std::vector<vertex_t> & path ){
return std::find(cycles.begin(), cycles.end(), path) == cycles.end();
};
vertex_t start_node = sub_path[0];
vertex_t next_node;
// visit each edge and each node of each edge
for(auto edge : edges)
{
if( edge.has(start_node) )
{
vertex_t node1 = edge.v1, node2 = edge.v2;
if(node1 == start_node)
next_node = node2;
else
next_node = node1;
if( !visisted(next_node, sub_path) )
{
// neighbor node not on path yet
std::vector<vertex_t> sub;
sub.push_back(next_node);
sub.insert(sub.end(), sub_path.begin(), sub_path.end());
findNewCycles( sub );
}
else if( sub_path.size() > 2 && next_node == sub_path.back() )
{
// cycle found
auto p = rotate_to_smallest(sub_path);
auto inv = invert(p);
if( isNew(p) && isNew(inv) )
cycles.push_back( p );
}
}
}
};
for(auto edge : edges)
{
findNewCycles( std::vector<vertex_t>(1,edge.v1) );
findNewCycles( std::vector<vertex_t>(1,edge.v2) );
}
}
vertex_t是什么？ - Denis受到@LetterRip和@Axel Kemper的启发，这里是Java的简化版本:
public static int[][] graph =
{
{1, 2}, {2, 3}, {3, 4}, {2, 4},
{3, 5}
};
public static Set<List<Integer>> cycles = new HashSet<>();
static void findNewCycles(ArrayList<Integer> path) {
int start = path.get(0);
int next = -1;
for (int[] edge : graph) {
if (start == edge[0] || start == edge[1]) {
next = (start == edge[0]) ? edge[1] : edge[0];
if (!path.contains(next)) {
ArrayList<Integer> newPath = new ArrayList<>();
newPath.add(next);
newPath.addAll((path));
findNewCycles(newPath);
} else if (path.size() > 2 && next == path.get(path.size() - 1)) {
List<Integer> normalized = new ArrayList<>(path);
Collections.sort(normalized);
cycles.add(normalized);
}
}
}
}
public static void detectCycle() {
for (int i = 0; i < graph.length; i++)
for (int j = 0; j < graph[i].length; j++) {
ArrayList<Integer> path = new ArrayList<>();
path.add(graph[i][j]);
findNewCycles(path);
}
for (List<Integer> c : cycles) {
System.out.println(c);
}
}
对于任何需要的人，这里只是一种非常基本的MATLAB版本，从上面的python代码中改编而来。
function cycleList = searchCycles(edgeMap)
tic
global graph cycles numCycles;
graph = edgeMap;
numCycles = 0;
cycles = {};
for i = 1:size(graph,1)
for j = 1:2
findNewCycles(graph(i,j))
end
end
% print out all found cycles
for i = 1:size(cycles,2)
cycles{i}
end
% return the result
cycleList = cycles;
toc
function findNewCycles(path)
global graph cycles numCycles;
startNode = path(1);
nextNode = nan;
sub = [];
% visit each edge and each node of each edge
for i = 1:size(graph,1)
node1 = graph(i,1);
node2 = graph(i,2);
if node1 == startNode
nextNode = node2;
elseif node2 == startNode
nextNode = node1;
end
if ~(visited(nextNode, path))
% neighbor node not on path yet
sub = nextNode;
sub = [sub path];
% explore extended path
findNewCycles(sub);
elseif size(path,2) > 2 && nextNode == path(end)
% cycle found
p = rotate_to_smallest(path);
inv = invert(p);
if isNew(p) && isNew(inv)
numCycles = numCycles + 1;
cycles{numCycles} = p;
end
end
end
function inv = invert(path)
inv = rotate_to_smallest(path(end:-1:1));
% rotate cycle path such that it begins with the smallest node
function new_path = rotate_to_smallest(path)
[~,n] = min(path);
new_path = [path(n:end), path(1:n-1)];
function result = isNew(path)
global cycles
result = 1;
for i = 1:size(cycles,2)
if size(path,2) == size(cycles{i},2) && all(path == cycles{i})
result = 0;
break;
end
end
function result = visited(node,path)
result = 0;
if isnan(node) && any(isnan(path))
result = 1;
return
end
for i = 1:size(path,2)
if node == path(i)
result = 1;
break
end
end
const graph = [[1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [2, 6], [4, 6], [8, 7], [8, 9], [9, 7]]
let cycles = []
function main() {
for (const edge of graph) {
for (const node of edge) {
findNewCycles([node])
}
}
for (cy of cycles) {
console.log(cy.join(','))
}
}
function findNewCycles(path) {
const start_node = path[0]
let next_node = null
let sub = []
// visit each edge and each node of each edge
for (const edge of graph) {
const [node1, node2] = edge
if (edge.includes(start_node)) {
next_node = node1 === start_node ? node2 : node1
}
if (notVisited(next_node, path)) {
// eighbor node not on path yet
sub = [next_node].concat(path)
// explore extended path
findNewCycles(sub)
} else if (path.length > 2 && next_node === path[path.length - 1]) {
// cycle found
const p = rotateToSmallest(path)
const inv = invert(p)
if (isNew(p) && isNew(inv)) {
cycles.push(p)
}
}
}
}
function invert(path) {
return rotateToSmallest([...path].reverse())
}
// rotate cycle path such that it begins with the smallest node
function rotateToSmallest(path) {
const n = path.indexOf(Math.min(...path))
return path.slice(n).concat(path.slice(0, n))
}
function isNew(path) {
const p = JSON.stringify(path)
for (const cycle of cycles) {
if (p === JSON.stringify(cycle)) {
return false
}
}
return true
}
function notVisited(node, path) {
const n = JSON.stringify(node)
for (const p of path) {
if (n === JSON.stringify(p)) {
return false
}
}
return true
}
main()
上面的Python代码的VB .net版本如下：
Module Module1
' Graph modelled as list of edges
Public graph As Integer(,) = {{{1, 2}, {1, 3}, {1, 4}, {2, 3},
{3, 4}, {2, 6}, {4, 6}, {7, 8},
{8, 9}, {9, 7}}
Public cycles As New List(Of Integer())()
Sub Main()
For i As Integer = 0 To graph.GetLength(0) - 1
For j As Integer = 0 To graph.GetLength(1) - 1
findNewCycles(New Integer() {graph(i, j)})
Next
Next
For Each cy As Integer() In cycles
Dim s As String
s = cy(0)
For i As Integer = 1 To cy.Length - 1
s = s & "," & cy(i)
Next
Console.WriteLine(s)
Debug.Print(s)
Next
End Sub
Private Sub findNewCycles(path As Integer())
Dim n As Integer = path(0)
Dim x As Integer
Dim [sub] As Integer() = New Integer(path.Length) {}
For i As Integer = 0 To graph.GetLength(0) - 1
For y As Integer = 0 To 1
If graph(i, y) = n Then
' edge referes to our current node
x = graph(i, (y + 1) Mod 2)
If Not visited(x, path) Then
' neighbor node not on path yet
[sub](0) = x
Array.Copy(path, 0, [sub], 1, path.Length)
' explore extended path
findNewCycles([sub])
ElseIf (path.Length > 2) AndAlso (x = path(path.Length - 1)) Then
' cycle found
Dim p As Integer() = normalize(path)
Dim inv As Integer() = invert(p)
If isNew(p) AndAlso isNew(inv) Then
cycles.Add(p)
End If
End If
End If
Next
Next
End Sub
Private Function equals(a As Integer(), b As Integer()) As Boolean
Dim ret As Boolean = (a(0) = b(0)) AndAlso (a.Length = b.Length)
Dim i As Integer = 1
While ret AndAlso (i < a.Length)
If a(i) <> b(i) Then
ret = False
End If
i += 1
End While
Return ret
End Function
Private Function invert(path As Integer()) As Integer()
Dim p As Integer() = New Integer(path.Length - 1) {}
For i As Integer = 0 To path.Length - 1
p(i) = path(path.Length - 1 - i)
Next
Return normalize(p)
End Function
' rotate cycle path such that it begins with the smallest node
Private Function normalize(path As Integer()) As Integer()
Dim p As Integer() = New Integer(path.Length - 1) {}
Dim x As Integer = smallest(path)
Dim n As Integer
Array.Copy(path, 0, p, 0, path.Length)
While p(0) <> x
n = p(0)
Array.Copy(p, 1, p, 0, p.Length - 1)
p(p.Length - 1) = n
End While
Return p
End Function
Private Function isNew(path As Integer()) As Boolean
Dim ret As Boolean = True
For Each p As Integer() In cycles
If equals(p, path) Then
ret = False
Exit For
End If
Next
Return ret
End Function
Private Function smallest(path As Integer()) As Integer
Dim min As Integer = path(0)
For Each p As Integer In path
If p < min Then
min = p
End If
Next
Return min
End Function
Private Function visited(n As Integer, path As Integer()) As Boolean
Dim ret As Boolean = False
For Each p As Integer In path
If p = n Then
ret = True
Exit For
End If
Next
Return ret
End Function
结束模块
global graph cycles numCycles;
startNode = path(1);
nextNode = nan;
sub = [];
% visit each edge and each node of each edge
for i = 1:size(graph,1)
node1 = graph(i,1);
node2 = graph(i,2);
if (node1 == startNode) || (node2==startNode) %% this if is required
if node1 == startNode
nextNode = node2;
elseif node2 == startNode
nextNode = node1;
end
if ~(visited(nextNode, path))
% neighbor node not on path yet
sub = nextNode;
sub = [sub path];
% explore extended path
findNewCycles(sub);
elseif size(path,2) > 2 && nextNode == path(end)
% cycle found
p = rotate_to_smallest(path);
inv = invert(p);
if isNew(p) && isNew(inv)
numCycles = numCycles + 1;
cycles{numCycles} = p;
end
end
end
end
看起来上面的循环查找器存在一些问题。C#版本无法找到一些循环。我的图形是：
{2,8},{4,8},{5,8},{1,9},{3,9},{4,9},{5,9},{6,9},{1,10},
{4,10},{5,10},{6,10},{7,10},{1,11},{4,11},{6,11},{7,11},
{1,12},{2,12},{3,12},{5,12},{6,12},{2,13},{3,13},{4,13},
{6,13},{7,13},{2,14},{5,14},{7,14}
1-9-3-12-5-10 没有被找到。
我也尝试了C++版本，它返回非常大（数百万）的循环次数，显然是错误的。可能是因为它无法匹配这些循环。