{ "id": "graphs/dijkstra", "version": 1, "deterministic": true, "url": "/dsa/graphs/dijkstra", "topic": { "slug": "graphs", "title": "Graphs" }, "title": "Dijkstra's Shortest Path", "tagline": "The cheapest route through a weighted map — always finish the nearest place first.", "vizKind": "graph", "complexity": { "time": { "best": "O(E log V)", "average": "O(E log V)", "worst": "O(E log V)" }, "space": "O(V)", "growth": "nlogn", "note": "With a priority queue (a heap!) to grab the nearest unfinished node, it's O(E log V). Without one you scan every node each round, giving O(V²) — still fine for small dense graphs. This is a real payoff from the Heap lesson." }, "code": { "javascript": "function dijkstra(graph, start) {\n const dist = {};\n for (const v of graph.nodes) dist[v] = Infinity;\n dist[start] = 0;\n const done = new Set();\n\n while (done.size < graph.nodes.length) {\n // the unfinished node with the smallest known distance\n const u = closest(dist, done);\n if (u === null) break;\n done.add(u);\n\n for (const [v, w] of graph.edges[u]) {\n if (dist[u] + w < dist[v]) {\n dist[v] = dist[u] + w;\n }\n }\n }\n return dist;\n}", "python": "import heapq\n\ndef dijkstra(graph, start):\n dist = {v: float('inf') for v in graph}\n dist[start] = 0\n done = set()\n\n pq = [(0, start)]\n while pq:\n d, u = heapq.heappop(pq)\n if u in done: continue\n done.add(u)\n\n for v, w in graph[u]:\n if dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\n heapq.heappush(pq, (dist[v], v))\n\n return dist\n", "java": "Map dijkstra(Graph g, String start) {\n Map dist = new HashMap<>();\n for (String v : g.nodes) dist.put(v, Integer.MAX_VALUE);\n dist.put(start, 0);\n Set done = new HashSet<>();\n\n while (done.size() < g.nodes.size()) {\n String u = closest(dist, done);\n if (u == null) break;\n done.add(u);\n\n for (Edge e : g.edges.get(u)) {\n if (dist.get(u) + e.w < dist.get(e.to)) {\n dist.put(e.to, dist.get(u) + e.w);\n }\n }\n }\n return dist;\n}", "cpp": "map dijkstra(Graph& g, string start) {\n map dist;\n for (auto& v : g.nodes) dist[v] = INT_MAX;\n dist[start] = 0;\n set done;\n\n while ((int)done.size() < (int)g.nodes.size()) {\n string u = closest(dist, done);\n if (u.empty()) break;\n done.insert(u);\n\n for (auto& [v, w] : g.edges[u]) {\n if (dist[u] + w < dist[v]) {\n dist[v] = dist[u] + w;\n }\n }\n }\n return dist;\n}" }, "defaultInput": [ 1 ], "defaultTarget": null, "inputHint": "Pick a start node: 1 = A, up to 7 = G. Numbers on the roads are their cost.", "pitfalls": [ { "wrong": "Using Dijkstra on a graph with negative edge weights.", "right": "It will confidently return the wrong answer. 'Nearest unfinished node is final' is only true when travelling further can't make things cheaper. Use Bellman–Ford for negative edges." }, { "wrong": "Re-finalising a node that's already done.", "right": "Once a node is finalised, skip it forever. Popping stale entries out of the priority queue without checking is a classic source of wrong answers and infinite work." }, { "wrong": "Confusing Dijkstra with BFS.", "right": "BFS finds the fewest *edges*; Dijkstra finds the lowest *total weight*. On an unweighted graph they agree — and Dijkstra reduces to BFS exactly." }, { "wrong": "Scanning every node to find the closest, on a big graph.", "right": "That's O(V²). Use a priority queue so 'get me the closest' costs O(log V) instead of O(V)." } ], "quiz": [ { "q": "Why can Dijkstra declare the nearest unfinished node's distance final?", "options": [ "Because it checked every possible route first", "Because any other route would first have to pass somewhere even further away, so it would cost more", "Because it's a guess that usually works", "It can't — it revisits nodes later" ], "answer": 1, "why": "That's the whole proof. If a node is the closest thing left, every alternative route to it detours through something more distant — which already costs more. So no cheaper route can exist." }, { "q": "What breaks Dijkstra?", "options": [ "Very large graphs", "Negative edge weights", "Cycles", "Disconnected nodes" ], "answer": 1, "why": "Negative edges destroy the argument: travelling further could make a route cheaper, so 'nearest so far is final' stops being true. Bellman–Ford handles that case." }, { "q": "What data structure makes Dijkstra fast?", "options": [ "A stack", "A hash map", "A priority queue (heap)", "A linked list" ], "answer": 2, "why": "It constantly needs the closest unfinished node. A heap gives you that in O(log V) instead of scanning all V nodes — turning O(V²) into O(E log V)." } ], "problems": [ { "name": "Network Delay Time", "difficulty": "Medium", "url": "https://leetcode.com/problems/network-delay-time/" }, { "name": "Path with Maximum Probability", "difficulty": "Medium", "url": "https://leetcode.com/problems/path-with-maximum-probability/" }, { "name": "Cheapest Flights Within K Stops", "difficulty": "Medium", "url": "https://leetcode.com/problems/cheapest-flights-within-k-stops/" } ], "trace": { "note": "Each frame is one immutable step: data snapshot, pointers, per-index roles, executing code line, and a one-sentence explanation.", "length": 28, "frames": [ { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.1, "y": 0.5, "role": "active" }, { "id": "B", "label": "B ∞", "x": 0.35, "y": 0.18 }, { "id": "C", "label": "C ∞", "x": 0.35, "y": 0.82 }, { "id": "D", "label": "D ∞", "x": 0.62, "y": 0.5 }, { "id": "E", "label": "E ∞", "x": 0.62, "y": 0.12 }, { "id": "F", "label": "F ∞", "x": 0.9, "y": 0.35 }, { "id": "G", "label": "G ∞", "x": 0.9, "y": 0.8 } ], "edges": [ { "from": "A", "to": "B", "weight": 4 }, { "from": "A", "to": "C", "weight": 2 }, { "from": "B", "to": "D", "weight": 5 }, { "from": "B", "to": "E", "weight": 10 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "C", "to": "G", "weight": 12 }, { "from": "D", "to": "E", "weight": 2 }, { "from": "D", "to": "F", "weight": 6 }, { "from": "E", "to": "F", "weight": 3 }, { "from": "F", "to": "G", "weight": 4 } ], "directed": false, "visited": [] }, "pointers": [], "highlights": [], "codeLine": 4, "variables": { "start": "A", "A": 0 }, "explanation": "Find the cheapest route from A to everywhere. Every node starts at ∞ — unreachable as far as we know — except A, which costs 0 to reach from itself.", "stats": { "comparisons": 0, "swaps": 0 }, "accent": "active" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.1, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B ∞", "x": 0.35, "y": 0.18 }, { "id": "C", "label": "C ∞", "x": 0.35, "y": 0.82 }, { "id": "D", "label": "D ∞", "x": 0.62, "y": 0.5 }, { "id": "E", "label": "E ∞", "x": 0.62, "y": 0.12 }, { "id": "F", "label": "F ∞", "x": 0.9, "y": 0.35 }, { "id": "G", "label": "G ∞", "x": 0.9, "y": 0.8 } ], "edges": [ { "from": "A", "to": "B", "weight": 4 }, { "from": "A", "to": "C", "weight": 2 }, { "from": "B", "to": "D", "weight": 5 }, { "from": "B", "to": "E", "weight": 10 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "C", "to": "G", "weight": 12 }, { "from": "D", "to": "E", "weight": 2 }, { "from": "D", "to": "F", "weight": 6 }, { "from": "E", "to": "F", "weight": 3 }, { "from": "F", "to": "G", "weight": 4 } ], "directed": false, "visited": [ "A" ] }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "finalising": "A", "cost": 0 }, "explanation": "A is the closest unfinished node, at cost 0. Nothing can ever reach it more cheaply — any other route would have to pass through something even further away. So A's distance is final.", "stats": { "comparisons": 1, "swaps": 0 }, "accent": "sorted" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.1, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B ∞", "x": 0.35, "y": 0.18, "role": "compare" }, { "id": "C", "label": "C ∞", "x": 0.35, "y": 0.82 }, { "id": "D", "label": "D ∞", "x": 0.62, "y": 0.5 }, { "id": "E", "label": "E ∞", "x": 0.62, "y": 0.12 }, { "id": "F", "label": "F ∞", "x": 0.9, "y": 0.35 }, { "id": "G", "label": "G ∞", "x": 0.9, "y": 0.8 } ], "edges": [ { "from": "A", "to": "B", "weight": 4, "role": "compare" }, { "from": "A", "to": "C", "weight": 2 }, { "from": "B", "to": "D", "weight": 5 }, { "from": "B", "to": "E", "weight": 10 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "C", "to": "G", "weight": 12 }, { "from": "D", "to": "E", "weight": 2 }, { "from": "D", "to": "F", "weight": 6 }, { "from": "E", "to": "F", "weight": 3 }, { "from": "F", "to": "G", "weight": 4 } ], "directed": false, "visited": [ "A" ] }, "pointers": [], "highlights": [], "codeLine": 13, "variables": { "from": "A", "to": "B", "edge": 4, "viaU": 4, "known": "∞" }, "explanation": "Going A → B costs 0 + 4 = 4, which beats the unknown we had. That's a better route.", "stats": { "comparisons": 2, "swaps": 0 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.1, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B 4", "x": 0.35, "y": 0.18, "role": "compare" }, { "id": "C", "label": "C ∞", "x": 0.35, "y": 0.82 }, { "id": "D", "label": "D ∞", "x": 0.62, "y": 0.5 }, { "id": "E", "label": "E ∞", "x": 0.62, "y": 0.12 }, { "id": "F", "label": "F ∞", "x": 0.9, "y": 0.35 }, { "id": "G", "label": "G ∞", "x": 0.9, "y": 0.8 } ], "edges": [ { "from": "A", "to": "B", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 2 }, { "from": "B", "to": "D", "weight": 5 }, { "from": "B", "to": "E", "weight": 10 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "C", "to": "G", "weight": 12 }, { "from": "D", "to": "E", "weight": 2 }, { "from": "D", "to": "F", "weight": 6 }, { "from": "E", "to": "F", "weight": 3 }, { "from": "F", "to": "G", "weight": 4 } ], "directed": false, "visited": [ "A" ] }, "pointers": [], "highlights": [], "codeLine": 14, "variables": { "B": 4 }, "explanation": "Update B to 4. This is called \"relaxing\" the edge — we found a shortcut and wrote it down.", "stats": { "comparisons": 2, "swaps": 1 }, "accent": "found" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.1, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B 4", "x": 0.35, "y": 0.18, "role": "pivot" }, { "id": "C", "label": "C ∞", "x": 0.35, "y": 0.82, "role": "compare" }, { "id": "D", "label": "D ∞", "x": 0.62, "y": 0.5 }, { "id": "E", "label": "E ∞", "x": 0.62, "y": 0.12 }, { "id": "F", "label": "F ∞", "x": 0.9, "y": 0.35 }, { "id": "G", "label": "G ∞", "x": 0.9, "y": 0.8 } ], "edges": [ { "from": "A", "to": "B", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 2, "role": "compare" }, { "from": "B", "to": "D", "weight": 5 }, { "from": "B", "to": "E", "weight": 10 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "C", "to": "G", "weight": 12 }, { "from": "D", "to": "E", "weight": 2 }, { "from": "D", "to": "F", "weight": 6 }, { "from": "E", "to": "F", "weight": 3 }, { "from": "F", "to": "G", "weight": 4 } ], "directed": false, "visited": [ "A" ] }, "pointers": [], "highlights": [], "codeLine": 13, "variables": { "from": "A", "to": "C", "edge": 2, "viaU": 2, "known": "∞" }, "explanation": "Going A → C costs 0 + 2 = 2, which beats the unknown we had. That's a better route.", "stats": { "comparisons": 3, "swaps": 1 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.1, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B 4", "x": 0.35, "y": 0.18, "role": "pivot" }, { "id": "C", "label": "C 2", "x": 0.35, "y": 0.82, "role": "compare" }, { "id": "D", "label": "D ∞", "x": 0.62, "y": 0.5 }, { "id": "E", "label": "E ∞", "x": 0.62, "y": 0.12 }, { "id": "F", "label": "F ∞", "x": 0.9, "y": 0.35 }, { "id": "G", "label": "G ∞", "x": 0.9, "y": 0.8 } ], "edges": [ { "from": "A", "to": "B", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 2, "role": "sorted" }, { "from": "B", "to": "D", "weight": 5 }, { "from": "B", "to": "E", "weight": 10 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "C", "to": "G", "weight": 12 }, { "from": "D", "to": "E", "weight": 2 }, { "from": "D", "to": "F", "weight": 6 }, { "from": "E", "to": "F", "weight": 3 }, { "from": "F", "to": "G", "weight": 4 } ], "directed": false, "visited": [ "A" ] }, "pointers": [], "highlights": [], "codeLine": 14, "variables": { "C": 2 }, "explanation": "Update C to 2. This is called \"relaxing\" the edge — we found a shortcut and wrote it down.", "stats": { "comparisons": 3, "swaps": 2 }, "accent": "found" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.1, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B 4", "x": 0.35, "y": 0.18, "role": "pivot" }, { "id": "C", "label": "C 2", "x": 0.35, "y": 0.82, "role": "sorted" }, { "id": "D", "label": "D ∞", "x": 0.62, "y": 0.5 }, { "id": "E", "label": "E ∞", "x": 0.62, "y": 0.12 }, { "id": "F", "label": "F ∞", "x": 0.9, "y": 0.35 }, { "id": "G", "label": "G ∞", "x": 0.9, "y": 0.8 } ], "edges": [ { "from": "A", "to": "B", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 2, "role": "sorted" }, { "from": "B", "to": "D", "weight": 5 }, { "from": "B", "to": "E", "weight": 10 }, { "from": "C", "to": "D", "weight": 8 }, { "from": "C", "to": "G", "weight": 12 }, { "from": "D", "to": "E", "weight": 2 }, { "from": "D", "to": "F", "weight": 6 }, { "from": "E", "to": "F", "weight": 3 }, { "from": "F", "to": "G", "weight": 4 } ], "directed": false, "visited": [ "A", "C" ] }, "pointers": [], "highlights": [], "codeLine": 9, "variables": { "finalising": "C", "cost": 2 }, "explanation": "C is the closest unfinished node, at cost 2. Nothing can ever reach it more cheaply — any other route would have to pass through something even further away. So C's distance is final.", "stats": { "comparisons": 4, "swaps": 2 }, "accent": "sorted" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.1, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B 4", "x": 0.35, "y": 0.18, "role": "pivot" }, { "id": "C", "label": "C 2", "x": 0.35, "y": 0.82, "role": "sorted" }, { "id": "D", "label": "D ∞", "x": 0.62, "y": 0.5, "role": "compare" }, { "id": "E", "label": "E ∞", "x": 0.62, "y": 0.12 }, { "id": "F", "label": "F ∞", "x": 0.9, "y": 0.35 }, { "id": "G", "label": "G ∞", "x": 0.9, "y": 0.8 } ], "edges": [ { "from": "A", "to": "B", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 2, "role": "sorted" }, { "from": "B", "to": "D", "weight": 5 }, { "from": "B", "to": "E", "weight": 10 }, { "from": "C", "to": "D", "weight": 8, "role": "compare" }, { "from": "C", "to": "G", "weight": 12 }, { "from": "D", "to": "E", "weight": 2 }, { "from": "D", "to": "F", "weight": 6 }, { "from": "E", "to": "F", "weight": 3 }, { "from": "F", "to": "G", "weight": 4 } ], "directed": false, "visited": [ "A", "C" ] }, "pointers": [], "highlights": [], "codeLine": 13, "variables": { "from": "C", "to": "D", "edge": 8, "viaU": 10, "known": "∞" }, "explanation": "Going C → D costs 2 + 8 = 10, which beats the unknown we had. That's a better route.", "stats": { "comparisons": 5, "swaps": 2 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.1, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B 4", "x": 0.35, "y": 0.18, "role": "pivot" }, { "id": "C", "label": "C 2", "x": 0.35, "y": 0.82, "role": "sorted" }, { "id": "D", "label": "D 10", "x": 0.62, "y": 0.5, "role": "compare" }, { "id": "E", "label": "E ∞", "x": 0.62, "y": 0.12 }, { "id": "F", "label": "F ∞", "x": 0.9, "y": 0.35 }, { "id": "G", "label": "G ∞", "x": 0.9, "y": 0.8 } ], "edges": [ { "from": "A", "to": "B", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 2, "role": "sorted" }, { "from": "B", "to": "D", "weight": 5 }, { "from": "B", "to": "E", "weight": 10 }, { "from": "C", "to": "D", "weight": 8, "role": "sorted" }, { "from": "C", "to": "G", "weight": 12 }, { "from": "D", "to": "E", "weight": 2 }, { "from": "D", "to": "F", "weight": 6 }, { "from": "E", "to": "F", "weight": 3 }, { "from": "F", "to": "G", "weight": 4 } ], "directed": false, "visited": [ "A", "C" ] }, "pointers": [], "highlights": [], "codeLine": 14, "variables": { "D": 10 }, "explanation": "Update D to 10. This is called \"relaxing\" the edge — we found a shortcut and wrote it down.", "stats": { "comparisons": 5, "swaps": 3 }, "accent": "found" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.1, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B 4", "x": 0.35, "y": 0.18, "role": "pivot" }, { "id": "C", "label": "C 2", "x": 0.35, "y": 0.82, "role": "sorted" }, { "id": "D", "label": "D 10", "x": 0.62, "y": 0.5, "role": "pivot" }, { "id": "E", "label": "E ∞", "x": 0.62, "y": 0.12 }, { "id": "F", "label": "F ∞", "x": 0.9, "y": 0.35 }, { "id": "G", "label": "G ∞", "x": 0.9, "y": 0.8, "role": "compare" } ], "edges": [ { "from": "A", "to": "B", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 2, "role": "sorted" }, { "from": "B", "to": "D", "weight": 5 }, { "from": "B", "to": "E", "weight": 10 }, { "from": "C", "to": "D", "weight": 8, "role": "sorted" }, { "from": "C", "to": "G", "weight": 12, "role": "compare" }, { "from": "D", "to": "E", "weight": 2 }, { "from": "D", "to": "F", "weight": 6 }, { "from": "E", "to": "F", "weight": 3 }, { "from": "F", "to": "G", "weight": 4 } ], "directed": false, "visited": [ "A", "C" ] }, "pointers": [], "highlights": [], "codeLine": 13, "variables": { "from": "C", "to": "G", "edge": 12, "viaU": 14, "known": "∞" }, "explanation": "Going C → G costs 2 + 12 = 14, which beats the unknown we had. That's a better route.", "stats": { "comparisons": 6, "swaps": 3 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "A", "label": "A 0", "x": 0.1, "y": 0.5, "role": "sorted" }, { "id": "B", "label": "B 4", "x": 0.35, "y": 0.18, "role": "pivot" }, { "id": "C", "label": "C 2", "x": 0.35, "y": 0.82, "role": "sorted" }, { "id": "D", "label": "D 10", "x": 0.62, "y": 0.5, "role": "pivot" }, { "id": "E", "label": "E ∞", "x": 0.62, "y": 0.12 }, { "id": "F", "label": "F ∞", "x": 0.9, "y": 0.35 }, { "id": "G", "label": "G 14", "x": 0.9, "y": 0.8, "role": "compare" } ], "edges": [ { "from": "A", "to": "B", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 2, "role": "sorted" }, { "from": "B", "to": "D", "weight": 5 }, { "from": "B", "to": "E", "weight": 10 }, { "from": "C", "to": "D", "weight": 8, "role": "sorted" }, { "from": "C", "to": "G", "weight": 12, "role": "sorted" }, { "from": "D", "to": "E", "weight": 2 }, { "from": "D", "to": "F", "weight": 6 }, { "from": "E", "to": "F", "weight": 3 }, { "from": "F", "to": "G", "weight": 4 } ], "directed": false, "visited": [ "A", "C" ] }, "pointers": [], "highlights": [], "codeLine": 14, "variables": { "G": 14 }, "explanation": "Update G to 14. 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