794. Valid Tic-Tac-Toe State

A Tic-Tac-Toe board is given as a string array board. Return True if and only if it is possible to reach this board position during the course of a valid tic-tac-toe game.

The board is a 3 x 3 array, and consists of characters " ", "X", and "O". The " " character represents an empty square.

Here are the rules of Tic-Tac-Toe:

• Players take turns placing characters into empty squares (" ").

• The first player always places "X" characters, while the second player always places "O" characters.

• "X" and "O" characters are always placed into empty squares, never filled ones.

• The game ends when there are 3 of the same (non-empty) character filling any row, column, or diagonal.

• The game also ends if all squares are non-empty.

• No more moves can be played if the game is over.

Example 1:
Input: board = ["O  ", "   ", "   "]
Output: false
Explanation: The first player always plays "X".

Example 2:
Input: board = ["XOX", " X ", "   "]
Output: false
Explanation: Players take turns making moves.

Example 3:
Input: board = ["XXX", "   ", "OOO"]
Output: false

Example 4:
Input: board = ["XOX", "O O", "XOX"]
Output: true

Note:

• board is a length-3 array of strings, where each string board[i] has length 3.

• Each board[i][j] is a character in the set {" ", "X", "O"}.

bool validTicTacToe(vector<string>& board) { // time: O(1); space: O(1)
    bool X_win = false, O_win = false;
    vector<int> rows(3, 0), cols(3, 0);
    int diag = 0, antidiag = 0, turns = 0;
    for (int i = 0; i < 3; ++i) {
        for (int j = 0; j < 3; ++j) {
            if (board[i][j] == 'X') {
                ++rows[i];
                ++cols[j];
                ++turns;
                if (i == j) ++diag;
                if (i + j == 2) ++antidiag;
            } else if (board[i][j] == 'O') {
                --rows[i];
                --cols[j];
                --turns;
                if (i == j) --diag;
                if (i + j == 2) --antidiag;
            }
        }
    }
    X_win = rows[0] == 3 || rows[1] == 3 || rows[2] == 3 || 
            cols[0] == 3 || cols[1] == 3 || cols[2] == 3 ||
            diag == 3 || antidiag == 3;
    O_win = rows[0] == -3 || rows[1] == -3 || rows[2] == -3 || 
            cols[0] == -3 || cols[1] == -3 || cols[2] == -3 ||
            diag == -3 || antidiag == -3;
    if ((X_win && turns == 0) || (O_win && turns == 1)) return false;
    return (turns == 0 || turns == 1) && (!X_win || !O_win);
}