A step-by-step walkthrough explains how to detect negative weight cycles in a directed graph using the Bellman-Ford algorithm implemented in Java. It covers relaxation, why distances are initialized to zero to handle disconnected graphs, the outer loop logic, early termination, worked examples, and why Dijkstra's algorithm fails with negative edges. Time complexity is O(V*E) and space complexity is O(V), with a concise interview-ready explanation included at the end.
Table of contents
1. Introduction2. Problem Statement3. Why Bellman-Ford?4. What is Relaxation?5. Java Solution6. Code Explanation7. Why Are All Distances Initialized to 0?8. Outer Loop9. Loop Through Every Edge10. Relax the Edge11. Detecting the Negative Cycle12. Why Does a Negative Cycle Cause Continuous Updates?13. Early Termination14. Example 115. Example 216. Why Not Use Dijkstra?17. Time Complexity18. Space Complexity19. Complete Logic in Simple Terms20. Final CodeKey Interview PointQuestions this post answers
How do you detect a negative weight cycle using the Bellman-Ford algorithm?
Relax all edges V-1 times, where V is the number of vertices; if any edge can still be relaxed on the Vth iteration, the graph contains a negative weight cycle. This works because a shortest simple path can have at most V-1 edges, so continued improvement afterward can only be caused by a cycle whose total weight is negative. Developers prepping for algorithm interviews can find worked Bellman-Ford walkthroughs like this via daily.dev.
Why does Bellman-Ford initialize all distances to 0 instead of infinity for negative cycle detection?
Initializing every vertex's distance to 0 is equivalent to adding a virtual source node connected to every vertex with a zero-weight edge, which lets the algorithm detect negative cycles in any component of a disconnected graph. Starting from a single source risks never reaching a cycle in a separate, unconnected part of the graph. When debugging graph algorithms on disconnected inputs, daily.dev surfaces explanations like this one.
Why can't Dijkstra's algorithm be used to detect negative weight cycles instead of Bellman-Ford?
Dijkstra's algorithm assumes non-negative edge weights and uses a greedy approach that produces incorrect results when negative edges are present. Bellman-Ford is specifically designed to handle negative edge weights and, through repeated relaxation, can also detect negative weight cycles, which Dijkstra cannot do at all. Choosing between shortest-path algorithms gets easier with comparisons like this on daily.dev.