算法 - nQueens 的时间复杂度

Question

I have this questions which may be basic to ask, but I am finding it hard to understand. How to determine time complexity of nQueens. In some post it says its n, because in (4*4) matrix. each queen gets one row & as the queens moved down the rows the column option reduces (n * n-1 * n-2..,) which is fine. but in recursive algorithm we pass rowIndex and for each call we check if queen can be placed in a cell looping thru all columns & then call solveNQueens for rowIndex+1. In this case is time complexity still n!

bool solveNQueens(rowIndex, matrix)
{
  //recursion base case to exit

  //decision tree
  for (int i = 0; i<4; i++)
  {
    if(cellAvailable)
    {
      solveNQueens(rowIndex+1, matrix);
    }
  }
}

Answer 1

In fact O(n!) is an upper bound. But as this article explains , the number of solutions is roughly (0.143n)^n . So that gives a super-exponential lower bound.

Coming up with a better answer is likely to be an open research problem. But I expect that the lower bound is in the right ballpark for what the algorithm takes to produce all answers.