8 Puzzles # nodes expended of 1000 solvable instances
Conclusion:
Nodes expended: BiDirectional A* << A* (Manhattan) <= Bidirectional BFS < A* Hamming << BFS
Running time: BiDirectional A* < Bidirectional BFS <= A* (Manhattan) < A* Hamming << BFS
Code:
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# Author: Huahua from collections import deque import heapq import numpy as np import random import sys N = 3 def Manhattan(s1, s2): dist = 0 for i1, d in enumerate(s1): i2 = s2.index(d) x1 = i1 % N y1 = i1 // N x2 = i2 % N y2 = i2 // N dist += abs(x1 - x2) + abs(y1 - y2) return dist def Hamming(s1, s2): return sum(x != y for x, y in zip(s1, s2)) class Node: dirs = [0, -1, 0, 1, 0] def __init__(self, state, parent = None, h = 0): self.state = state self.parent = parent self.g = parent.g + 1 if parent else 0 self.h = h self.f = self.g + self.h def getMoves(self): moves = [] index = self.state.index(0) x = index % N y = index // N for i in range(4): tx = x + Node.dirs[i] ty = y + Node.dirs[i + 1] if tx < 0 or ty < 0 or tx == N or ty == N: continue i = ty * N + tx move = list(self.state) move[index] = move[i] move[i] = 0 moves.append(tuple(move)) return moves def print(self): print(np.reshape(self.state, (N, N))) def __lt__(self, other): return self.f < other.f def AStarSearch(start_state, end_state, heuristic): def h(s): return heuristic(s, end_state) q = [] s = Node(start_state, h=h(start_state)) heapq.heappush(q, s) opened = {s.state : s.f} closed = dict() while q: n = heapq.heappop(q) if n.state == end_state: return n, len(opened), len(closed) if n.state in closed: continue closed[n.state] = n.f for move in n.getMoves(): node = Node(move, n, h(move)) if move in opened and opened[move] <= node.f: continue opened[node.state] = node.f heapq.heappush(q, node) return None, len(opened), len(closed) def BFS(start_state, end_state): q = deque() q.append(Node(start_state)) opened = set(start_state) closed = 0 while q: n = q.popleft() if n.state == end_state: return n, len(opened), closed closed += 1 for move in n.getMoves(): if move in opened: continue opened.add(move) q.append(Node(move, n)) return None, len(opened), closed def getRootNode(n): return getRootNode(n.parent) if n.parent else n def BidirectionalBFS(start_state, end_state): def constructPath(p, o): while o: t = o.parent o.parent = p p, o = o, t return p ns = Node(start_state) ne = Node(end_state) q = [deque([ns]), deque([ne])] opened = [{start_state : ns}, {end_state: ne}] closed = [0, 0] while q[0]: l = len(q[0]) while l > 0: p = q[0].popleft() closed[0] += 1 for move in p.getMoves(): n = Node(move, p) if move in opened[1]: o = opened[1][move] if getRootNode(n).state == end_state: o, n = n, o n = constructPath(n, o.parent) return n, len(opened[0]) + len(opened[1]), closed[0] + closed[1] if move in opened[0]: continue opened[0][move] = n q[0].append(n) l -= 1 q.reverse() opened.reverse() closed.reverse() return None, len(opened[0]) + len(opened[1]), closed[0] + closed[1] def print_path(n): if not n: return print_path(n.parent) n.print() def solvable(state): inv = 0 for i in range(N*N): for j in range(i + 1, N*N): if all((state[i] > 0, state[j] > 0, state[i] > state[j])): inv += 1 return inv % 2 == 0 if __name__ == '__main__': s = [1, 2, 3, 4, 5, 6, 7, 8, 0] e = [1, 2, 3, 4, 5, 6, 7, 8, 0] i = 0 while True: random.shuffle(s) if not solvable(s): continue n1, opened1, closed1 = AStarSearch(tuple(s), tuple(e), heuristic=Manhattan) n2, opened2, closed2 = AStarSearch(tuple(s), tuple(e), heuristic=Hamming) n3, opened3, closed3 = BFS(tuple(s), tuple(e)) n4, opened4, closed4 = BidirectionalBFS(tuple(s), tuple(e)) print("%d\t%d\t%d\t%d" % (closed1, closed2, closed3, closed4)) sys.stdout.flush() # print("A*") # print_path(n1) # print('---------------') # print("BFS") # print_path(n3) # print('---------------') # print("Bi-BFS") # print_path(n4) # print('---------------') i += 1 if i == 1000: break |
C++ Version
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#include <algorithm> #include <array> #include <chrono> #include <deque> #include <functional> #include <iostream> #include <memory> #include <queue> #include <unordered_map> #include <unordered_set> #include <vector> using namespace std; using chrono::high_resolution_clock; constexpr int N = 3; constexpr int dirs[] = {0, 1, 0, -1, 0}; struct Node; typedef string State; typedef shared_ptr<Node> NodePtr; typedef function<int(const State& s, const State& t)> Heuristic; struct Node { Node(State s, NodePtr p = nullptr, int h = 0) : s(s), p(p), h(h), g(p ? p->g + 1 : 0), f(this->h + g) {} vector<State> GetNextStates() const { vector<State> states; int index = find(s.begin(), s.end(), '0') - s.begin(); int x = index % N; int y = index / N; for (int i = 0; i < 4; ++i) { int tx = x + dirs[i]; int ty = y + dirs[i + 1]; if (tx < 0 || ty < 0 || tx == N || ty == N) continue; int new_index = ty * N + tx; State next(s); next[index] = s[new_index]; next[new_index] = '0'; states.push_back(move(next)); } return states; } NodePtr p; int h; int g; int f; State s; }; bool Sovlable(const State& s) { int inv_count = 0; for (int i = 0; i < N * N; ++i) { for (int j = i + 1; j < N * N; ++j) { if (s[i] != '0' && s[j] != '0' && s[i] > s[j]) { ++inv_count; } } } return inv_count % 2 == 0; } NodePtr GetRoot(NodePtr n) { return n->p ? GetRoot(n->p) : n; } NodePtr WirePath(NodePtr p, NodePtr n) { while (n) { NodePtr t = n->p; n->p = p; p = n; n = t; } return p; } void ConstructPath(NodePtr node, vector<State>* path) { while (node) { path->push_back(node->s); node = node->p; } reverse(path->begin(), path->end()); } class Solver { public: virtual bool Solve(State start, State target, vector<State>* path, int* opened, int* closed) = 0; }; class BFSSolver : public Solver { public: bool Solve(State start, State target, vector<State>* path, int* opened, int* closed) override { int expended = 0; unordered_set<string> seen; deque<NodePtr> q{make_shared<Node>(start)}; while (!q.empty()) { auto cur = q.front(); q.pop_front(); ++expended; if (cur->s == target) { ConstructPath(cur, path); *opened = seen.size(); *closed = expended; return true; } for (const auto& s : cur->GetNextStates()) { auto r = seen.insert(s); if (!r.second) continue; q.push_back(make_shared<Node>(s, cur)); } } return false; } }; class BidirectionalBFSSolver : public Solver { public: bool Solve(State start, State target, vector<State>* path, int* opened, int* closed) override { auto start_node = make_shared<Node>(start); auto target_node = make_shared<Node>(target); unordered_map<string, NodePtr> seen0{{start, start_node}}; unordered_map<string, NodePtr> seen1{{target, target_node}}; deque<NodePtr> q0{start_node}; deque<NodePtr> q1{target_node}; int expended = 0; while (!q0.empty()) { size_t size = q0.size(); while (size--) { auto cur = q0.front(); q0.pop_front(); ++expended; for (const auto& s : cur->GetNextStates()) { auto n = make_shared<Node>(s, cur); auto it = seen1.find(s); if (it != seen1.end()) { auto p = it->second; if (GetRoot(n)->s == target) swap(n, p); n = WirePath(n, p->p); ConstructPath(n, path); *opened = seen0.size() + seen1.size(); *closed = expended; return true; } if (seen0.count(s)) continue; seen0[s] = n; q0.push_back(n); } } if (q1.size() < q0.size()) { swap(q0, q1); swap(seen0, seen1); } } return false; } private: }; struct NodeCompare : public binary_function<NodePtr, NodePtr, bool> { bool operator()(const NodePtr& x, const NodePtr& y) const { return x->f > y->f; } }; int ManhattanDistance(const State& s, const State& t) { int h = 0; for (int i1 = 0; i1 < N * N; ++i1) { int i2 = t.find(s[i1]); int x1 = i1 % N; int y1 = i1 / N; int x2 = i2 % N; int y2 = i2 / N; h += abs(x1 - x2) + abs(y1 - y2); } return h; } int HammingDistance(const State& s, const State& t) { int h = 0; for (size_t i = 0; i < s.size(); ++i) { if (s[i] != t[i]) ++h; } return h; } class AStarSolver : public Solver { public: explicit AStarSolver(Heuristic heuristic) : heuristic_(heuristic) {} bool Solve(State start, State target, vector<State>* path, int* opened, int* closed) override { unordered_map<string, NodePtr> o; unordered_set<string> c; priority_queue<NodePtr, vector<NodePtr>, NodeCompare> q; q.emplace(new Node(start, nullptr, heuristic_(start, target))); while (!q.empty()) { auto cur = q.top(); q.pop(); if (cur->s == target) { ConstructPath(cur, path); *opened = o.size(); *closed = c.size(); return true; } if (!c.insert(cur->s).second) continue; for (const auto& s : cur->GetNextStates()) { auto it = o.find(s); auto n = make_shared<Node>(s, cur, heuristic_(s, target)); if (it != o.end() && n->f >= it->second->f) continue; o[s] = n; q.push(n); } } return false; } private: Heuristic heuristic_; }; class BiAStarSolver : public Solver { public: explicit BiAStarSolver(Heuristic heuristic) : heuristic_(heuristic) {} bool Solve(State start, State target, vector<State>* path, int* opened, int* closed) override { unordered_map<string, NodePtr> o0, o1; unordered_map<string, NodePtr> c0, c1; priority_queue<NodePtr, vector<NodePtr>, NodeCompare> q0, q1; q0.emplace(new Node(start, nullptr, heuristic_(start, target))); q1.emplace(new Node(target, nullptr, heuristic_(target, start))); bool farward = true; while (!q0.empty()) { auto size = q0.size(); while (size--) { auto cur = q0.top(); q0.pop(); if (c0.count(cur->s)) continue; c0[cur->s] = cur; if (c1.count(cur->s)) { auto p = c1[cur->s]; if (GetRoot(cur)->s == target) swap(cur, p); cur = WirePath(cur, p->p); ConstructPath(cur, path); *opened = o0.size() + o1.size(); *closed = c0.size() + c1.size(); return true; } for (const auto& s : cur->GetNextStates()) { auto it = o0.find(s); auto n = make_shared<Node>(s, cur, heuristic_(s, farward ? target : start)); if (it != o0.end() && n->f >= it->second->f) continue; o0[s] = n; q0.push(n); } } if (q1.size() < q0.size()) { swap(q0, q1); swap(c0, c1); swap(o0, o1); farward = !farward; } } return false; } private: Heuristic heuristic_; }; bool VerifyPath(const vector<State>& path, const State& s, const State& t) { if (path.empty()) return false; if (path.front() != s || path.back() != t) return false; for (size_t i = 1; i < path.size(); ++i) { Node n(path[i - 1]); auto states = n.GetNextStates(); if (find(begin(states), end(states), path[i]) == end(states)) { return false; } } return true; } void StatisticsMode() { State s{"123456780"}; State t{"123456780"}; vector<unique_ptr<Solver>> solvers; solvers.emplace_back(new BFSSolver); solvers.emplace_back(new AStarSolver(HammingDistance)); solvers.emplace_back(new AStarSolver(ManhattanDistance)); solvers.emplace_back(new BidirectionalBFSSolver); solvers.emplace_back(new BiAStarSolver(ManhattanDistance)); for (int i = 0; i < 1000;) { random_shuffle(s.begin(), s.end()); if (!Sovlable(s)) continue; ++i; for (auto& solver : solvers) { vector<State> path; int opened; int closed; auto t0 = high_resolution_clock::now(); bool sovlable = solver->Solve(s, t, &path, &opened, &closed); auto t1 = high_resolution_clock::now(); if (!VerifyPath(path, s, t)) { cerr << "Invalid path!" << endl; return; } auto time_span = chrono::duration_cast<chrono::duration<double>>(t1 - t0); cout << closed << "\t" << time_span.count() * 1000 << "\t"; } cout << endl; } } int main() { StatisticsMode(); } |
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