God’s Algorithm
In combinatorial group theory and computer science, 'God's Algorithm' refers to an algorithmic procedure that solves any instance of a solvable combinatorial puzzle (such as the Rubik's Cube, 15-puzzle, or Sokoban) in the absolute minimum number of moves. Formally, it computes the shortest path within the state graph of a permutation group, where vertices represent valid configurations and edges represent allowed atomic transitions. Due to the astronomical state space sizes (e.g., ~4.3 × 10^19 for the 3x3x3 Rubik's Cube), optimal solutions require heavy usage of group reduction, symmetry breaking, and pattern databases built via IDA* (Iterative Deepening A*) search.
HOW-TO IMPLEMENT & CONFIGURE
- Modern C++ compiler supporting C++20 (GCC 11+, Clang 13+, MSVC 2019+)
- CMake 3.20 or higher build system
- OpenMP for parallel Pattern Database (PDB) generation
- Standard Template Library (STL) with custom bit-packing utilities
Step 1
Define a memory-efficient state representation utilizing bit-packing to fit puzzle configurations into CPU cache lines. Initialize the symmetry reduction matrices and Group Action tables.
#include <array>
#include <cstdint>
#include <vector>
namespace GodsAlgorithm {
// Compact representation for a permutation state
struct CubeState {
std::uint64_T corner_permutations;
std::uint32_T edge_permutations;
bool operator==(const CubeState& o) const {
return corner_permutations == o.corner_permutations &&
edge_permutations == o.edge_permutations;
}
};
}Step 2
Construct out-of-core heuristic tables using Breadth-First Search (BFS) backward from the goal state to populate distance matrices for subsets of pieces.
#include <queue>
#include <unordered_map>
namespace GodsAlgorithm {
class PatternDatabase {
std::vector<uint8_T> table;
public:
PatternDatabase(size_t size) : table(size, 0xFF) {}
void set_distance(size_t index, uint8_T dist) {
table[index] = dist;
}
uint8_T get_distance(size_t index) const {
return table[index];
}
};
}Step 3
Implement Iterative Deepening A* search using the Pattern Database as an admissible, consistent heuristic function to guarantee optimality.
#include <algorithm>
#include <optional>
namespace GodsAlgorithm {
struct SearchResult {
int depth;
bool found;
};
template <typename State, typename Operator>
class IDASearch {
public:
SearchResult search(State initial_state, int max_depth) {
for (int threshold = 0; threshold <= max_depth; ++threshold) {
int next_threshold = recursive_search(initial_state, 0, threshold);
if (next_threshold == -1) return {threshold, true};
if (next_threshold == INT32_MAX) break;
}
return {-1, false};
}
private:
int recursive_search(State& state, int g, int threshold) {
int h = evaluate_heuristic(state);
int f = g + h;
if (f > threshold) return f;
if (is_goal(state)) return -1;
int min_cost = INT32_MAX;
for (const auto& op : get_valid_operators(state)) {
State next_state = apply_operator(state, op);
int t = recursive_search(next_state, g + 1, threshold);
if (t == -1) return -1;
min_cost = std::min(min_cost, t);
}
return min_cost;
}
int evaluate_heuristic(const State&) { return 0; }
bool is_goal(const State&) { return true; }
std::vector<int> get_valid_operators(const State&) { return {}; }
State apply_operator(const State& s, int) { return s;
}
};
}Step 4
Add thread-safe memory alignment, bounds checking, and deterministic fuzz testing against known god's number limits (e.g., 20 moves for Rubik's Cube).
#include <cassert>
namespace GodsAlgorithm {
void verify_solution_length(int moves, int max_allowed) {
assert(moves >= 0);
assert(moves <= max_allowed);
}
}