Tuesday, August 17, 2010

Print numbers by spiral

Recently I came across simple yet interesting coding problem. So here is the deal. You are given positive integer N. Print first N ^ 2 positive integers in matrix form in a such a way that within matrix numbers form spiral starting from its center and goring clockwise. For example, for N = 5 matrix to be printed is:

21 22 23 24 25
20 7 8 9 10
19 6 1 2 11
18 5 4 3 12
17 16 15 14 13

Optimize it for speed and space.

One way you can approach it is to create N x N matrix and fill it with numbers that form spiral and then print whole matrix row by row. But this solution will be of N ^ 2 space complexity. Let’s try to reach O(1) space complexity.

The key observation here is how matrix changes when N changes by 1.

N = 1.

1

N = 2.

1 2
4 3

N = 3.

7 8 9
6 1 2
5 4 3

N = 4.

7 8 9 10
6 1 2 11
5 4 3 12
16 15 14 13

Can you see the pattern here? At every step we extend previous matrix (P) with additional column and row (C). If N is even we extend previous matrix of size N – 1 with right column and bottom row

P C
C C

and with left column and top row if it is odd

C C
C P

This leads us to naturally recursive algorithm. We have three cases:

  1. Print whole row of the current matrix (top when N is odd or bottom when N is even).
  2. Print row from previous matrix of size N - 1 first and then print value that belongs to current matrix (when N is even).
  3. Print value that belongs to current matrix and then print row from previous matrix of size N - 1 (when N is odd).
  4. Print matrix line by line.

So basically to print a row we need to know matrix size N and row index. Here goes the solution.

static void Print(int n)
{
	for(int i = 0; i < n; i++)
	{
		PrintLine(n, i);
		Console.WriteLine();
	}
}

static void PrintLine(int n, int i)
{
	// Number of integers in current matrix
	var n2 = n*n;
	// Number of itegers in previous matrix of size n - 1
	var m2 = (n - 1)*(n - 1);

	if (n % 2 == 0)
	{
		if (i == n - 1)
		{
			// n is even and we are at the last row so just 
			// print it
			for(int k = n2; k > n2 - n; k--)
			{
				PrintNum(k);
			}
		}
		else
		{
			// Print row from previous matrix of size n - 1 
			// first and then print value that belongs to current 
			// matrix. Previous matrix is at the top left corner 
			// so no need to adjust row index
			PrintLine(n - 1, i);
			// Skip all integers from previous matrix and upper 
			// ones in this columnas integers must form clockwise 
			// spiral
			PrintNum(m2 + 1 + i);
		}
	}
	else
	{
		if (i == 0)
		{
			// n is odd and we are at the first row so just 
			// print it
			for(int k = m2 + n; k <= n2; k++)
			{
				PrintNum(k);
			}
		}
		else
		{
			// Print value that belongs to current matrix and
			// then print row from previous matrix of size n - 1
			// Skip all integers from previous matric and bottom
			// ones in this column as integers must form clockwise
			// spiral
			PrintNum(m2 + n - i);
			// Previous matrix is at the bottom right corner so
			// row index must be reduced by 1
			PrintLine(n - 1, i - 1);
		}
	}
}

static void PrintNum(int n)
{
	Console.Write("{0, -4}  ", n);
}

If stack is not considered then this solution has O(1) space complexity otherwise O(N).

2 comments:

Nayefc said...
This comment has been removed by the author.
priyanka singh said...

sir. Would u be a bit more clear.... !!!!...... A detail algorithm is required........... Please help...