# Lecture 15: Dijkstra’s Algorithm and Huffman’s coding.

First i would say this is the kind of thinking(Trying to fit an efficient algorithm) you need to succeed in graph theory. That said, the approach you mentioned will. So Dijkstra's algorithm doesn't work for negative edges. So it's different. This algorithm is not subsumed by Dijkstra. That's important to understand. So Dijkstra's algorithm works for graphs with cycles. But all of the edge ways have to be either 0 or positive. This algorithm works for DAGs that can have negative edges. But you can't have.

## Why doesn't Dijkstra work with negative weight graphs? - Quora.

Actually, Dijkstra's algorithm fails to work for most of the negative weight edged graphs, but sometimes it works with some of the graphs with negative weighted edges too provided the graph doesn't have negative weight cycles, This is one case in which dijkstra's algorithm works fine and finds the shortest path between whatever the point u give.The trick is easy, Dijkstra algorithm doesn't work for negative weights, so we will force every weight to be in positive, and that by adding to each edge, the inverse of min negative weight, by that we have forced the graph to contains only positive weights, then we proceced with Dijkstra's algorithm, at the end we substract the value which we added it.Dijkstra doesn't work for Graphs with negative weight edges, Bellman-Ford works for such graphs. Bellman-Ford is also simpler than Dijkstra and suites. But Bellman-Ford algorithm does not work if a graph contains negative weight cycles although it can detect a negative weight cycle and it is slower than Dijkstra's Algorithm. Time complexity of bellman ford algorithm is O(V.E) in worst case.

Dijkstra's Algorithm. Dijkstra's algorithm solves the single-source shortest-path problem when all edges have non-negative weights.It is a greedy algorithm and similar to Prim's algorithm. Algorithm starts at the source vertex, s, it grows a tree, T, that ultimately spans all vertices reachable from S. Vertices are added to T in order of distance i.e., first S, then the vertex closest to S.Dijkstra was great. And this was for nonnegative edge weights. So, with negative edge weights, somehow we'd like to get the same running time. Now, how might I get the same running time? Well, it would be really nice if I could use Dijkstra. Of course, Dijkstra doesn't work with negative weights. So what could I do? What would I hope to do? Dijkstra's Algorithm fails when in a graph we have edges with negative weights. However, to this rule there is an exception: If In a directed acyclic graph only the edges that leave the source node are negative (all the other edges are positive), then we can successfully use Dijkstra's Algorithm. Are negative edge weights allowed? Are negative cycles allowed? Before we go on, convince yourself that even for graphs with all positive weights, the pure greedy solution to even the single-pair shortest path does not work. Why doesn't it? Dijkstra's Algorithm. For: single-source, no negative edges. Implement with a priority queue. Starting. Dijkstra’s algorithm doesn’t work on graphs with negative edge weights. Here is one attempt to x it: 1.Add a large number M to every edge so that there are no negative weights left. 2.Run Dijkstra to nd the shortest path in the new graph. 3.Return the path Dijkstra found, but with the old edge weights (i.e. subtract M from the weight of.

## Dijkstra's Shortest Path Algorithm - Cornell University. Negative edge weights Dijkstra’s algorithm (for shortest paths) fails Financial arbitrage corresponds to negative weight cycles Minimum spanning tree algorithms don’t care Can you fix Dijkstra’s to work with negative weights? Errata Last week, we suggested that you could make the dummy-node algorithm for shortest paths (replace edges with weight n by n unweighted edges) work for non. Edge weights may represent, distances, costs, etc.. Dijkstra’s Algorithm The distance of a vertex v from a vertex s is the length of a shortest path between s and v Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex s Assumptions: the graph is connected the edges are directed the edge weights are nonnegative We grow a “cloud” of vertices. Dijkstra’s algorithm doesn’t work on graphs with negative edge weights. Here is one attempt to x it: 1.Add a large number M to every edge so that there are no negative weights left. 2.Run Dijkstra’s to nd the shortest path in the new graph. 3.Return the path found by Dijkstra’s, but with the old edge weights (i.e. subtract M from the weight of each edge). Show that this algorithm doesn. Djikstra's algorithm works for additive weights, in graph's with no cycle of negative weight. Since you don't have additive weights, there is no reason to believe you can use Djikstra's algorthm. You can't apply a theorem unless the hypotheses are satisfied. The other answers so far demonstrate pretty well why Dijkstra's algorithm cannot handle negative weights on paths. But the question itself is maybe based on a wrong understanding of the weight of paths. If negative weights on paths would be allowed in pathfinding algorithms in general, then you would get permanent loops that would not stop.

## Detecting negative cycle using Bellman Ford algorithm. If a graph has a negative edge weight (but not a negative cycle), there is a (finite) shortest path between each pair of nodes (that is, provided there's a path between those two nodes to begin with), but the presence of a negative edge means that Dijkstra's algorithm isn't guaranteed to work; it won't necessarily find that shortest path. Dijkstra's algorithm just doesn't work in the presence of, presence of negative weights. So say, this weight from two to three is -nine. What Dijkstra's algorithm will do is just immediately select vertex three and never revisit that decision. Saying the shortest way to get from zero to three is, is, Is to take this edge of three which has weight two. But the actual shortest path goes over to. This helped me realize why dijkstra's doesn't work with negative edge weights, as since there are no cycles I've always assumed it would work with negative edges anyways. If you ever get stuck in a situation where it has to choose between two equal edge weights, if there's a negative edge in between them and it doesn't choose the one that connects to a negative edge, it will come out with the.