If you notice, there is a distinct pattern to the adjacency matrices you are creating. Specifically, they are symmetric and banded. You can take advantage of this fact to easily create your matrices using the diag function (or the spdiags function if you want to make a sparse matrix). Here is how you can create … Read more
I was able to prove this with graph-theory. Might want to add that tag in 🙂 Create a graph with n vertices. Create an edge from node n_i to n_j if the element in position i should be in position j in the correct ordering. You will now have a graph consisting of several non-intersecting … Read more
Everyone else comparing this to the Travelling Salesman Problem probably hasn’t read your question carefully. In TSP, the objective is to find the shortest cycle that visits all the vertices (a Hamiltonian cycle) — it corresponds to having every node labelled ‘mustpass’. In your case, given that you have only about a dozen labelled ‘mustpass’, … Read more
Recall that in Dijkstra’s algorithm, once a vertex is marked as “closed” (and out of the open set) – the algorithm found the shortest path to it, and will never have to develop this node again – it assumes the path developed to this path is the shortest. But with negative weights – it might … Read more
Finding all possible paths is a hard problem, since there are exponential number of simple paths. Even finding the kth shortest path [or longest path] are NP-Hard. One possible solution to find all paths [or all paths up to a certain length] from s to t is BFS, without keeping a visited set, or for … Read more
It appears that this can be accomplished with a depth-first search of the graph. The depth-first search will find all non-cyclical paths between two nodes. This algorithm should be very fast and scale to large graphs (The graph data structure is sparse so it only uses as much memory as it needs to). I noticed … Read more
I found this page in my search and since cycles are not same as strongly connected components, I kept on searching and finally, I found an efficient algorithm which lists all (elementary) cycles of a directed graph. It is from Donald B. Johnson and the paper can be found in the following link: http://www.cs.tufts.edu/comp/150GA/homeworks/hw1/Johnson%2075.PDF A … Read more
According to Lemma 22.11 of Cormen et al., Introduction to Algorithms (CLRS): A directed graph G is acyclic if and only if a depth-first search of G yields no back edges. This has been mentioned in several answers; here I’ll also provide a code example based on chapter 22 of CLRS. The example graph is … Read more
How about defining path/4 like this? path(R_2, Xs, A,Z) :- % A path `Xs` from `A` to `Z` is … walk(R_2, Xs, A,Z), % … a walk `Xs` from `A` to `Z` … all_dif(Xs). % … with no duplicates in `Xs`. To aid universal termination, we swap the two goals in above conjunction … path(R_2, … Read more