27.2.4. Negative Cycles

Try these first

1. 1.

   What is the primary purpose of Dijkstra's algorithm?

   Hint

   Think of how a navigator finds the quickest route.

2. 2.

   What do we call the local choices made in greedy algorithms?

   Hint

   Consider what happens when you always pick the smallest option.

3. 3.

   What does Dijkstra's algorithm primarily find?

   • Shortest paths from multiple sources
   • Shortest paths from one source
   • All paths in a graph

   Hint

   Recall which vertex serves as the starting point.

4. 4.

   True or False: Negative cycles can exist without negative edges.

   • True
   • False

   Hint

   Think about the costs involved in each cycle.

5. 5.

   Create a directed graph with 5 vertices and establish both positive and negative weights. Identify if the graph contains a negative cycle, and apply Dijkstra’s algorithm to see how it handles paths.

   Hint

   Trace paths and calculate costs to either reveal a negative cycle or establish valid shortest paths.

6. 6.

   Design an example illustrating how Dijkstra's algorithm efficiently manages a scenario without negative weights and compare the result when negative weights are added.

   Hint

   Keep track of which edges optimize paths best.