{ "id": "graphs/graph-bfs", "version": 1, "deterministic": true, "url": "/dsa/graphs/graph-bfs", "topic": { "slug": "graphs", "title": "Graphs" }, "title": "Breadth-First Search", "tagline": "Explore a network in rings, nearest first — using a queue.", "vizKind": "graph", "complexity": { "time": { "best": "O(V + E)", "average": "O(V + E)", "worst": "O(V + E)" }, "space": "O(V)", "growth": "linear", "note": "It touches every node (V) once and looks along every edge (E) once, so the cost is V + E. The space is the queue plus the 'seen' set, which in the worst case holds every node: O(V)." }, "code": { "javascript": "function bfs(graph, start) {\n const queue = [start];\n const seen = new Set([start]);\n while (queue.length > 0) {\n const node = queue.shift();\n visit(node);\n for (const next of graph[node]) {\n if (!seen.has(next)) {\n seen.add(next);\n queue.push(next);\n }\n }\n }\n}", "python": "from collections import deque\n\ndef bfs(graph, start):\n queue = deque([start])\n seen = {start}\n while queue:\n node = queue.popleft()\n visit(node)\n for nxt in graph[node]:\n if nxt not in seen:\n seen.add(nxt)\n queue.append(nxt)\n", "java": "void bfs(Map> graph, String start) {\n Deque queue = new ArrayDeque<>(List.of(start));\n Set seen = new HashSet<>(List.of(start));\n while (!queue.isEmpty()) {\n String node = queue.pollFirst();\n visit(node);\n for (String next : graph.get(node)) {\n if (!seen.contains(next)) {\n seen.add(next);\n queue.addLast(next);\n }\n }\n }\n}", "cpp": "void bfs(map>& graph, string start) {\n queue q; q.push(start);\n set seen{start};\n while (!q.empty()) {\n string node = q.front(); q.pop();\n visit(node);\n for (auto& next : graph[node]) {\n if (!seen.count(next)) {\n seen.insert(next);\n q.push(next);\n }\n }\n }\n}" }, "defaultInput": [ 1 ], "defaultTarget": null, "inputHint": "Pick a start node: 1 = A, up to 8 = H. Watch it explore in rings.", "pitfalls": [ { "wrong": "Forgetting to mark nodes as seen when you add them to the queue.", "right": "Mark a node the moment it enters the queue, not when you visit it. Otherwise the same node gets queued many times and you may loop forever on a cycle." }, { "wrong": "Using a stack instead of a queue.", "right": "A stack turns BFS into DFS — it goes deep instead of wide. If you want rings, you need first-in-first-out: a queue." }, { "wrong": "Checking 'seen' only when popping, not when pushing.", "right": "Too late — the duplicates are already in the queue. Check and mark before pushing, so a node can only ever be added once." } ], "quiz": [ { "q": "What data structure drives breadth-first search?", "options": [ "A stack", "A queue", "A sorted list", "Another graph" ], "answer": 1, "why": "A queue is first-in-first-out, so nodes are visited in the order they were discovered — which means nearer nodes always come out before farther ones. That's what produces the ring-by-ring exploration." }, { "q": "In an unweighted graph, what does BFS find?", "options": [ "The longest path", "The shortest path from the start to every node", "A random path", "Nothing useful" ], "answer": 1, "why": "Because BFS reaches nodes in order of distance, the first time it arrives at a node is by the fewest steps possible — that's the shortest path." }, { "q": "Why must you mark a node 'seen' as soon as it enters the queue?", "options": [ "To make it look nicer", "So it isn't added again and again, which could loop forever on a cycle", "To count the nodes", "You don't need to mark it at all" ], "answer": 1, "why": "Graphs have cycles. Without marking on entry, a node reachable by several paths gets queued multiple times, wasting work and risking an infinite loop." } ], "problems": [ { "name": "Number of Islands", "difficulty": "Medium", "url": "https://leetcode.com/problems/number-of-islands/" }, { "name": "Rotting Oranges", "difficulty": "Medium", "url": "https://leetcode.com/problems/rotting-oranges/" }, { "name": "Binary Tree Level Order Traversal", "difficulty": "Medium", "url": "https://leetcode.com/problems/binary-tree-level-order-traversal/" } ], "trace": { "note": "Each frame is one immutable step: data snapshot, pointers, per-index roles, executing code line, and a one-sentence explanation.", "length": 21, "frames": [ { "data": { "nodes": [ { "id": "A", "label": "A", "x": 0.5, "y": 0.12, "role": "pivot" }, { "id": "B", "label": "B", "x": 0.24, "y": 0.37 }, { "id": "C", "label": "C", "x": 0.76, "y": 0.37 }, { "id": "D", "label": "D", "x": 0.12, "y": 0.66 }, { "id": "E", "label": "E", "x": 0.4, "y": 0.66 }, { "id": "F", "label": "F", "x": 0.6, "y": 0.66 }, { "id": "G", "label": "G", "x": 0.88, "y": 0.66 }, { "id": "H", "label": "H", "x": 0.5, "y": 0.92 } ], "edges": [ { "from": "A", "to": "B" }, { "from": "A", "to": "C" }, { "from": "B", "to": "D" }, { "from": "B", "to": "E" }, { "from": "C", "to": "F" }, { "from": "C", "to": "G" }, { "from": "E", "to": "H" }, { "from": "F", "to": "H" } ], "directed": false, "frontier": { "label": "queue", "ids": [ "A" ] }, "visited": [] }, "pointers": [], "highlights": [], "codeLine": 2, "variables": { "start": "A", "queue": "A" }, "explanation": "Breadth-first search fans out in rings from A, using a queue. A queue is first-in-first-out, so nearer nodes always come off before farther ones — that's why BFS explores level by level.", "stats": { "comparisons": 0, "swaps": 0 }, "accent": "active" }, { "data": { "nodes": [ { "id": "A", "label": "A", "x": 0.5, "y": 0.12, "role": "active" }, { "id": "B", "label": "B", "x": 0.24, "y": 0.37 }, { "id": "C", "label": "C", "x": 0.76, "y": 0.37 }, { "id": "D", "label": "D", "x": 0.12, "y": 0.66 }, { "id": "E", "label": "E", "x": 0.4, "y": 0.66 }, { "id": "F", "label": "F", "x": 0.6, "y": 0.66 }, { "id": "G", "label": "G", "x": 0.88, "y": 0.66 }, { "id": "H", "label": "H", "x": 0.5, "y": 0.92 } ], "edges": [ { "from": "A", "to": "B" }, { "from": "A", "to": "C" }, { "from": "B", "to": "D" }, { "from": "B", "to": "E" }, { "from": "C", "to": "F" }, { "from": "C", "to": "G" }, { "from": "E", "to": "H" }, { "from": "F", "to": "H" } ], "directed": false, "frontier": { "label": "queue", "ids": [] }, "visited": [ "A" ] }, "pointers": [], "highlights": [], "codeLine": 6, "variables": { "visiting": "A", "order": "A" }, "explanation": "Take A off the front of the queue and visit it. Visit order so far: A.", "stats": { "comparisons": 1, "swaps": 0 }, "accent": "active" }, { "data": { "nodes": [ { "id": "A", "label": "A", "x": 0.5, "y": 0.12, "role": "active" }, { "id": "B", "label": "B", "x": 0.24, "y": 0.37, "role": "compare" }, { "id": "C", "label": "C", "x": 0.76, "y": 0.37 }, { "id": "D", "label": "D", "x": 0.12, "y": 0.66 }, { "id": "E", "label": "E", "x": 0.4, "y": 0.66 }, { "id": "F", "label": "F", "x": 0.6, "y": 0.66 }, { "id": "G", "label": "G", "x": 0.88, "y": 0.66 }, { "id": "H", "label": "H", "x": 0.5, "y": 0.92 } ], "edges": [ { "from": "A", "to": "B", "role": "active" }, { "from": "A", "to": "C" }, { "from": "B", "to": "D" }, { "from": "B", "to": "E" }, { "from": "C", "to": "F" }, { "from": "C", "to": "G" }, { "from": "E", "to": "H" }, { "from": "F", "to": "H" } ], "directed": false, "frontier": { "label": "queue", "ids": [ "B" ] }, "visited": [ "A" ] }, "pointers": [], "highlights": [], "codeLine": 10, "variables": { "from": "A", "discovered": "B", "queue": "B" }, "explanation": "B is a new neighbour of A. Mark it seen and add it to the queue.", "stats": { "comparisons": 1, "swaps": 0 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A", "x": 0.5, "y": 0.12, "role": "active" }, { "id": "B", "label": "B", "x": 0.24, "y": 0.37, "role": "pivot" }, { "id": "C", "label": "C", "x": 0.76, "y": 0.37, "role": "compare" }, { "id": "D", "label": "D", "x": 0.12, "y": 0.66 }, { "id": "E", "label": "E", "x": 0.4, "y": 0.66 }, { "id": "F", "label": "F", "x": 0.6, "y": 0.66 }, { "id": "G", "label": "G", "x": 0.88, "y": 0.66 }, { "id": "H", "label": "H", "x": 0.5, "y": 0.92 } ], "edges": [ { "from": "A", "to": "B" }, { "from": "A", "to": "C", "role": "active" }, { "from": "B", "to": "D" }, { "from": "B", "to": "E" }, { "from": "C", "to": "F" }, { "from": "C", "to": "G" }, { "from": "E", "to": "H" }, { "from": "F", "to": "H" } ], "directed": false, "frontier": { "label": "queue", "ids": [ "B", "C" ] }, "visited": [ "A" ] }, "pointers": [], "highlights": [], "codeLine": 10, "variables": { "from": "A", "discovered": "C", "queue": "B C" }, "explanation": "C is a new neighbour of A. Mark it seen and add it to the queue.", "stats": { "comparisons": 1, "swaps": 0 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A", "x": 0.5, "y": 0.12, "role": "sorted" }, { "id": "B", "label": "B", "x": 0.24, "y": 0.37, "role": "active" }, { "id": "C", "label": "C", "x": 0.76, "y": 0.37, "role": "pivot" }, { "id": "D", "label": "D", "x": 0.12, "y": 0.66 }, { "id": "E", "label": "E", "x": 0.4, "y": 0.66 }, { "id": "F", "label": "F", "x": 0.6, "y": 0.66 }, { "id": "G", "label": "G", "x": 0.88, "y": 0.66 }, { "id": "H", "label": "H", "x": 0.5, "y": 0.92 } ], "edges": [ { "from": "A", "to": "B" }, { "from": "A", "to": "C" }, { "from": "B", "to": "D" }, { "from": "B", "to": "E" }, { "from": "C", "to": "F" }, { "from": "C", "to": "G" }, { "from": "E", "to": "H" }, { "from": "F", "to": "H" } ], "directed": false, "frontier": { "label": "queue", "ids": [ "C" ] }, "visited": [ "A", "B" ] }, "pointers": [], "highlights": [], "codeLine": 6, "variables": { "visiting": "B", "order": "A · B" }, "explanation": "Take B off the front of the queue and visit it. 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Mark it seen and add it to the queue.", "stats": { "comparisons": 2, "swaps": 0 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A", "x": 0.5, "y": 0.12, "role": "sorted" }, { "id": "B", "label": "B", "x": 0.24, "y": 0.37, "role": "active" }, { "id": "C", "label": "C", "x": 0.76, "y": 0.37, "role": "pivot" }, { "id": "D", "label": "D", "x": 0.12, "y": 0.66, "role": "pivot" }, { "id": "E", "label": "E", "x": 0.4, "y": 0.66, "role": "compare" }, { "id": "F", "label": "F", "x": 0.6, "y": 0.66 }, { "id": "G", "label": "G", "x": 0.88, "y": 0.66 }, { "id": "H", "label": "H", "x": 0.5, "y": 0.92 } ], "edges": [ { "from": "A", "to": "B" }, { "from": "A", "to": "C" }, { "from": "B", "to": "D" }, { "from": "B", "to": "E", "role": "active" }, { "from": "C", "to": "F" }, { "from": "C", "to": "G" }, { "from": "E", "to": "H" }, { "from": "F", "to": "H" } ], "directed": false, "frontier": { "label": "queue", "ids": [ "C", "D", "E" ] }, "visited": [ "A", "B" ] }, "pointers": [], "highlights": [], "codeLine": 10, "variables": { "from": "B", "discovered": "E", "queue": "C D E" }, "explanation": "E is a new neighbour of B. Mark it seen and add it to the queue.", "stats": { "comparisons": 2, "swaps": 0 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A", "x": 0.5, "y": 0.12, "role": "sorted" }, { "id": "B", "label": "B", "x": 0.24, "y": 0.37, "role": "sorted" }, { "id": "C", "label": "C", "x": 0.76, "y": 0.37, "role": "active" }, { "id": "D", "label": "D", "x": 0.12, "y": 0.66, "role": "pivot" }, { "id": "E", "label": "E", "x": 0.4, "y": 0.66, "role": "pivot" }, { "id": "F", "label": "F", "x": 0.6, "y": 0.66 }, { "id": "G", "label": "G", "x": 0.88, "y": 0.66 }, { "id": "H", "label": "H", "x": 0.5, "y": 0.92 } ], "edges": [ { "from": "A", "to": "B" }, { "from": "A", "to": "C" }, { "from": "B", "to": "D" }, { "from": "B", "to": "E" }, { "from": "C", "to": "F" }, { "from": "C", "to": "G" }, { "from": "E", "to": "H" }, { "from": "F", "to": "H" } ], "directed": false, "frontier": { "label": "queue", "ids": [ "D", "E" ] }, "visited": [ "A", "B", "C" ] }, "pointers": [], "highlights": [], "codeLine": 6, "variables": { "visiting": "C", "order": "A · B · C" }, "explanation": "Take C off the front of the queue and visit it. Visit order so far: A · B · C.", "stats": { "comparisons": 3, "swaps": 0 }, "accent": "active" }, { "data": { "nodes": [ { "id": "A", "label": "A", "x": 0.5, "y": 0.12, "role": "sorted" }, { "id": "B", "label": "B", "x": 0.24, "y": 0.37, "role": "sorted" }, { "id": "C", "label": "C", "x": 0.76, "y": 0.37, "role": "active" }, { "id": "D", "label": "D", "x": 0.12, "y": 0.66, "role": "pivot" }, { "id": "E", "label": "E", "x": 0.4, "y": 0.66, "role": "pivot" }, { "id": "F", "label": "F", "x": 0.6, "y": 0.66, "role": "compare" }, { "id": "G", "label": "G", "x": 0.88, "y": 0.66 }, { "id": "H", "label": "H", "x": 0.5, "y": 0.92 } ], "edges": [ { "from": "A", "to": "B" }, { "from": "A", "to": "C" }, { "from": "B", "to": "D" }, { "from": "B", "to": "E" }, { "from": "C", "to": "F", "role": "active" }, { "from": "C", "to": "G" }, { "from": "E", "to": "H" }, { "from": "F", "to": "H" } ], "directed": false, "frontier": { "label": "queue", "ids": [ "D", "E", "F" ] }, "visited": [ "A", "B", "C" ] }, "pointers": [], "highlights": [], "codeLine": 10, "variables": { "from": "C", "discovered": "F", "queue": "D E F" }, "explanation": "F is a new neighbour of C. 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Mark it seen and add it to the queue.", "stats": { "comparisons": 3, "swaps": 0 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A", "x": 0.5, "y": 0.12, "role": "sorted" }, { "id": "B", "label": "B", "x": 0.24, "y": 0.37, "role": "sorted" }, { "id": "C", "label": "C", "x": 0.76, "y": 0.37, "role": "sorted" }, { "id": "D", "label": "D", "x": 0.12, "y": 0.66, "role": "active" }, { "id": "E", "label": "E", "x": 0.4, "y": 0.66, "role": "pivot" }, { "id": "F", "label": "F", "x": 0.6, "y": 0.66, "role": "pivot" }, { "id": "G", "label": "G", "x": 0.88, "y": 0.66, "role": "pivot" }, { "id": "H", "label": "H", "x": 0.5, "y": 0.92 } ], "edges": [ { "from": "A", "to": "B" }, { "from": "A", "to": "C" }, { "from": "B", "to": "D" }, { "from": "B", "to": "E" }, { "from": "C", "to": "F" }, { "from": "C", "to": "G" }, { "from": "E", "to": "H" }, { "from": "F", "to": "H" } ], "directed": false, "frontier": { "label": "queue", "ids": [ "E", "F", "G" ] }, "visited": [ "A", "B", "C", "D" ] }, "pointers": [], "highlights": [], "codeLine": 6, "variables": { "visiting": "D", "order": "A · B · C · D" }, "explanation": "Take D off the front of the queue and visit it. 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