{ "id": "graphs/a-star", "version": 1, "deterministic": true, "url": "/dsa/graphs/a-star", "topic": { "slug": "graphs", "title": "Graphs" }, "title": "A* Search", "tagline": "Dijkstra with a sense of direction — it guesses what's left, and aims.", "vizKind": "graph", "complexity": { "time": { "best": "O(E)", "average": "O(E log V)", "worst": "O(E log V)" }, "space": "O(V)", "growth": "nlogn", "note": "The same worst case as Dijkstra — with a useless heuristic it degrades to exactly Dijkstra. In practice a good heuristic slashes the number of nodes expanded, often dramatically. The complexity class doesn't change; the real-world work does." }, "code": { "javascript": "// Dijkstra explores in every direction. A* aims.\nfunction aStar(graph, start, goal) {\n const g = {}; // real cost from the start\n for (const v of graph.nodes) g[v] = Infinity;\n g[start] = 0;\n\n const open = new PriorityQueue();\n // f = what it cost to get here + a GUESS of what's left\n open.push(start, h(start, goal));\n\n while (!open.isEmpty()) {\n const u = open.pop(); // lowest f, not lowest g\n if (u === goal) return g[goal];\n\n for (const [v, w] of graph.edges[u]) {\n if (g[u] + w < g[v]) {\n g[v] = g[u] + w;\n open.push(v, g[v] + h(v, goal)); // f = g + h\n }\n }\n }\n return Infinity;\n}", "python": "# Dijkstra explores in every direction. A* aims.\nimport heapq\n\ndef a_star(graph, start, goal):\n g = {v: float('inf') for v in graph} # real cost from the start\n g[start] = 0\n\n # f = what it cost to get here + a GUESS of what's left\n open_set = [(h(start, goal), start)]\n\n while open_set:\n _, u = heapq.heappop(open_set) # lowest f, not lowest g\n if u == goal: return g[goal]\n\n for v, w in graph[u]:\n if g[u] + w < g[v]:\n g[v] = g[u] + w\n heapq.heappush(open_set, (g[v] + h(v, goal), v)) # f = g + h\n\n return float('inf')\n", "java": "// Dijkstra explores in every direction. A* aims.\nint aStar(Graph graph, String start, String goal) {\n Map g = new HashMap<>(); // real cost from the start\n for (String v : graph.nodes) g.put(v, INF);\n g.put(start, 0);\n\n PriorityQueue open = new PriorityQueue<>();\n // f = what it cost to get here + a GUESS of what's left\n open.add(new Node(start, h(start, goal)));\n\n while (!open.isEmpty()) {\n String u = open.poll().id; // lowest f, not lowest g\n if (u.equals(goal)) return g.get(goal);\n\n for (Edge e : graph.edges.get(u)) {\n if (g.get(u) + e.w < g.get(e.to)) {\n g.put(e.to, g.get(u) + e.w);\n open.add(new Node(e.to, g.get(e.to) + h(e.to, goal))); // f = g + h\n }\n }\n }\n return INF;\n}", "cpp": "// Dijkstra explores in every direction. A* aims.\nint aStar(Graph& graph, string start, string goal) {\n map g; // real cost from the start\n for (auto& v : graph.nodes) g[v] = INF;\n g[start] = 0;\n\n priority_queue, vector>, greater<>> open;\n // f = what it cost to get here + a GUESS of what's left\n open.push({h(start, goal), start});\n\n while (!open.empty()) {\n auto [f, u] = open.top(); open.pop(); // lowest f, not lowest g\n if (u == goal) return g[goal];\n\n for (auto& [v, w] : graph.edges[u]) {\n if (g[u] + w < g[v]) {\n g[v] = g[u] + w;\n open.push({g[v] + h(v, goal), v}); // f = g + h\n }\n }\n }\n return INF;\n}" }, "defaultInput": [ 1 ], "defaultTarget": null, "inputHint": "Each node shows g + h — the real cost so far, plus the guess of what's left.", "pitfalls": [ { "wrong": "Using a heuristic that overestimates.", "right": "Then A* can discard the genuinely best route because it guessed wrongly that the route was expensive. It stays fast and becomes wrong. The heuristic MUST be optimistic." }, { "wrong": "Expanding on the lowest g instead of the lowest f.", "right": "That's just Dijkstra with extra steps. The whole point is to prioritise by g + h, so the guess actually steers the search." }, { "wrong": "Using A* when there's no meaningful heuristic.", "right": "If you can't estimate the remaining distance, h = 0 — and A* is precisely Dijkstra. Use A* when the problem has real geometry to exploit." }, { "wrong": "Thinking A* has a better Big-O than Dijkstra.", "right": "It doesn't. Same worst case. What it improves is how many nodes it actually expands in practice, which is often the difference that matters." } ], "quiz": [ { "q": "What is f in A*?", "options": [ "The cost so far", "g + h — the real cost to get here, plus the estimated cost still to go", "The number of nodes expanded", "The final answer" ], "answer": 1, "why": "f = g + h. A* always expands the node with the smallest f, which is what makes it head towards the goal rather than spreading out blindly in all directions." }, { "q": "What must be true of the heuristic for A* to stay correct?", "options": [ "It must be exact", "It must never overestimate the remaining distance — it has to be optimistic", "It must be large", "It must be zero" ], "answer": 1, "why": "An optimistic (admissible) heuristic keeps A* provably correct. Overestimate, and it can throw away the true best route because it wrongly guessed that route was expensive." }, { "q": "You set h = 0 for every node. What is A* now?", "options": [ "Broken", "Exactly Dijkstra's algorithm", "Breadth-first search", "Faster than before" ], "answer": 1, "why": "With no guess, f = g, and A* expands purely by cost-so-far — which is Dijkstra, precisely. The guess is the entire difference between the two algorithms." } ], "problems": [ { "name": "Shortest Path in Binary Matrix", "difficulty": "Medium", "url": "https://leetcode.com/problems/shortest-path-in-binary-matrix/" }, { "name": "Sliding Puzzle", "difficulty": "Hard", "url": "https://leetcode.com/problems/sliding-puzzle/" }, { "name": "Minimum Knight Moves", "difficulty": "Medium", "url": "https://leetcode.com/problems/minimum-knight-moves/" } ], "trace": { "note": "Each frame is one immutable step: data snapshot, pointers, per-index roles, executing code line, and a one-sentence explanation.", "length": 19, "frames": [ { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "active" }, { "id": "A", "label": "A ∞+7", "x": 0.3, "y": 0.18 }, { "id": "B", "label": "B ∞+7", "x": 0.3, "y": 0.82 }, { "id": "C", "label": "C ∞+4", "x": 0.52, "y": 0.5 }, { "id": "D", "label": "D ∞+6", "x": 0.52, "y": 0.1 }, { "id": "E", "label": "E ∞+3", "x": 0.74, "y": 0.75 }, { "id": "F", "label": "F ∞+3", "x": 0.74, "y": 0.3 }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3 }, { "from": "S", "to": "B", "weight": 2 }, { "from": "A", "to": "D", "weight": 4 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4 }, { "from": "B", "to": "E", "weight": 6 }, { "from": "C", "to": "F", "weight": 3 }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "S" ] }, "visited": [] }, "pointers": [], "highlights": [], "codeLine": 8, "variables": { "start": "S", "goal": "G", "note": "labels show g + h" }, "explanation": "Get from S to G. Each node shows two numbers: **g**, what it actually cost to reach it, and **h**, a straight-line *guess* of what's left. A* always expands the node with the smallest g + h — so it aims at the goal instead of spreading out blindly like Dijkstra.", "stats": { "comparisons": 0, "swaps": 0 }, "accent": "active" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A ∞+7", "x": 0.3, "y": 0.18 }, { "id": "B", "label": "B ∞+7", "x": 0.3, "y": 0.82 }, { "id": "C", "label": "C ∞+4", "x": 0.52, "y": 0.5 }, { "id": "D", "label": "D ∞+6", "x": 0.52, "y": 0.1 }, { "id": "E", "label": "E ∞+3", "x": 0.74, "y": 0.75 }, { "id": "F", "label": "F ∞+3", "x": 0.74, "y": 0.3 }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3 }, { "from": "S", "to": "B", "weight": 2 }, { "from": "A", "to": "D", "weight": 4 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4 }, { "from": "B", "to": "E", "weight": 6 }, { "from": "C", "to": "F", "weight": 3 }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [] }, "visited": [ "S" ] }, "pointers": [], "highlights": [], "codeLine": 12, "variables": { "expanding": "S", "g": "0", "h": 9, "f": 9 }, "explanation": "Expand S: it has the lowest f = g + h = 0 + 9 = 9. Dijkstra would have picked purely on g and might have gone the wrong way; the h term is what makes A* aim.", "stats": { "comparisons": 1, "swaps": 0 }, "accent": "sorted" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A ∞+7", "x": 0.3, "y": 0.18, "role": "compare" }, { "id": "B", "label": "B ∞+7", "x": 0.3, "y": 0.82 }, { "id": "C", "label": "C ∞+4", "x": 0.52, "y": 0.5 }, { "id": "D", "label": "D ∞+6", "x": 0.52, "y": 0.1 }, { "id": "E", "label": "E ∞+3", "x": 0.74, "y": 0.75 }, { "id": "F", "label": "F ∞+3", "x": 0.74, "y": 0.3 }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "compare" }, { "from": "S", "to": "B", "weight": 2 }, { "from": "A", "to": "D", "weight": 4 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4 }, { "from": "B", "to": "E", "weight": 6 }, { "from": "C", "to": "F", "weight": 3 }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [] }, "visited": [ "S" ] }, "pointers": [], "highlights": [], "codeLine": 16, "variables": { "from": "S", "to": "A", "edge": 3, "via": 3, "known": "∞", "h": 7 }, "explanation": "S → A costs 0 + 3 = 3, better than the ∞ we had. Update g(A) = 3, so f(A) = 3 + 7 = 10.", "stats": { "comparisons": 2, "swaps": 0 }, "accent": "found" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "pivot" }, { "id": "B", "label": "B ∞+7", "x": 0.3, "y": 0.82, "role": "compare" }, { "id": "C", "label": "C ∞+4", "x": 0.52, "y": 0.5 }, { "id": "D", "label": "D ∞+6", "x": 0.52, "y": 0.1 }, { "id": "E", "label": "E ∞+3", "x": 0.74, "y": 0.75 }, { "id": "F", "label": "F ∞+3", "x": 0.74, "y": 0.3 }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "compare" }, { "from": "A", "to": "D", "weight": 4 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4 }, { "from": "B", "to": "E", "weight": 6 }, { "from": "C", "to": "F", "weight": 3 }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "A" ] }, "visited": [ "S" ] }, "pointers": [], "highlights": [], "codeLine": 16, "variables": { "from": "S", "to": "B", "edge": 2, "via": 2, "known": "∞", "h": 7 }, "explanation": "S → B costs 0 + 2 = 2, better than the ∞ we had. Update g(B) = 2, so f(B) = 2 + 7 = 9.", "stats": { "comparisons": 3, "swaps": 1 }, "accent": "found" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "pivot" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C ∞+4", "x": 0.52, "y": 0.5 }, { "id": "D", "label": "D ∞+6", "x": 0.52, "y": 0.1 }, { "id": "E", "label": "E ∞+3", "x": 0.74, "y": 0.75 }, { "id": "F", "label": "F ∞+3", "x": 0.74, "y": 0.3 }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4 }, { "from": "B", "to": "E", "weight": 6 }, { "from": "C", "to": "F", "weight": 3 }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "A" ] }, "visited": [ "S", "B" ] }, "pointers": [], "highlights": [], "codeLine": 12, "variables": { "expanding": "B", "g": "2", "h": 7, "f": 9 }, "explanation": "Expand B: it has the lowest f = g + h = 2 + 7 = 9. Dijkstra would have picked purely on g and might have gone the wrong way; the h term is what makes A* aim.", "stats": { "comparisons": 4, "swaps": 2 }, "accent": "sorted" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "pivot" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C ∞+4", "x": 0.52, "y": 0.5, "role": "compare" }, { "id": "D", "label": "D ∞+6", "x": 0.52, "y": 0.1 }, { "id": "E", "label": "E ∞+3", "x": 0.74, "y": 0.75 }, { "id": "F", "label": "F ∞+3", "x": 0.74, "y": 0.3 }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "compare" }, { "from": "B", "to": "E", "weight": 6 }, { "from": "C", "to": "F", "weight": 3 }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "A" ] }, "visited": [ "S", "B" ] }, "pointers": [], "highlights": [], "codeLine": 16, "variables": { "from": "B", "to": "C", "edge": 4, "via": 6, "known": "∞", "h": 4 }, "explanation": "B → C costs 2 + 4 = 6, better than the ∞ we had. Update g(C) = 6, so f(C) = 6 + 4 = 10.", "stats": { "comparisons": 5, "swaps": 2 }, "accent": "found" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "pivot" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "pivot" }, { "id": "D", "label": "D ∞+6", "x": 0.52, "y": 0.1 }, { "id": "E", "label": "E ∞+3", "x": 0.74, "y": 0.75, "role": "compare" }, { "id": "F", "label": "F ∞+3", "x": 0.74, "y": 0.3 }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "compare" }, { "from": "C", "to": "F", "weight": 3 }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "A", "C" ] }, "visited": [ "S", "B" ] }, "pointers": [], "highlights": [], "codeLine": 16, "variables": { "from": "B", "to": "E", "edge": 6, "via": 8, "known": "∞", "h": 3 }, "explanation": "B → E costs 2 + 6 = 8, better than the ∞ we had. Update g(E) = 8, so f(E) = 8 + 3 = 11.", "stats": { "comparisons": 6, "swaps": 3 }, "accent": "found" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "sorted" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "pivot" }, { "id": "D", "label": "D ∞+6", "x": 0.52, "y": 0.1 }, { "id": "E", "label": "E 8+3", "x": 0.74, "y": 0.75, "role": "pivot" }, { "id": "F", "label": "F ∞+3", "x": 0.74, "y": 0.3 }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4 }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "sorted" }, { "from": "C", "to": "F", "weight": 3 }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "C", "E" ] }, "visited": [ "S", "B", "A" ] }, "pointers": [], "highlights": [], "codeLine": 12, "variables": { "expanding": "A", "g": "3", "h": 7, "f": 10 }, "explanation": "Expand A: it has the lowest f = g + h = 3 + 7 = 10. Dijkstra would have picked purely on g and might have gone the wrong way; the h term is what makes A* aim.", "stats": { "comparisons": 7, "swaps": 4 }, "accent": "sorted" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "sorted" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "pivot" }, { "id": "D", "label": "D ∞+6", "x": 0.52, "y": 0.1, "role": "compare" }, { "id": "E", "label": "E 8+3", "x": 0.74, "y": 0.75, "role": "pivot" }, { "id": "F", "label": "F ∞+3", "x": 0.74, "y": 0.3 }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4, "role": "compare" }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "sorted" }, { "from": "C", "to": "F", "weight": 3 }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "C", "E" ] }, "visited": [ "S", "B", "A" ] }, "pointers": [], "highlights": [], "codeLine": 16, "variables": { "from": "A", "to": "D", "edge": 4, "via": 7, "known": "∞", "h": 6 }, "explanation": "A → D costs 3 + 4 = 7, better than the ∞ we had. Update g(D) = 7, so f(D) = 7 + 6 = 13.", "stats": { "comparisons": 8, "swaps": 4 }, "accent": "found" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "sorted" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "compare" }, { "id": "D", "label": "D 7+6", "x": 0.52, "y": 0.1, "role": "pivot" }, { "id": "E", "label": "E 8+3", "x": 0.74, "y": 0.75, "role": "pivot" }, { "id": "F", "label": "F ∞+3", "x": 0.74, "y": 0.3 }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 3, "role": "compare" }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "sorted" }, { "from": "C", "to": "F", "weight": 3 }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "C", "E", "D" ] }, "visited": [ "S", "B", "A" ] }, "pointers": [], "highlights": [], "codeLine": 16, "variables": { "from": "A", "to": "C", "edge": 3, "via": 6, "known": "6", "h": 4 }, "explanation": "A → C costs 6, no better than the 6 we know. Skip.", "stats": { "comparisons": 9, "swaps": 5 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "sorted" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "sorted" }, { "id": "D", "label": "D 7+6", "x": 0.52, "y": 0.1, "role": "pivot" }, { "id": "E", "label": "E 8+3", "x": 0.74, "y": 0.75, "role": "pivot" }, { "id": "F", "label": "F ∞+3", "x": 0.74, "y": 0.3 }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "sorted" }, { "from": "C", "to": "F", "weight": 3 }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "E", "D" ] }, "visited": [ "S", "B", "A", "C" ] }, "pointers": [], "highlights": [], "codeLine": 12, "variables": { "expanding": "C", "g": "6", "h": 4, "f": 10 }, "explanation": "Expand C: it has the lowest f = g + h = 6 + 4 = 10. Dijkstra would have picked purely on g and might have gone the wrong way; the h term is what makes A* aim.", "stats": { "comparisons": 10, "swaps": 5 }, "accent": "sorted" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "sorted" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "sorted" }, { "id": "D", "label": "D 7+6", "x": 0.52, "y": 0.1, "role": "pivot" }, { "id": "E", "label": "E 8+3", "x": 0.74, "y": 0.75, "role": "pivot" }, { "id": "F", "label": "F ∞+3", "x": 0.74, "y": 0.3, "role": "compare" }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "sorted" }, { "from": "C", "to": "F", "weight": 3, "role": "compare" }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "E", "D" ] }, "visited": [ "S", "B", "A", "C" ] }, "pointers": [], "highlights": [], "codeLine": 16, "variables": { "from": "C", "to": "F", "edge": 3, "via": 9, "known": "∞", "h": 3 }, "explanation": "C → F costs 6 + 3 = 9, better than the ∞ we had. Update g(F) = 9, so f(F) = 9 + 3 = 12.", "stats": { "comparisons": 11, "swaps": 5 }, "accent": "found" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "sorted" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "sorted" }, { "id": "D", "label": "D 7+6", "x": 0.52, "y": 0.1, "role": "pivot" }, { "id": "E", "label": "E 8+3", "x": 0.74, "y": 0.75, "role": "sorted" }, { "id": "F", "label": "F 9+3", "x": 0.74, "y": 0.3, "role": "pivot" }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5 } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "sorted" }, { "from": "C", "to": "F", "weight": 3, "role": "sorted" }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4 }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "D", "F" ] }, "visited": [ "S", "B", "A", "C", "E" ] }, "pointers": [], "highlights": [], "codeLine": 12, "variables": { "expanding": "E", "g": "8", "h": 3, "f": 11 }, "explanation": "Expand E: it has the lowest f = g + h = 8 + 3 = 11. Dijkstra would have picked purely on g and might have gone the wrong way; the h term is what makes A* aim.", "stats": { "comparisons": 12, "swaps": 6 }, "accent": "sorted" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "sorted" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "sorted" }, { "id": "D", "label": "D 7+6", "x": 0.52, "y": 0.1, "role": "pivot" }, { "id": "E", "label": "E 8+3", "x": 0.74, "y": 0.75, "role": "sorted" }, { "id": "F", "label": "F 9+3", "x": 0.74, "y": 0.3, "role": "pivot" }, { "id": "G", "label": "G ∞+0", "x": 0.94, "y": 0.5, "role": "compare" } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "sorted" }, { "from": "C", "to": "F", "weight": 3, "role": "sorted" }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4, "role": "compare" }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "D", "F" ] }, "visited": [ "S", "B", "A", "C", "E" ] }, "pointers": [], "highlights": [], "codeLine": 16, "variables": { "from": "E", "to": "G", "edge": 4, "via": 12, "known": "∞", "h": 0 }, "explanation": "E → G costs 8 + 4 = 12, better than the ∞ we had. Update g(G) = 12, so f(G) = 12 + 0 = 12.", "stats": { "comparisons": 13, "swaps": 6 }, "accent": "found" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "sorted" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "sorted" }, { "id": "D", "label": "D 7+6", "x": 0.52, "y": 0.1, "role": "pivot" }, { "id": "E", "label": "E 8+3", "x": 0.74, "y": 0.75, "role": "sorted" }, { "id": "F", "label": "F 9+3", "x": 0.74, "y": 0.3, "role": "sorted" }, { "id": "G", "label": "G 12+0", "x": 0.94, "y": 0.5, "role": "pivot" } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "sorted" }, { "from": "C", "to": "F", "weight": 3, "role": "sorted" }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4, "role": "sorted" }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "D", "G" ] }, "visited": [ "S", "B", "A", "C", "E", "F" ] }, "pointers": [], "highlights": [], "codeLine": 12, "variables": { "expanding": "F", "g": "9", "h": 3, "f": 12 }, "explanation": "Expand F: it has the lowest f = g + h = 9 + 3 = 12. Dijkstra would have picked purely on g and might have gone the wrong way; the h term is what makes A* aim.", "stats": { "comparisons": 14, "swaps": 7 }, "accent": "sorted" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "sorted" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "sorted" }, { "id": "D", "label": "D 7+6", "x": 0.52, "y": 0.1, "role": "compare" }, { "id": "E", "label": "E 8+3", "x": 0.74, "y": 0.75, "role": "sorted" }, { "id": "F", "label": "F 9+3", "x": 0.74, "y": 0.3, "role": "sorted" }, { "id": "G", "label": "G 12+0", "x": 0.94, "y": 0.5, "role": "pivot" } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "sorted" }, { "from": "C", "to": "F", "weight": 3, "role": "sorted" }, { "from": "D", "to": "F", "weight": 5, "role": "compare" }, { "from": "E", "to": "G", "weight": 4, "role": "sorted" }, { "from": "F", "to": "G", "weight": 2 } ], "directed": false, "frontier": { "label": "queue", "ids": [ "D", "G" ] }, "visited": [ "S", "B", "A", "C", "E", "F" ] }, "pointers": [], "highlights": [], "codeLine": 16, "variables": { "from": "F", "to": "D", "edge": 5, "via": 14, "known": "7", "h": 6 }, "explanation": "F → D costs 14, no better than the 7 we know. Skip.", "stats": { "comparisons": 15, "swaps": 7 }, "accent": "compare" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "sorted" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "sorted" }, { "id": "D", "label": "D 7+6", "x": 0.52, "y": 0.1, "role": "pivot" }, { "id": "E", "label": "E 8+3", "x": 0.74, "y": 0.75, "role": "sorted" }, { "id": "F", "label": "F 9+3", "x": 0.74, "y": 0.3, "role": "sorted" }, { "id": "G", "label": "G 12+0", "x": 0.94, "y": 0.5, "role": "compare" } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "sorted" }, { "from": "C", "to": "F", "weight": 3, "role": "sorted" }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4, "role": "sorted" }, { "from": "F", "to": "G", "weight": 2, "role": "compare" } ], "directed": false, "frontier": { "label": "queue", "ids": [ "D", "G" ] }, "visited": [ "S", "B", "A", "C", "E", "F" ] }, "pointers": [], "highlights": [], "codeLine": 16, "variables": { "from": "F", "to": "G", "edge": 2, "via": 11, "known": "12", "h": 0 }, "explanation": "F → G costs 9 + 2 = 11, better than the 12 we had. Update g(G) = 11, so f(G) = 11 + 0 = 11.", "stats": { "comparisons": 16, "swaps": 7 }, "accent": "found" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "sorted" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "sorted" }, { "id": "D", "label": "D 7+6", "x": 0.52, "y": 0.1, "role": "pivot" }, { "id": "E", "label": "E 8+3", "x": 0.74, "y": 0.75, "role": "sorted" }, { "id": "F", "label": "F 9+3", "x": 0.74, "y": 0.3, "role": "sorted" }, { "id": "G", "label": "G 11+0", "x": 0.94, "y": 0.5, "role": "sorted" } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "sorted" }, { "from": "C", "to": "F", "weight": 3, "role": "sorted" }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4, "role": "sorted" }, { "from": "F", "to": "G", "weight": 2, "role": "sorted" } ], "directed": false, "frontier": { "label": "queue", "ids": [ "D" ] }, "visited": [ "S", "B", "A", "C", "E", "F", "G" ] }, "pointers": [], "highlights": [], "codeLine": 13, "variables": { "reached": "G", "cost": 11, "expanded": 7 }, "explanation": "Reached G, at a true cost of 11. A* expanded only 7 of 8 nodes — the guess kept it pointed at the goal, so it never wandered off exploring the far side of the map.", "stats": { "comparisons": 17, "swaps": 8 }, "accent": "found" }, { "data": { "nodes": [ { "id": "S", "label": "S 0+9", "x": 0.08, "y": 0.5, "role": "sorted" }, { "id": "A", "label": "A 3+7", "x": 0.3, "y": 0.18, "role": "sorted" }, { "id": "B", "label": "B 2+7", "x": 0.3, "y": 0.82, "role": "sorted" }, { "id": "C", "label": "C 6+4", "x": 0.52, "y": 0.5, "role": "sorted" }, { "id": "D", "label": "D 7+6", "x": 0.52, "y": 0.1, "role": "pivot" }, { "id": "E", "label": "E 8+3", "x": 0.74, "y": 0.75, "role": "sorted" }, { "id": "F", "label": "F 9+3", "x": 0.74, "y": 0.3, "role": "sorted" }, { "id": "G", "label": "G 11+0", "x": 0.94, "y": 0.5, "role": "sorted" } ], "edges": [ { "from": "S", "to": "A", "weight": 3, "role": "sorted" }, { "from": "S", "to": "B", "weight": 2, "role": "sorted" }, { "from": "A", "to": "D", "weight": 4, "role": "sorted" }, { "from": "A", "to": "C", "weight": 3 }, { "from": "B", "to": "C", "weight": 4, "role": "sorted" }, { "from": "B", "to": "E", "weight": 6, "role": "sorted" }, { "from": "C", "to": "F", "weight": 3, "role": "sorted" }, { "from": "D", "to": "F", "weight": 5 }, { "from": "E", "to": "G", "weight": 4, "role": "sorted" }, { "from": "F", "to": "G", "weight": 2, "role": "sorted" } ], "directed": false, "frontier": { "label": "queue", "ids": [ "D" ] }, "visited": [ "S", "B", "A", "C", "E", "F", "G" ] }, "pointers": [], "highlights": [], "codeLine": 20, "variables": { "cost": "11", "expanded": 7, "neverTouched": "D" }, "explanation": "Cheapest route to G: 11. It never even looked at D — the heuristic said they were pointing away from the goal. With h set to 0, A* *is* Dijkstra. The whole difference is the guess — and it only stays correct while that guess never overestimates.", "stats": { "comparisons": 17, "swaps": 8 }, "accent": "sorted" } ] } }