Reweighted graphs in MatlabBGL

Matlab sparse matrices only support non-zero values. Because the structure of the sparse matrix is used to infer the edges of an underlying graph this limitation prevents MatlabBGL from trivially addressing graphs with 0-weight edges.

To fix this problem, codes that accept a weighted graph allow the user to specify a vector of edge weights for each edge in the graph using the optional 'weights' parameter. Using the 'weights' parameter correctly can be difficult due to issues with how edges are indexed in MatlabBGL.

Contents

Disclaimer

The details of this section are complicated. This means their implementation is error-prone. If you get strange behavior, please let me know.


 

"I just want the simpliest solution..."

new in version 4.0 In this section, we'll see the really-easy but somewhat expensive way of reweighting a graph. I'll run through all the cases detailed below with the simple code. If you just need something to work and don't necessarily need to know about all the details, this section is for you!

Let's compute shortest paths in a cycle graph with only one weighted edge. The simple case requires a structural and weight matrix.

n will be the total size of the graph, and u and v will be the special vertices connected with a weight one edge.

n = 8; % it's just an example, so let's make it small.
u = 1;
v = 2;

These commands create an undirected cycle graph. The cycle is ... n - 1 - 2 - ... - n-1 - n - 1 ... where the weight on every edge is 0 except for the edge between vertex u,v. Notice that the edge list is already symmetric.

This setup means that while there is a weight 1 edge between u and v, the shortest path, or smallest weight path, is actually the path from u, circling every vertex except v and so d(v) should be 0.

E = [1:n 2:n 1; 2:n 1 1:n]';
w = [1 zeros(1,n-1) 1 zeros(1,n-1)]';

A = sparse(E(:,1), E(:,2), w, n, n); % create a weighted sparse matrix
As = sparse(E(:,1), E(:,2), true, n, n); % create a structural sparse matrix

The relationship between As and A is that As should have a non-zero value for every edge, but the values of As will be ignored and the computation will proceed with the values in the corresponding spots in the matrix A.

The wrong way to compute shortest paths.

[d pred] = shortest_paths(A,u);
d(v)
Undefined function or method 'shortest_paths' for input arguments of type 'double'.

Error in ==> reweighted_graphs at 61
[d pred] = shortest_paths(A,u);

The right way to compute shortest paths.

[d pred] = shortest_paths(As,u,struct('edge_weight',edge_weight_vector(As,A)));
d(v)

That's better, d(v) = 0 as expected.


 

A first attempt

These commands create an undirected cycle graph. The cycle is ... n - 1 - 2 - ... - n-1 - n - 1 ... where the weight on every edge is 0 except for the edge between vertex u,v. Notice that the edge list is already symmetric.

E = [1:n 2:n 1; 2:n 1 1:n]';
w = [1 zeros(1,n-1) 1 zeros(1,n-1)]';

A = sparse(E(:,1), E(:,2), w, n, n);

The shortest weighted path between u and v is then through the vertex n because traversing the cycle in the other direction will use the u,v edge of weight 1. Let's check this with the shortest_paths function.

[d pred] = shortest_paths(A,u);
d(v)

That is weird, there is a u-v path of length 0 in the graph! Let's see what path the shortest path algorithm chose.

path_from_pred(pred,v)

The path it chose was from u to v directly, taking the weight 1 edge. Let's look at the sparse matrix.

A

There are only two edges in the matrix corresponding to our symmetric weight 1 edge between u and v. This happens because Matlab removes all 0 weight edges from the graph.


 

A first solution

The solution to the problem is to use the 'edge_weight' optional parameter to the shortest_paths function to give it a set of weights to use for each edge.

help shortest_paths

Well, shortest_paths says to read this document, so you are on the right track! It also has a pointer to the function edge_weight_index. Let's look at that function

help edge_weight_index

This function claims to help us. It requires building a structural matrix which has a non-zero for each edge in the graph. Let's do that.

As = sparse(E(:,1), E(:,2), 1, n, n)

Now the matrix has all of the required edges. According to the edge_weight_index function, it returns both a matrix and an index vector. The index vector is a way to permute an intelligently ordered set of edge weights to the order that MatlabBGL requires the edge weights.

[ei Ei] = edge_weight_index(As);

full(Ei)
ei

Now let's create a new edge weight vector for this graph corresponding to all the edges we want. Each non-zero in the matrix should have an associated edge weight. Most the edge weights in this case are 0, so it makes it simple.

ew = zeros(nnz(As),1);
ew(Ei(u,v)) = 1;
ew(Ei(v,u)) = 1;

[d pred] = shortest_paths(As,u,struct('edge_weight', ew(ei)));

path_from_pred(pred,v)

Excellent, now the shorest path avoids the edge (u,v) as we would expect it.


 

A simplified solution

The current example is somewhat tedious because we have to create the sparse matrix, then create the edge index matrix, and finally create and edit the edge weight array.

The indexed_sparse function makes the process easier.

Recall that using the sparse function directly generated an incorrect graph adjacency matrix.

A = sparse(E(:,1), E(:,2), w, n, n)

The indexed_sparse function is designed as a replacement for sparse where the adjacency matrix must be indexed using the edge_weight_index or contains 0 weight edges.

help indexed_sparse

From the documentation of indexed_sparse, the first two return values are the structural sparse matrix (As) and the sparse matrix (A) that sparse would have returned. The final two return values are the edge index list that edge_weight_index returns as well as the edge index matrix.

% save the old Ei as an example
old_Ei = Ei;

[As A eil Ei] = indexed_sparse(E(:,1), E(:,2), w, n, n);

fprintf('old_Ei = \n\n');
disp(full(old_Ei));
fprintf('Ei = \n\n');
disp(full(Ei))

Note that the edge indices changed between the two calls. The reason for this change is that indexed_sparse generates edge indices based on order of E(:,1) and E(:,2). Consequently, this function is much easier to use when you already have a set of weighted edges.

In this case, we don't have to create the ew array again! (Note that the call uses eil instead of ei.)

[d pred] = shortest_paths(As,u,struct('edge_weight', w(eil)));

path_from_pred(pred,v)

 

An undirected solution

The situation for undirected graphs is more complicated. The trouble with the previous solution is that each directed edge had its own weight in the vector w. For an undirected graph, we really want each undirected edge to have a single weight, so the natural length of v would be nnz(A)/2 instead of nnz(A).

However, MatlabBGL really treats all problems as directed graphs, so it will need a vector w of length nnz(A), but that vector should satisfy the requirement w(ei1) = w(ei2) if the edges corresponding to ei1 and ei2 are (i,j) and (j,i), respectively.

Again, the edge_weight_index function provides a solution to this problem. We just have to tell edge_weight_index we have an undirected graph.

Let's start with the same sparse matrix

As = sparse(E(:,1), E(:,2), 1, n, n)

Here we use the edge_weight_index

[ei Ei] = edge_weight_index(As,struct('undirected',1));
full(Ei) % look at the matrix

Let's create the edge weight vector

ew = zeros(nnz(As)/2,1); % only half as many zeros here.
ew(Ei(u,v)) = 1; % and we only needed to set one entry to 0

[d pred] = shortest_paths(As,u,struct('edge_weight', ew(ei)));

path_from_pred(pred,v)

And we get the same output as before!


 

The undirected simplification

You can probably guess that the simplification for undirected graphs will use the indexed_sparse call again too.

help indexed_sparse

From the documentation, we find that indexed_sparse has an option called "undirected" which is set to 0 by default.

[As A eil Ei] = indexed_sparse(E(:,1), E(:,2), w, n, n, struct('undirected',1));

fprintf('Ei = \n\n');
disp(full(Ei))

In this case, the indexed_sparse routine only issued edge indices that were between 1 and 8, rather than 1 and 16 as in the previous case.

[d pred] = shortest_paths(As,u,struct('edge_weight', w(eil)));

path_from_pred(pred,v)

 

Summary

The functions that support reweighted edges as of MatlabBGL 3.0 are shortest_paths, all_shortest_paths, dijkstra_sp, bellman_ford_sp, dag_sp, betweenness_centrality, astar_search, johnson_all_sp, floyd_warshall_all_sp, mst, kruskal_mst, and prim_mst. Note that max_flow does not support these indices.

The functions that assist working with the edge indices for the edge_weight vector are edge_weight_index, indexed_sparse, and edge_weight_vector.