Given a 2D square matrix, find the sum of elements in Principal and Secondary diagonals. For example, consider the following 4 X 4 input matrix.
A00 A01 A02 A03
A10 A11 A12 A13
A20 A21 A22 A23
A30 A31 A32 A33
The primary diagonal is formed by the elements A00, A11, A22, A33.
- Condition for Principal Diagonal: The row-column condition is row = column.
The secondary diagonal is formed by the elements A03, A12, A21, A30. - Condition for Secondary Diagonal: The row-column condition is row = numberOfRows - column -1.
Examples :
Input:
4
1 2 3 4
4 3 2 1
7 8 9 6
6 5 4 3
Output :
Principal Diagonal: 16
Secondary Diagonal: 20
Input :
3
1 1 1
1 1 1
1 1 1
Output :
Principal Diagonal: 3
Secondary Diagonal: 3
Method 1 - Native Approach(O(n ^ 2) ):
In this method, we use two loops i.e. a loop for columns and a loop for rows and in the inner loop we check for the condition stated above:
<?php
// A simple PHP program to
// find sum of diagonals
$MAX = 100;
function printDiagonalSums($mat, $n)
{
global $MAX;
$principal = 0;
$secondary = 0;
for ($i = 0; $i < $n; $i++)
{
for ($j = 0; $j < $n; $j++)
{
// Condition for
// principal diagonal
if ($i == $j)
$principal += $mat[$i][$j];
// Condition for
// secondary diagonal
if (($i + $j) == ($n - 1))
$secondary += $mat[$i][$j];
}
}
echo "Principal Diagonal:" ,
$principal ,"
";
echo "Secondary Diagonal:",
$secondary ,"
";
}
// Driver code
$a = array (array ( 1, 2, 3, 4 ),
array ( 5, 6, 7, 8 ),
array ( 1, 2, 3, 4 ),
array ( 5, 6, 7, 8 ));
printDiagonalSums($a, 4);
// This code is contributed by ajit
?>
Output
Principal Diagonal:18 Secondary Diagonal:18
Complexity Analysis:
- Time Complexity: O(N*N), as we are using a loop to traverse N*N times.
- Auxiliary Space: O(1), as we are not using any extra space.
Method 2 - Efficient Approach (O(n) ):
In this method we use one loop i.e. a loop for calculating sum of both the principal and secondary diagonals:
<?php
// An efficient PHP program
// to find sum of diagonals
$MAX = 100;
function printDiagonalSums($mat, $n)
{
global $MAX;
$principal = 0; $secondary = 0;
for ($i = 0; $i < $n; $i++)
{
$principal += $mat[$i][$i];
$secondary += $mat[$i][$n - $i - 1];
}
echo "Principal Diagonal:" ,
$principal ,"
";
echo "Secondary Diagonal:" ,
$secondary ,"
";
}
// Driver Code
$a = array(array(1, 2, 3, 4),
array(5, 6, 7, 8),
array(1, 2, 3, 4),
array(5, 6, 7, 8));
printDiagonalSums($a, 4);
// This code is contributed by aj_36
?>
Output
Principal Diagonal:18 Secondary Diagonal:18
Complexity Analysis:
- Time Complexity: O(N), as we are using a loop to traverse N times.
- Auxiliary Space: O(1), as we are not using any extra space.
Please refer complete article on Efficiently compute sums of diagonals of a matrix for more details!
