Question

When using Dijkstra’s algorithm, why does the presence of negative edge weights invalidate the greedy assumption used to finalize a node&#x27;s shortest path distance?

Accepted Answer

Dijkstra’s algorithm operates on the greedy assumption that once a node is marked as visited, its current distance estimate is the absolute shortest path from the source. This assumption relies on the principle that adding edges to a path will only ever increase the total path cost or keep it the same, never decrease it. When all edge weights are non-negative, any path extending from a visited node will have a cost equal to or greater than the cost of the path already established, ensuring that no future updates could possibly find a better route. Negative edge weights break this fundamental logic because they allow a path that appears longer at an early stage to become shorter later by incorporating a negative value. For instance, consider a source node A connected to nodes B and C. A path A to B costs 5, and a path A to C costs 10. Dijkstra’s algorithm selects B first because 5 is the smallest available distance. However, if there is an edge from B to C with a weight of -7, the actual path from A to B to C would cost only -2. Because Dijkstra’s algorithm finalized node C at a cost of 10 when it was first reached, it ignores the possibility that a later, cheaper path could exist through B. The algorithm does not revisit or update nodes once they are marked as finalized, meaning it misses the discovery of the true shortest path, rendering the greedy choice incorrect.

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When using Dijkstra’s algorithm, why does the presence of negative edge weights invalidate the greedy assumption used to finalize a node's shortest path distance?