How does the Dijkstra Algorithm handle graphs with negative cycles?

The Dijkstra Algorithm is a popular algorithm used to find the shortest path between two nodes in a graph. However, it is designed to work only with graphs that do not contain negative cycles. A negative cycle is a cycle in the graph where the sum of the weights of the edges is negative.

When the Dijkstra Algorithm encounters a graph with negative cycles, it fails to produce correct results. This is because negative cycles can lead to an infinite loop in the algorithm, causing it to keep revisiting the same nodes and decreasing the distance to them indefinitely.

To handle graphs with negative cycles, an alternative algorithm called the Bellman-Ford Algorithm can be used. The Bellman-Ford Algorithm is capable of detecting negative cycles and can be used to find the shortest path in graphs that contain them.

The Bellman-Ford Algorithm works by iteratively relaxing the edges of the graph. It starts by initializing the distance of all nodes to infinity, except for the source node which is set to 0. Then, it relaxes each edge in the graph repeatedly, updating the distance of the destination node if a shorter path is found. This process is repeated for a number of iterations equal to the number of nodes in the graph minus one.

After the initial iterations, the Bellman-Ford Algorithm performs one additional iteration to check for negative cycles. If during this iteration, a shorter path is found for any node, it means that the graph contains a negative cycle. This is because a negative cycle can be traversed repeatedly, decreasing the distance to the destination node indefinitely.

If a negative cycle is detected, the Bellman-Ford Algorithm can be used to identify the nodes that are part of the negative cycle. This can be done by performing additional iterations and marking the nodes that are updated. These marked nodes will be part of the negative cycle.

In conclusion, the Dijkstra Algorithm is not suitable for handling graphs with negative cycles. Instead, the Bellman-Ford Algorithm should be used, as it is capable of detecting negative cycles and finding the shortest path in such graphs.