Choosing the correct shortest path algorithm depends entirely on the graph's structure and the nature of its edge weights.

  • Scenario 1: Cyclic Graph, Positive Weights. This is the classic shortest path problem, solvable only by Dijkstra's Algorithm.
  • Scenario 2: DAG, Positive Weights. Both Dijkstra's and Topo Sort work, but the DAG Shortest Path approach is significantly faster, achieving $O(V+E)$.
  • Scenario 3: DAG, Negative Weights. Dijkstra's fails with negative weights, but the DAG Shortest Path algorithm handles them correctly because there are no cycles.
  • Scenario 4: Cyclic Graph, Negative Weights. This scenario requires the Bellman-Ford Algorithm to detect and handle potential negative cycles. (Note: Bellman-Ford is beyond the scope of this course.)
The "Which Algorithm?" Decision Tree
1# Decision Logic for Shortest Path Algorithms (Bellman-Ford is out of scope)
2def choose_algorithm(graph):
3    if graph.has_negative_cycle():
4        return 'Bellman-Ford' # O(V*E). (Out of scope)
5    elif graph.is_dag():
6        return 'DAG Shortest Path' # O(V+E) - Fastest
7    elif graph.has_negative_weights():
8        return 'Bellman-Ford' # Handles negative weights (Out of scope)
9    else:
10        return 'Dijkstra\'s' # O(E log V) - Standard positive weights