Dijkstra’s algorithm works correctly on graphs with negative-weight edges, as long as there are no negative-weight cycles.

Question

BlackTom AI · Accepted Answer

The statement to evaluate is: 'Dijkstra’s algorithm works correctly on graphs with negative-weight edges, as long as there are no negative-weight cycles.'

First, note that the provided data shows there are no answer options to analyze beyond the single true/false claim. In such true/false assessments, we examine the properties of the algorithm itself.

Consider how Dijkstra’s algorithm operates: it maintains a priority queue of vertices by their currently known shortest distance from the source, and when a vertex is extracted as the closest unvisited vertex, its distance is considered final. This relies on the assumption that all edge weights are non-negative.

If there are negative-weight edges, this key invariant can fail. A path that is discovered later may offer a shorter route to an already-visited vertex, meaning that the previously finalized distance could be improved. Consequently, negative weights can lead to incorrect shortest-path results regardless of the presence or absence of negative cycles.

Regarding negative cycles, Dijkstra’s algorithm can also behave unpredictably or fail to terminate in a meaningful way when cycles with negative total weight exist, because continuing to traverse the cycle could keep reducing the path length indefinitely. However, the core issue remains: the algorithm’s correctness is not guaranteed whenever negative weights are present, even if one tries to assume there are no negative cycles.

In contrast, there are algorithms designed to handle graphs with negative weights safely. The Bellman-Ford algorithm, for instance, accommodates negative edge weights and can detect negative-weight cycles. It relaxes edges repeatedly and does not rely on the same non-negative-weight invariant as Dijkstra’s.

In summary, the claim that Dijkstra’s algorithm works correctly with negative-weight edges provided there are no negative cycles is misleading because the core correctness condition (non-negative edge weights) is violated by negative weights, which can cause incorrect shortest-path results even absent negative cycles. Bellman-Ford provides a correct alternative under these circumstances.

Note: If there are any multiple-choice options to consider, we would evaluate each against these principles, highlighting why non-negative weights are essential for Dijkstra’s correctness and why negative weights—even without negative cycles—break the guarantees.

Dijkstra’s algorithm works correctly on graphs with negative-weight edges, as long as there are no negative-weight cycles.True/False

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