Find all possible paths in a graph using Matlab Brute Force Search - matlab

Question

I need to find all paths with a graph, and save these paths. My starting nodes are A, B or C, and the final node is G. My graphs have maximum 16 unweighted vertices.
I made the Matlab code below, but this has problems with bifurcations. Also, I don't know how to impose the starting and the final nodes. Can anyone help me with this?
path = cell(1,10) ; % initialize
% one_graph ={'AH','BO','CN','EG','EN','EO','HO','KN'} % (Graph example)
one_graph ={'AH','BN','DH','DN','GN'} % (Graph example)
for p = 1:length(one_graph)
edge = one_graph(p);
% In each graph there is only 1:1 conections
% detect node 1
existing_node1 = edge{1}(1) ;
Index_existing_node1 = strfind(allnodes, existing_node1) ;
[row1,col1] = find(ismember(allnodes, existing_node1));
% detect node 2
existing_node2 = edge{1}(2) ;
Index_existing_node2 = strfind(allnodes, existing_node2);
[row2,col2] = find(ismember(allnodes, existing_node2));
path_nonz = path(~cellfun('isempty',path)) ;
t = length(path_nonz) ;
if t>0 % save the first 2 nodes in the path
ttt = strcmp(allnodes(row1), path{t});
ttt2 = strcmp(allnodes(row2), path{t});
end;
if t==0
path{t+1} = allnodes{row1} ;
path{t+2} = allnodes{row2} ;
elseif ttt == 1
% disp('connect right')
path{t+1} = allnodes{row2} ;
elseif ttt2 == 1
% disp('connect right')
path{t+1} = allnodes{row1} ;
else
disp('Not next vertex')
end
end
For example, for
one_graph ={'AH','BN','DH','DN','GN'} % (Graph example)
I should save the following paths:
path1 = AHDNG
path2 = BNG
and for
one_graph ={'AH','BO','CN','EG','EN','EO','HO','KN'} % (Graph example)
I should save the following paths:
path1 = AHOEG
path2 = BOEG
path3 = CNEG
UPDATE 1:
From the adjacency matrix B(:,:,1)
B =
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
I derive the proper adjacency list:
Asparse = sparse(B(:,:,1));
Asparse =
(8,1) 1
(14,2) 1
(8,4) 1
(14,4) 1
(14,7) 1
(1,8) 1
(4,8) 1
(2,14) 1
(4,14) 1
(7,14) 1
Then, I tried to use the BFS algorithm found on Matlab Website
[distances,times,pred] = bfs(Asparse,1);
But, this doesn't save the paths. It just saves the previous node of each current node (in pred) and the distance from the initial node to each node (in distances). Any idea, how to save each path?

Answer 1

I've had to write a custom function to do this since 1) most BFS/DFS functions stop when the goal is reached and 2) they explicitly ignore cycles, which are required for multiple paths to the same target.
I believe this will get you what you need. I've made a slight modification to the adjacency matrix in your example to create an edge from {2,7} and {7,2} so that there would be two paths from 2 to 14. Note that this is a recursive function, so if you get around 500 nodes or so you're going to have problems and we'll have to come up with a version that uses an explicit stack.
function paths = findpaths(Adj, nodes, currentPath, start, target)
paths = {};
nodes(start) = 0;
currentPath = [currentPath start];
childAdj = Adj(start,:) & nodes;
childList = find(childAdj);
childCount = numel(childList);
if childCount == 0 || start == target
if start == target
paths = [paths; currentPath];
end
return;
end
for idx = 1:childCount
currentNode = childList(idx);
newNodes = nodes;
newNodes(currentNode) = 0;
newPaths = findpaths(Adj, newNodes, currentPath, currentNode, target);
paths = [paths; newPaths];
end
end
If you call this function like this:
A =[
0 0 0 0 0 0 0 1 0 0 0 0 0 0;
0 0 0 0 0 0 1 0 0 0 0 0 0 1;
0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 1 0 0 0 0 0 1;
0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 1 0 0 0 0 0 0 0 0 0 0 0 1;
1 0 0 1 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 1 0 1 0 0 1 0 0 0 0 0 0 0];
unusedNodes=ones(1,size(A,1));
start=2;
target=14;
emptyPath=[];
allPaths = findpaths(A, unusedNodes, emptyPath, start, target)
the output should be:
allPaths =
{
[1,1] =
2 7 14
[2,1] =
2 14
}
Naturally, you need to call this for each starting node.
Actually, you don't have to call this multiple times. There was one more tip I forgot to tell you. If your graph has n nodes and you introduce a new node n+1 that has edges only to your candidate start nodes, you can call the function once with the new node as the start.
So if I add node 15 to the graph above with edges:
{15,1}, {15,2}
%// I wouldn't bother with {1,15} and {2,15}, they're totally unnecessary
and call the function with start = 15, here's what I get:
allPaths =
{
[1,1] =
15 1 8 4 14
[2,1] =
15 2 7 14
[3,1] =
15 2 14
}
You now have all of the paths with one call, although you need to remove the new node 15 from the head of each path.