Auxiliary Space Complexity of Bellman–Ford Algorithm
The auxiliary space complexity of the Bellman-Ford algorithm is O(V), where V is the number of vertices in the graph, primarily due to the need to store distances from the source vertex to all other vertices.
- A distance array, storing distances from the source vertex to every other vertex, requiring O(V) space.
- Additional data structures, such as for tracking predecessors or relaxation updates, contributing linearly to space complexity.
- Optional use of queues or stacks for vertex relaxation, which typically require minimal space compared to primary data structures.
Time and Space Complexity of Bellman–Ford Algorithm
The Bellman-Ford algorithm has a time complexity of O(V*E), where V is the number of vertices and E is the number of edges in the graph. In the worst-case scenario, the algorithm needs to iterate through all edges for each vertex, resulting in this time complexity. The space complexity of the Bellman-Ford algorithm is O(V), where V is the number of vertices in the graph. This space complexity is mainly due to storing the distances from the source vertex to all other vertices in the graph.
| Operation | Time Complexity | Space Complexity |
|---|---|---|
| Initialization | O(V) | O(V) |
| Relaxation | O(V*E) | O(1) |
| Overall Complexity | O(V*E) | O(V) |
Let’s explore the detailed time and space complexity of the Bellman–Ford Algorithm: