Diagnostic Quiz

Problems

1. Tower of Hanoi

Disk i-1 is moved twice the number of times of Disk i

It is correct. This can be reasoned using the recurrence relation $T(n)=2T(n-1)+1$ for the Tower of Hanoi problem.

5. Time Complexity for Nqueens

Consider the solution to N-Queens given in class. Suppose the running time of nqueens is $T(n)$. What is $T(1)$?

This is an awesome question. $T(1)$ happens when we reach the base case, so here we need to think about what will happen in the base case.

The base case is:

if (row == n - 1) {
  if (!threaten_each_other_diagonally(queens, n - 1)) {
    cs1010_println_string(queens);
    return true;
  }
  return false;
}

It calls threaten_each_other_diagonally which runs in $O(n)$.

bool threaten_each_other_diagonally(char queens[], size_t last_row) {
  for (size_t curr_row = 0; curr_row <= last_row; curr_row += 1) {
    if (has_a_queen_in_diagonal(queens, curr_row, last_row)) {
      return true;
    }
  }
  return false;
}

And print a string with length n takes $O(n)$. So, the overall running time for $T(1)$ is $O(n^2)$.