EDIT: See simon’s answer, where he fixed the formula presented here. Actually there is an O(1) formula This is an image that I’ve made to visualize it ( Squares a knight can reach on Nth move are painted with same color ). Can you notice the pattern here? Although we can see the pattern, it … Read more
Technically, Breadth-first search (BFS) by itself does not let you find the shortest path, simply because BFS is not looking for a shortest path: BFS describes a strategy for searching a graph, but it does not say that you must search for anything in particular. Dijkstra’s algorithm adapts BFS to let you find single-source shortest … Read more
This is laborous case work: A < B and 0 and N-1 are joined by A -> trivial. B < A and 0 and N-1 are joined by B -> trivial. B < A and 0 and N-1 are joined by A -> Do BFS on graph with only K edges. A < B and … Read more
NEW ANSWER Given the recent update, you could try A* with the Hamming distance as a heuristic. It’s an admissible heuristic since it’s not going to overestimate the distance OLD ANSWER You can modify the dynamic-program used to compute the Levenshtein distance to obtain the sequence of operations. EDIT: If there are a constant number … Read more
Dijkstra‘s algorithm finds the shortest path between a node and every other node in the graph. You’d run it once for every node. Weights must be non-negative, so if necessary you have to normalise the values in the graph first. Floyd-Warshall calculates the shortest routes between all pairs of nodes in a single run! Cycle … Read more
As a caveat, remember that there can be exponentially many shortest paths between two nodes in a graph. Any algorithm for this will potentially take exponential time. That said, there are a few relatively straightforward algorithms that can find all the paths. Here’s two. BFS + Reverse DFS When running a breadth-first search over a … Read more
The algorithm you have suggested will indeed find the shortest path in this graph, but not all graphs in general. For example, consider this graph: Let’s trace through the execution of your algorithm. First, you set d(A) to 0 and the other distances to ∞. You then expand out node A, setting d(B) to 1, … 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
If you implemented it using a priority queue, see Djikstra’s algorithm, you only need to put an “infinite” weight on the path that has reached the dead end and it won’t be visited again. A “min” priority queue is a datastructure that pops the element with the least weight attached to it. So when you’re … Read more