Detect a negative cycle in a Graph

Given a directed weighted graph, your task is to find whether the given graph contains any negative cycles that are reachable from the source vertex (e.g., node 0).

Note: A negative-weight cycle is a cycle in a graph whose edges sum to a negative value.

Examples:

Input: V = 4, edges[][] = [[0, 3, 6], [1, 0, 4], [1, 2, 6], [3, 1, 2]]

👁 Example1

Output: 0
Explanation : Cycle 1 → 0 → 3 → 1 has total weight 6 + 4 + 2 = 12, which is positive, so no negative weight cycle exists.

Input : V = 4, edges[][] = [[1, 0, 4], [3, 1, -2], [1, 2, -6], [2, 3, 5]]
👁 Example2
Output: 1
Explanation : There is a cycle 1 → 2 → 3 → 1 with total weight -3, which is negative, so a negative weight cycle exists.

Detecting Negative Weight Cycle using Bellman-Ford – O(V * E) Time and O(V) Space

The idea is to use the Bellman-Ford Algorithm where we relax all edges V-1 times to compute shortest paths. In a normal graph, distances stabilize after these iterations. If we can still relax any edge after that, it means the distance is continuously decreasing, which happens only when there is a negative weight cycle in the graph.

• Initialize a distance array dist with all values as 0
• Perform edge relaxation (n - 1) times:
  • For each edge (u, v, wt)
  • If dist[u] + wt < dist[v], update dist[v]
• Run one more iteration over all edges:
  • If any edge still relaxes → return 1 (negative cycle exists)
• If no relaxation happens → return 0

true

Related articles:

• Detecting negative cycle using Floyd Warshall
• Dijkstra’s Algorithm
• Bellman–Ford Algorithm