I’m working on a graph-related project where I need to find the shortest paths using Dijkstra’s algorithm. However, the graph I’m dealing with has some unique properties:

1. It’s a sparse graph, meaning it has relatively few edges compared to the number of vertices.
2. While most edge weights are non-negative, there are a few edges with negative weights. However, there are no negative cycles in the graph.

I know that Dijkstra’s algorithm is not typically designed to handle graphs with negative weights, but in my case, negative weights are rare and do not form cycles. I want to explore any potential optimizations or tweaks to Dijkstra’s algorithm that could help it work efficiently in this context.

Initially, I tried using the standard implementation of Dijkstra’s algorithm, but it failed to handle the negative weights correctly, resulting in incorrect shortest path calculations. I then considered switching to the Bellman-Ford algorithm, which can handle negative weights, but it seems overkill given that the graph is sparse and the negative weights are rare.

I was expecting that there might be a way to modify or augment Dijkstra’s algorithm to handle these negative weights efficiently, perhaps by preprocessing the graph or selectively applying Bellman-Ford to the affected edges.

Would there be a way to combine the efficiency of Dijkstra’s algorithm with some adjustments or a hybrid approach that accounts for these negative weights without sacrificing performance? Any insights or suggestions for alternative approaches would be greatly appreciated.