Given a graph that has negative weights but we know for sure it has no negative cycle:
Add a big enough constant to all weights so they are now positive and use Dijkstra's algorithm to find the smallest path.
Is the above correct if we don't have negative cycles? If we have negative cycles we can't use that algorithm since it will need to revisit the nodes Dijkstra marked as completed.
By adding a constant to all the weights you shall make paths with more number of edges more costly and paths with less number of edges relatively less costly hence disrupting the original problem.
So you can't apply Dijkstra even if there is no negative weight cycle.
Related
There are two functions that I wish to minimize:
a. the number of "obstacles" on the path (assume each obstacle increases the cost); and
b. total number of edges between the source and the destination.
If I had to minimize just (a), I would have used Dijkstra's algorithm; if I had to minimize just (b), I would have used BFS.
But given that I have to minimize both, can I use Dijkstra's algorithm only? In other words, if I find the path with the least cost from the obstacles, does Dijkstra's algorithm also guarantee that the path length thus obtained (between source and destination) would be the shortest?
When discussing paths on weighted graphs, the term "shortest path" means the path with the lowest total cost. Think of the weights as distances. This is the path that Dijkstra's algorithm will find.
You can use any cost function you like, as long as the cost to get from one vertex to another is always positive or zero. As mentioned in comments, however, you can only minimize one function at a time. This is a general fact that has nothing to do with Dijkstra's algorithm.
The cost function that you seem to suggesting is perfectly fine -- the cost to move to a normal vertex is 1, while the cost to move to an "obstacle" vertex is higher. Dijkstra's algorithm is the appropriate way to find a path with lowest total cost.
I have programmed the minimum Hungarian algorithm for a bipartite graph, with Dijkstra's algorithm to find the minimum cost of a maximum matching. However, I want to use such an algorithm to implement the maximum Hungarian algorithm and don't know if it's correct to just negate the edges, because I don't know if the algorithm will handle it.
My implementation is based on the explanation on the following site: https://www.ics.uci.edu/~eppstein/163/lecture6b.pdf
Given G=(AUB, E), the idea is to label the vertices via an artificial start vertex s which has edges with unsaturated nodes in A, and run Dijkstra's algorithm from s in order to label each vertex, then after labeling each, the edges will be reweighted by their original weight minus the labels of the edge's endpoints.
I have read a lot of articles, and the only I could see is that a minimum Hungarian algorithm can be handled well with maximum cost by negating each edge, however, I am afraid that due to the fact that Dijkstra's algorithm doesn't handle negative edges well, it won't work.
First find the maximum weight in your graph. Then negate all of the weights and add the maximum weight to them. Adding the original maximum to all of the negated values makes them all positive.
You can also use INT_MAX (or whatever is equivalent to it in the programming language you're using) instead of the maximum weight. This skips the step of finding the maximum weight, but could make the first iteration of the Hungarian Algorithm take longer, or cause you to need an extra iteration of the algorithm to get the result. It probably doesn't make much of a difference either way and the performance difference will vary based on the particular weights in your graph.
I study on Greedy Algorithm. summarize some important aspects about Dijkstra's Algorithms, that will be TRUE. i suspect about (4) and (1), anyone could help me?
I) if all edges weight be negative, the Dijkstra's algorithm, works well.
II) if in graph we have a negative cycle, Dijkstra's get into a infinite loop and never end.
III) if a graph has a one edge with negative weight, but hasn't a negative cycle, the algorithm doesn't works well.
IV) if graph hasn't a negative cycle, the algorithms work well.
Dijkstra's algorithm only works on graphs with non-negative edges. This is because it assumes that the first time a node is popped off the queue we have found the shortest path to that node and this is not necessarily true as soon as you have even one negative weight.
Therefore I is false, II is false (because the negative cycle may not necessarily be reachable), III is true, IV is false (it may still have a negative edge even without a negative cycle).
If I remember it correctly Dijkstra's algorithm (at least the classic version of it) has a built in assumption that all weights in a graph are non-negative. So we have no guarantee that it will work correctly if we have negative edges in our graph.
I) In the first case we will most likely get a longest possible chain of edges with highest absolute values of weights as the shortest path, technically this will be the correct shortest path. (I assume the graph has no negative cycles)
II) False as negative cycle may be unreachable (as Peter de Rivaz correctly said)
III-IV) I assume that the 3rd paragraph is about graph having one edge with negative weight, but hasn't a negative cycle. In this case there is a possibility of algorithm stucking in a loop, because it can find a cycle with a negative-weight edge that will seem to an algorithm as a good path prolongation beacuse at some point in algorithm we decide on the next step on the weight of one edge. The negative one will always be the first choice no doubt, so even with a non-negative cycle we getting a possibility of infinite looping.
I have a directed graph that has all non-negative edges except the edge(s) that leave the source (S). There are no edges from any other vertices to the source. To find the shortest distance from source (S) to a vertex (T) in the graph, can I use Dijkstra's shortest path algorithm even though the edges leaving the source is negative?
Assuming only source-adjecent edges can have negative weights and there is no path back to the source from any of the source-adjecent nodes (as mentioned in the comment), you can just add a constant C onto all edges leaving the source to make them all non-negative. Then subtract C from the final result.
On a more general note, Dijkstra can be used to solve shortest-path in any graph with negative edge weights (but no negative cycles) after applying Johnson's reweighting algorithm (which is essentially Bellman-Ford, but needs to be performed only once).
Yes, you can use Dijkstra on that type of directed graph.
If you use already finished alghoritm for Dijsktra and it cannot use negative values, it can be good practise to find the lowest negative edge and add that number to all starting edges, therefore there is no-negative number at all. You substract that number after finishing.
If you code it yourself (which is acutally pretty easy and I recommend it to you), you almost does not change anything, just start with lowest value (as usual for Dijkstra) and allow it, that lowest value can be negative. It will work in your case.
The reason you generally can't use Dijkstra's algorithm for (directed) graphs with negative links is that Dijkstra's algorithm is greedy. It assumes that once you pick a vertex with minimum distance, there is no way it can later be reached by a smaller paths.
In your particular graph, after the very first step, you traverse all possible negative edges and Dijkstra's assumption actually holds from now on. Regardless of the fact that those vertices directly connected to start now have negative values, once you identify which has the minimum distance, it can never be reached again with a smaller distance (since all edges you would traverse from this point on would have a positive distance).
If you think about the conditions that dijkstra's algorithm puts upon the edges for the algorithm to work it is only that they are never decreasing after initialisation.
Thus, it actually doesn't matter if the first step is negative as from those several points onwards the function is constantly increasing and thus the correct output will be found (provided there is no way to get back to the start square.).
Given a weighted graph (directed or undirected) I need to find the cycle of the graph with the maximum weight.
The weight of a cycle being the sum of the weight of the edges of the graph.
It can be any cycle, not just base cycle for which we can
find all base cycle (see Algorithms to Identify All the Cycle Bases in a UnDirected Graph )
compute the weight of each base cycle and find the maximum
I could try to enumerate all cycles of the graph and then compute the maximum but the total number of cycles can be really big (if the graph is complete then any sequence of vertices where the first and last one are identical is a cycle).
Do you have any idea to find that maximum weight cycle without enumerating all cycles ?
If you need hypothesis on the graph (positives weights for example) please indicates them.
This is NP-Hard.
Hamiltonian Cycle problem can be reduced to this.
Given a graph for which we need to check if there exists a Hamiltonian Cycle or not, assign weight 1 to each edge.
Now run your algorithm to get the maximum weight cycle. If the weight is < n, then the original graph has no Hamiltonian cycle, otherwise it does.
If you can find the minimum weighted path in your specific case, just reverse the signs of all the weights and apply your algorithm. Of course you are making some unstated assumptions because Moron's argument is correct (no pun intended). The assumptions you are making could be positive weights or no negative weight cycles. I think you should make an effort to state them instead of letting people search in the infinite space of possible assumptions. As to hardness results, this is also hard to approximate in a number of way, check out this paper. The same paper contains several positive results for important types of graphs, but it's concerned with longest unweighted paths so my guess is that most algorithms in the paper won't directly help in your case. If you search for "Heavy cycles" you will find a number of interesting papers, but they are more mathematical in character. If your weights are small integers (up to a polynomial in the size of the graph), you can try and replace every edge with an unweighted path to reduce your problem to the unweighted case. I hope this helps to some degree, but you might have an open research problem on your hands.