How does the Dijkstra Algorithm handle negative edge weights?

The Dijkstra Algorithm is primarily designed to work with non-negative edge weights. However, it can be modified to handle negative edge weights as well, but with certain limitations and considerations.

By default, the Dijkstra Algorithm assumes that all edge weights are non-negative. This assumption is crucial for the algorithm's correctness and efficiency. The algorithm maintains a priority queue of vertices, and at each step, it selects the vertex with the minimum distance from the source vertex. This selection is based on the assumption that the minimum distance found so far is the actual minimum distance.

When negative edge weights are present, the Dijkstra Algorithm may not produce correct results or may fail to terminate. This is because the algorithm's greedy nature relies on the assumption that once a vertex is visited, its distance is finalized and will not be updated further. However, negative edge weights can create cycles that continuously decrease the distance, leading to an infinite loop.

To handle negative edge weights, we can use a modified version of the Dijkstra Algorithm called the Bellman-Ford Algorithm. The Bellman-Ford Algorithm can handle negative edge weights and detect negative cycles in the graph. It works by iteratively relaxing the edges in the graph, updating the distance of each vertex until no further updates are possible or a negative cycle is detected.

The Bellman-Ford Algorithm guarantees to find the shortest path from a single source vertex to all other vertices in the presence of negative edge weights, as long as there are no negative cycles reachable from the source vertex. If a negative cycle is detected, it indicates that the graph has no well-defined shortest paths, as we can keep traversing the cycle to decrease the distance indefinitely.

In summary, the Dijkstra Algorithm is not directly suitable for handling negative edge weights. Instead, the Bellman-Ford Algorithm should be used in such cases to ensure correct results and handle negative cycles.