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The backtracking approach is a classic method to solve problems like Sudoku that require exploring all configurations to find a valid solution. This involves recursively trying all numbers from 1 to 9 in each cell and verifying if they meet Sudoku rules. If a number fits, move to the next cell. If stuck, backtrack to the previous cell and try a different number. This method ensures all constraints are respected at each step.
Time Complexity: O(9^(N*N)), where N = 9 (the size of the board), as each empty cell has 9 possibilities.
Space Complexity: O(N*N) for the recursion stack.
1#include <stdbool.h>
2
3bool isValid(char board[9][9], int row, int col, char num) {
4 for (int x = 0; x < 9; x++) {
5 if (board[row][x] == num || board[x][col] == num ||
6 board[3 * (row / 3) + x / 3][3 * (col / 3) + x % 3] == num) {
7 return false;
8 }
9 }
10 return true;
11}
12
13bool solveSudokuHelper(char board[9][9]) {
14 for (int row = 0; row < 9; row++) {
15 for (int col = 0; col < 9; col++) {
16 if (board[row][col] == '.') {
17 for (char num = '1'; num <= '9'; num++) {
18 if (isValid(board, row, col, num)) {
19 board[row][col] = num;
20 if (solveSudokuHelper(board)) return true;
21 board[row][col] = '.'; // Backtrack
22 }
23 }
24 return false; // Trigger backtracking
25 }
26 }
27 }
28 return true;
29}
30
31void solveSudoku(char board[9][9]) {
32 solveSudokuHelper(board);
33}This solution implements a backtracking algorithm in C. The isValid function checks if placing a digit in a particular cell violates Sudoku rules. The solveSudokuHelper function tries placing digits and uses recursion to backtrack if necessary until the board is completely filled.
This approach enhances the basic backtracking method by introducing constraint propagation. Before choosing a number for a cell, it checks constraints upfront, reducing unnecessary exploration by propagating constraints once a number is filled. This technique decreases the number of options that need to be tried, thereby optimizing the backtracking process.
Time Complexity: Expected better than O(9^(N*N)), depending on effectiveness of constraint propagation reducing possible combinations.
Space Complexity: O(N*N), influenced by recursion.
1class Solution:
2 def solveSudoku(self, board: List[List
This Python solution incorporates constraint propagation into the backtracking method. It attempts to minimize choices by evaluating constraints before any number is placed, helping to reduce the branching factor in the search tree. If a valid scenario is found using propagation, it progresses further in solving the Sudoku.