It works!

But some optimizations left to do.

main.cpp

@@ -1,79 +1,168 @@
 #include <iostream>
 #include <list>
+#include <vector>
 #include <chrono>
 #include <algorithm>
+#include <iterator>
 
 #define BOARD_SIZE 81
 #define BOARD_DIMENSION 9
 
-class Sudoku {
-private:
-    int board[BOARD_SIZE];
+typedef std::pair<std::vector<int>, std::list<int>> move;
 
+class Board {
 public:
- 
-    Sudoku(std::string s) {
-        for (unsigned int i = 0; i < s.length(); i++) {
-            board[i] = (int) (s[i] - '0');
-        }
-    }
- 
-    void solve() {
-        // Very naive first attempt: Fails because it sets stuff wrongly and doesn't backtrack
-        for (int y = 0; y < BOARD_DIMENSION; y++) {
-            for (int x = 0; x < BOARD_DIMENSION; x++) {
-                if (get(x, y) == 0) {
-                    auto possibilities = get_possibilities(x, y);
-
-                    set(x, y, possibilities.front());
-                }
-            }
-        }
-
-        // Real search:
-        // 1. Sort possibilities by how many new constraints they create
-        // 2. Start with the one creating the smallest amount of new constraints
-        // 3. Continue until dead end
-        // 4. If dead end: Try the next one
-    }
+    int values[BOARD_SIZE];
 
     int coordinates_to_id(int x, int y) {
         return y * BOARD_DIMENSION + x;
     }
 
     int get(int x, int y) {
-        return board[coordinates_to_id(x, y)];
+        return values[coordinates_to_id(x, y)];
     }
 
     void set(int x, int y, int value) {
-        board[coordinates_to_id(x, y)] = value;
+        values[coordinates_to_id(x, y)] = value;
+    }
+};
+
+class Cell {
+public:
+    Board *board;
+    int x, y;
+
+    int value;
+    std::list<int> possible_values;
+
+    Cell(Board *board, int x, int y) : board(board), x(x), y(y), value(board->get(x, y)) {}
+
+    void set_value(int new_value) {
+        value = new_value;
+        update_possible_values();
     }
 
-    std::list<int> get_possibilities(int x, int y) {
+    void update_possible_values() {
         std::list<int> possibilities{1, 2, 3, 4, 5, 6, 7, 8 ,9};
 
         // Remove all from this row
         for (int dx = 0; dx < BOARD_DIMENSION; dx++) {
-            possibilities.remove(get(dx, y));
+            possibilities.remove(board->get(dx, y));
         }
 
         // Remove all from this column
         for (int dy = 0; dy < BOARD_DIMENSION; dy++) {
-            possibilities.remove(get(x, dy));
+            possibilities.remove(board->get(x, dy));
         }
 
         // Remove all from this quadrant
         for (int dy = 0; dy < 3; dy++) {
             for (int dx = 0; dx < 3; dx++) {
-                possibilities.remove(get((x - x % 3) + dx, (y - y % 3) + dy));
+                possibilities.remove(board->get((x - x % 3) + dx, (y - y % 3) + dy));
             }
         }
 
-        return possibilities;
+        possible_values = possibilities;
+    }
+};
+
+class Sudoku {
+private:
+    Board *board = new Board();
+    std::vector<Cell> heap;
+ 
+public:
+ 
+    Sudoku(std::string s) {
+        int x = 0;
+        int y = 0;
+
+        for (unsigned int i = 0; i < s.length(); i++) {
+            board->values[i] = (int) (s[i] - '0');
+
+            if (board->values[i] == 0) {
+                Cell cell(board, x, y);
+                heap.emplace_back(cell);
+            }
+
+            x++;
+            if (x == BOARD_DIMENSION) {
+                x -= BOARD_DIMENSION;
+                y++;
+            }
+        }
+
+        // Update values of cells initially
+        for (Cell &cell : heap) {
+            cell.update_possible_values();
+        }
+
+        std::make_heap(heap.begin(), heap.end(), sort_cells);
+    }
+
+    static bool sort_cells(const Cell& a, const Cell& b) {
+        return a.possible_values.size() < b.possible_values.size(); 
+    }
+
+    void update_all_cells() {
+        for (Cell &cell : heap) {
+            cell.update_possible_values();
+        }
+
+        std::sort_heap(heap.begin(), heap.end(), sort_cells);
+    }
+
+    Cell get_next_best_empty_cell() {
+		std::pop_heap(heap.begin(), heap.end(), sort_cells);
+		Cell c = heap.back();
+        heap.pop_back();
+        
+		return c;
+	}
+ 
+    void solve() {
+        solve_recurse();
+
+        print_board();
+    }
+
+    bool solve_recurse() {
+        // If there are no more possible moves in the heap...
+        if (heap.empty()) {
+            // Check whether there are empty fields left
+            for (int y = 0; y < BOARD_DIMENSION; y++) {
+                for (int x = 0; x < BOARD_DIMENSION; x++) {
+                    if (board->get(x, y) == 0) return false;
+                }
+            }
+
+            // If not, we're done!
+            return true;
+        }
+        if (heap.empty()) return true;
+
+        Cell cell = get_next_best_empty_cell();
+
+        for (int value : cell.possible_values) {
+            board->set(cell.x, cell.y, value);
+            update_all_cells();
+
+            if (solve_recurse()) {
+                return true;
+            }
+
+            board->set(cell.x, cell.y, 0);
+            update_all_cells();
+        }
+
+        heap.push_back(cell);
+        std::push_heap(heap.begin(), heap.end(), sort_cells);
+
+        return false;
     }
 
     void print_board() {
-        std::cout << board[0] << " ";
+        std::cout << board->values[0] << " ";
 		for (unsigned int i = 1; i < BOARD_SIZE; i++) {
 			if (i % 3 == 0 && i % 9 != 0) {
 				std::cout << "| ";
@@ -84,7 +173,7 @@ public:
 			if (i % 27 == 0) {
 				std::cout << "------+-------+------" << std::endl;
 			}
-			std::cout << board[i] << " ";
+			std::cout << board->values[i] << " ";
 		}
 
         std::cout << std::endl;
@@ -98,7 +187,7 @@ int main() {
 	Sudoku solver("006001849030000000000020006000400320400003000600008000010060003000005004029074005");
 	//Sudoku solver("850002400720000009004000000000107002305000900040000000000080070017000000000036040");
 	solver.solve();
-	solver.print_board();
+	//solver.print_board();
 
     return 0;
 }