Dijkstra vs Bellman-Ford

Jul 2, 2026knoxiboy

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Dijkstra's Algorithm and the Bellman-Ford Algorithm are both designed to solve the Single-Source Shortest Path (SSSP) problem, but they have major differences in constraints and efficiency.

Comparison Table

Variables:

• $V$: Number of vertices (nodes)
• $E$: Number of edges

Feature	Dijkstra's Algorithm	Bellman-Ford Algorithm
Time Complexity	$O((V + E) \log V)$ (with Min-Heap)	$O(V \cdot E)$
Negative Edges	May produce incorrect results (greedy assumption is violated).	Handles negative edge weights safely.
Negative Cycles	Cannot detect (may loop infinitely or return incorrect distances).	Detects negative cycles and reports them.
Approach	Greedy.	Dynamic Programming (relaxation of all edges).
Applicability	Single-source shortest path.	Single-source shortest path, routing protocols.

How Relaxation Differs

• Dijkstra greedily selects the closest unvisited vertex and relaxes its outgoing edges. It visits each vertex once.
• Bellman-Ford relaxes all $E$ edges in the graph $V-1$ times. It does not make any greedy assumptions, allowing it to correctly propagate negative weights.

Decision Criteria

• Choose Dijkstra's for most general shortest path applications (like maps or routing) where all edge weights are non-negative, as it is much faster.
• Choose Bellman-Ford only when negative edge weights are present or when you need to detect negative cycles (e.g., in financial arbitrage detection).