{ "id": "graphs/bellman-ford", "version": 1, "deterministic": true, "url": "/dsa/graphs/bellman-ford", "topic": { "slug": "graphs", "title": "Graphs" }, "title": "Bellman–Ford", "tagline": "Slower than Dijkstra, and it can do the one thing Dijkstra can't.", "vizKind": "graph", "complexity": { "time": { "best": "O(E)", "average": "O(V × E)", "worst": "O(V × E)" }, "space": "O(V)", "growth": "quadratic", "note": "V−1 rounds, each sweeping all E edges: O(V × E). That's markedly slower than Dijkstra's O(E log V). You pay that price for two things Dijkstra cannot give you: negative edges, and negative-cycle detection." }, "code": { "javascript": "function bellmanFord(nodes, edges, start) {\n const dist = {};\n for (const v of nodes) dist[v] = Infinity;\n dist[start] = 0;\n\n // Relax EVERY edge, n-1 times. No cleverness at all.\n for (let round = 1; round < nodes.length; round++) {\n for (const [u, v, w] of edges) {\n if (dist[u] + w < dist[v]) dist[v] = dist[u] + w;\n }\n }\n\n // One more round. If anything still improves, a negative\n // cycle exists — and \"shortest\" has no meaning.\n for (const [u, v, w] of edges) {\n if (dist[u] + w < dist[v]) return null; // negative cycle\n }\n\n return dist;\n}", "python": "def bellman_ford(nodes, edges, start):\n dist = {v: float('inf') for v in nodes}\n dist[start] = 0\n\n # Relax EVERY edge, n-1 times. No cleverness at all.\n for _ in range(len(nodes) - 1):\n for u, v, w in edges:\n if dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\n\n\n # One more round. If anything still improves, a negative\n # cycle exists — and \"shortest\" has no meaning.\n for u, v, w in edges:\n if dist[u] + w < dist[v]:\n return None # negative cycle\n\n return dist\n", "java": "Map bellmanFord(List nodes, List edges, String start) {\n Map dist = new HashMap<>();\n for (String v : nodes) dist.put(v, INF);\n dist.put(start, 0);\n\n // Relax EVERY edge, n-1 times. No cleverness at all.\n for (int round = 1; round < nodes.size(); round++) {\n for (Edge e : edges) {\n if (dist.get(e.u) + e.w < dist.get(e.v))\n dist.put(e.v, dist.get(e.u) + e.w);\n }\n }\n\n // One more round. If anything still improves, a negative\n // cycle exists — and \"shortest\" has no meaning.\n for (Edge e : edges) {\n if (dist.get(e.u) + e.w < dist.get(e.v)) return null;\n }\n\n return dist;\n}", "cpp": "map bellmanFord(vector& nodes, vector& edges, string start) {\n map dist;\n for (auto& v : nodes) dist[v] = INF;\n dist[start] = 0;\n\n // Relax EVERY edge, n-1 times. No cleverness at all.\n for (int round = 1; round < (int)nodes.size(); round++) {\n for (auto& [u, v, w] : edges) {\n if (dist[u] + w < dist[v]) dist[v] = dist[u] + w;\n }\n }\n\n // One more round. If anything still improves, a negative\n // cycle exists — and \"shortest\" has no meaning.\n for (auto& [u, v, w] : edges) {\n if (dist[u] + w < dist[v]) return {}; // negative cycle\n }\n\n return dist;\n}" }, "defaultInput": [ 1 ], "defaultTarget": null, "inputHint": "The pink edge C→B costs −4. Watch a negative edge make a longer route cheaper.", "pitfalls": [ { "wrong": "Reaching for Dijkstra on a graph with negative edges.", "right": "It returns a wrong answer without any error — the worst failure mode there is. Negative weights mean Bellman–Ford." }, { "wrong": "Skipping the final extra round.", "right": "That round is the negative-cycle detector, and it's half the reason to use this algorithm. Without it you'll happily report distances that don't exist." }, { "wrong": "Running only a few rounds because 'it looks settled'.", "right": "You need V−1 rounds in the worst case. Do add the early exit — if a whole round changes nothing, stop — but don't guess a smaller number." }, { "wrong": "Thinking a negative cycle just means 'a very cheap route'.", "right": "It means there is *no* shortest path. You could loop forever, getting cheaper each time. The right answer is 'this question has no answer'." } ], "quiz": [ { "q": "Why does Dijkstra fail on negative edges?", "options": [ "It crashes", "It finalises a node as soon as it's nearest, assuming a longer route can never be cheaper — which a negative edge makes false", "It runs forever", "It doesn't fail" ], "answer": 1, "why": "Dijkstra's proof depends on distances only ever growing as you travel. A negative edge breaks that, so the 'final' distance it commits to can later turn out to be beatable — and it never re-checks." }, { "q": "How does Bellman–Ford detect a negative cycle?", "options": [ "It counts the edges", "It runs one extra round — if anything STILL improves, a negative cycle exists", "It looks for negative weights", "It can't detect one" ], "answer": 1, "why": "After V−1 rounds everything should be final. If another sweep still finds an improvement, there must be a loop you can ride round to keep getting cheaper — so no shortest path exists." }, { "q": "What do you give up by using Bellman–Ford instead of Dijkstra?", "options": [ "Correctness", "Speed — O(V × E) instead of O(E log V)", "The ability to handle cycles", "Nothing" ], "answer": 1, "why": "It's markedly slower, because it brute-forces every edge V−1 times instead of being greedy. You pay that for negative edges and cycle detection." } ], "problems": [ { "name": "Cheapest Flights Within K Stops", "difficulty": "Medium", "url": "https://leetcode.com/problems/cheapest-flights-within-k-stops/" }, { "name": "Network Delay Time", "difficulty": "Medium", "url": "https://leetcode.com/problems/network-delay-time/" }, { "name": "Find the City With the Smallest Number of Neighbors", "difficulty": "Medium", "url": "https://leetcode.com/problems/find-the-city-with-the-smallest-number-of-neighbors-at-a-threshold-distance/" } ], "trace": { "note": "Each frame is one immutable step: data snapshot, pointers, per-index roles, executing code line, and a one-sentence explanation.", "length": 17, "frames": [ { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B ∞", "x": 0.38, "y": 0.2 }, { "id": "C", "label": "C ∞", "x": 0.38, "y": 0.8 }, { "id": "D", "label": "D ∞", "x": 0.66, "y": 0.5 }, { "id": "E", "label": "E ∞", "x": 0.9, "y": 0.5 } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 4, "variables": { "start": "A", "negativeEdge": "C → B = −4" }, "explanation": "Look at the pink edge: C → B costs **−4**. Dijkstra would go badly wrong here — it finalises B early on the cheap-looking direct route and never reconsiders. Bellman–Ford makes no assumptions at all, so it copes.", "stats": { "comparisons": 0, "swaps": 0 }, "accent": "active" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B ∞", "x": 0.38, "y": 0.2 }, { "id": "C", "label": "C ∞", "x": 0.38, "y": 0.8 }, { "id": "D", "label": "D ∞", "x": 0.66, "y": 0.5 }, { "id": "E", "label": "E ∞", "x": 0.9, "y": 0.5 } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 7, "variables": { "round": 1, "of": 4 }, "explanation": "Round 1 of 4. Try to improve *every* edge, in order. No cleverness — just sweep the lot.", "stats": { "comparisons": 0, "swaps": 0 }, "accent": "pivot" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "active" }, { "id": "B", "label": "B ∞", "x": 0.38, "y": 0.2, "role": "found" }, { "id": "C", "label": "C ∞", "x": 0.38, "y": 0.8 }, { "id": "D", "label": "D ∞", "x": 0.66, "y": 0.5 }, { "id": "E", "label": "E ∞", "x": 0.9, "y": 0.5 } ], "edges": [ { "from": "A", "to": "B", "weight": 6, "role": "found" }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "edge": "A→B", "weight": 6, "via": 6, "known": "∞" }, "explanation": "A→B: 0 + 6 = 6, better than ∞. Improve it.", "stats": { "comparisons": 1, "swaps": 0 }, "accent": "found" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "active" }, { "id": "B", "label": "B 6", "x": 0.38, "y": 0.2, "role": "sorted" }, { "id": "C", "label": "C ∞", "x": 0.38, "y": 0.8, "role": "found" }, { "id": "D", "label": "D ∞", "x": 0.66, "y": 0.5 }, { "id": "E", "label": "E ∞", "x": 0.9, "y": 0.5 } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3, "role": "found" }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "edge": "A→C", "weight": 3, "via": 3, "known": "∞" }, "explanation": "A→C: 0 + 3 = 3, better than ∞. Improve it.", "stats": { "comparisons": 2, "swaps": 1 }, "accent": "found" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B 6", "x": 0.38, "y": 0.2, "role": "found" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "active" }, { "id": "D", "label": "D ∞", "x": 0.66, "y": 0.5 }, { "id": "E", "label": "E ∞", "x": 0.9, "y": 0.5 } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "found" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "edge": "C→B", "weight": -4, "via": -1, "known": "6" }, "explanation": "C→B: 3 + -4 = -1, better than 6. Improve it. Notice a *negative* edge just made a route cheaper — this is precisely what Dijkstra cannot cope with.", "stats": { "comparisons": 3, "swaps": 2 }, "accent": "found" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B -1", "x": 0.38, "y": 0.2, "role": "active" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "sorted" }, { "id": "D", "label": "D ∞", "x": 0.66, "y": 0.5, "role": "found" }, { "id": "E", "label": "E ∞", "x": 0.9, "y": 0.5 } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2, "role": "found" }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "edge": "B→D", "weight": 2, "via": 1, "known": "∞" }, "explanation": "B→D: -1 + 2 = 1, better than ∞. Improve it.", "stats": { "comparisons": 4, "swaps": 3 }, "accent": "found" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B -1", "x": 0.38, "y": 0.2, "role": "sorted" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "active" }, { "id": "D", "label": "D 1", "x": 0.66, "y": 0.5, "role": "compare" }, { "id": "E", "label": "E ∞", "x": 0.9, "y": 0.5 } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8, "role": "compare" }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "edge": "C→D", "weight": 8, "via": 11, "known": "1" }, "explanation": "C→D: 3 + 8 = 11, no better than 1. Leave it.", "stats": { "comparisons": 5, "swaps": 4 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B -1", "x": 0.38, "y": 0.2, "role": "sorted" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "sorted" }, { "id": "D", "label": "D 1", "x": 0.66, "y": 0.5, "role": "active" }, { "id": "E", "label": "E ∞", "x": 0.9, "y": 0.5, "role": "found" } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3, "role": "found" } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "edge": "D→E", "weight": 3, "via": 4, "known": "∞" }, "explanation": "D→E: 1 + 3 = 4, better than ∞. Improve it.", "stats": { "comparisons": 6, "swaps": 4 }, "accent": "found" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B -1", "x": 0.38, "y": 0.2, "role": "sorted" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "sorted" }, { "id": "D", "label": "D 1", "x": 0.66, "y": 0.5, "role": "sorted" }, { "id": "E", "label": "E 4", "x": 0.9, "y": 0.5, "role": "sorted" } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 7, "variables": { "round": 2, "of": 4 }, "explanation": "Round 2 of 4. Try to improve *every* edge, in order. No cleverness — just sweep the lot.", "stats": { "comparisons": 6, "swaps": 5 }, "accent": "pivot" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "active" }, { "id": "B", "label": "B -1", "x": 0.38, "y": 0.2, "role": "compare" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "sorted" }, { "id": "D", "label": "D 1", "x": 0.66, "y": 0.5, "role": "sorted" }, { "id": "E", "label": "E 4", "x": 0.9, "y": 0.5, "role": "sorted" } ], "edges": [ { "from": "A", "to": "B", "weight": 6, "role": "compare" }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "edge": "A→B", "weight": 6, "via": 6, "known": "-1" }, "explanation": "A→B: 0 + 6 = 6, no better than -1. Leave it.", "stats": { "comparisons": 7, "swaps": 5 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "active" }, { "id": "B", "label": "B -1", "x": 0.38, "y": 0.2, "role": "sorted" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "compare" }, { "id": "D", "label": "D 1", "x": 0.66, "y": 0.5, "role": "sorted" }, { "id": "E", "label": "E 4", "x": 0.9, "y": 0.5, "role": "sorted" } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3, "role": "compare" }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "edge": "A→C", "weight": 3, "via": 3, "known": "3" }, "explanation": "A→C: 0 + 3 = 3, no better than 3. Leave it.", "stats": { "comparisons": 8, "swaps": 5 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B -1", "x": 0.38, "y": 0.2, "role": "compare" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "active" }, { "id": "D", "label": "D 1", "x": 0.66, "y": 0.5, "role": "sorted" }, { "id": "E", "label": "E 4", "x": 0.9, "y": 0.5, "role": "sorted" } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "compare" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "edge": "C→B", "weight": -4, "via": -1, "known": "-1" }, "explanation": "C→B: 3 + -4 = -1, no better than -1. Leave it.", "stats": { "comparisons": 9, "swaps": 5 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B -1", "x": 0.38, "y": 0.2, "role": "active" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "sorted" }, { "id": "D", "label": "D 1", "x": 0.66, "y": 0.5, "role": "compare" }, { "id": "E", "label": "E 4", "x": 0.9, "y": 0.5, "role": "sorted" } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2, "role": "compare" }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "edge": "B→D", "weight": 2, "via": 1, "known": "1" }, "explanation": "B→D: -1 + 2 = 1, no better than 1. Leave it.", "stats": { "comparisons": 10, "swaps": 5 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B -1", "x": 0.38, "y": 0.2, "role": "sorted" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "active" }, { "id": "D", "label": "D 1", "x": 0.66, "y": 0.5, "role": "compare" }, { "id": "E", "label": "E 4", "x": 0.9, "y": 0.5, "role": "sorted" } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8, "role": "compare" }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "edge": "C→D", "weight": 8, "via": 11, "known": "1" }, "explanation": "C→D: 3 + 8 = 11, no better than 1. Leave it.", "stats": { "comparisons": 11, "swaps": 5 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B -1", "x": 0.38, "y": 0.2, "role": "sorted" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "sorted" }, { "id": "D", "label": "D 1", "x": 0.66, "y": 0.5, "role": "active" }, { "id": "E", "label": "E 4", "x": 0.9, "y": 0.5, "role": "compare" } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3, "role": "compare" } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "edge": "D→E", "weight": 3, "via": 4, "known": "4" }, "explanation": "D→E: 1 + 3 = 4, no better than 4. Leave it.", "stats": { "comparisons": 12, "swaps": 5 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B -1", "x": 0.38, "y": 0.2, "role": "sorted" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "sorted" }, { "id": "D", "label": "D 1", "x": 0.66, "y": 0.5, "role": "sorted" }, { "id": "E", "label": "E 4", "x": 0.9, "y": 0.5, "role": "sorted" } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 7, "variables": { "round": 2, "settled": true }, "explanation": "A whole round with nothing improved. Everything has settled, so we can stop early — the remaining rounds would change nothing.", "stats": { "comparisons": 12, "swaps": 5 }, "accent": "sorted" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.12, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B -1", "x": 0.38, "y": 0.2, "role": "sorted" }, { "id": "C", "label": "C 3", "x": 0.38, "y": 0.8, "role": "sorted" }, { "id": "D", "label": "D 1", "x": 0.66, "y": 0.5, "role": "sorted" }, { "id": "E", "label": "E 4", "x": 0.9, "y": 0.5, "role": "sorted" } ], "edges": [ { "from": "A", "to": "B", "weight": 6 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "C", "to": "B", "weight": -4, "role": "pivot" }, { "from": "B", "to": "D", "weight": 2 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "D", "to": "E", "weight": 3 } ], "directed": true }, "pointers": [], "highlights": [], "codeLine": 15, "variables": { "result": "A=0 B=-1 C=3 D=1 E=4", "negativeCycle": false }, "explanation": "Done: A=0 B=-1 C=3 D=1 E=4. No negative cycle — one extra sweep improved nothing, which proves these distances are final. Bellman–Ford is slower than Dijkstra (O(V·E) against O(E log V)), and in exchange it handles negative edges and detects the impossible case.", "stats": { "comparisons": 12, "swaps": 5 }, "accent": "sorted" } ] } }