The reader is advised to read all the lessons (tutorials) in this full course, in the order presented.

Problem

Given a matrix of 2D array of n rows and m columns, print this matrix in ZIG-ZAG fashion as shown in the following figure:

Example 1: 
Input: mat[][] = {{1, 2, 3}
                  {4, 5, 6}
                  {7, 8, 9}}

Output : 1, 2, 4, 7, 5, 3, 6, 8, 9

Example 2: 
Input: mat[][] = {{1,  2,  3,  4,  5}, 
                  {6,  7,  8,  9,  10},
                  {11, 12, 13, 14, 15},
                  {16, 17, 18, 19, 20}};

Output : 1, 2, 6, 11, 7, 3, 4, 8, 12, 16, 17, 13, 9, 5, 10, 14, 18, 19, 15, 20

Solution in O(n x m) Time

The algorithm is as follows:

- Visit the top-left element first. Visit the next element on the right.
- Continue visiting the current minor diagonal elements in the left-downwards direction until meeting the left edge or bottom edge of the matrix.
- If the left edge of the matrix is met and not the bottom edge, visit the next element below.
- Continue visiting the current minor diagonal elements in the right-upwards direction until meeting the top edge or right edge of the matrix.
- If the bottom edge of the matrix was met, and not the left edge, then visit the next element on the right at the bottom edge.
- Continue visiting the current minor diagonal elements in the right-upwards direction until meeting the top edge or right edge of the matrix.
- If the top edge of the matrix is met and not the right edge, visit the next element on the right.
- Continue visiting the current minor diagonal elements in the left-downwards direction until meeting the left edge or bottom edge of the matrix.
- If the right edge of the matrix was met, and not the top edge, then visit the next element below.
- Continue visiting the current minor diagonal elements in the left-downwards direction until meeting the left edge or bottom edge of the matrix.
- Continue repeating the above steps accordingly, until the element at the bottom-right corner of the matrix is met, which will always be met, either from the left or from above.
- Do not forget to consider the cases, where the visit is at the four corners.

There are eight cases to consider: when the visit is at the four corners and when the visit is at the four edges. The case of what happens after the visit of the bottom-right element, does not have to be addressed. This gives rise to seven code segments in a major while-loop.

Indexes

The top most row of the matrix is identified by i=0.
The leftmost column of the matrix is identified by j=0.
The bottommost row of the matrix is identified by i=n-1.
The rightmost column of the matrix is identified by j=m-1.

The program

The program is (read through the code and comments):

<?php

    require_once 'vendor/autoload.php';

    use Rindow\Math\Matrix\MatrixOperator;
    
    // Initialize the Matrix Operator
    $mo = new MatrixOperator();

    function zigZagMatrix($mat) {
        //Determine dimensions automatically via the shape() method
        $shape = $mat->shape();
        $n = $shape[0];
        $m = $shape[1];

        $i=0; $j=0;
    
        while ($i < $n && $j < $m) {
            if ($i == 0 && $j == 0) {    //top-left corner
                echo($mat[$i][$j] . ", ");
                $j++;    //go right once
                echo($mat[$i][$j] . ", ");
                while ($j > 0 && $i < $n-1) {    //end at left or bottom edge
                    $i++; $j--;    //go down and left
                    echo($mat[$i][$j] . ", ");
                }
            }
        
            if ($j == 0 && $i > 0 && $i < $n-1) {    //left edge and not top or bottom ends
                $i++;    //go down once
                echo($mat[$i][$j] . ", ");
                while ($i > 0 && $j < $m - 1) {    //end at top or right edge
                    $i--; $j++;   //go right and up
                    echo($mat[$i][$j] . ", ");
                }
            }
        
            if ($i == 0 && $j > 0 && $j < $m-1) {    //top edge and not left or right ends 
                $j++;    //go right once
                echo($mat[$i][$j] . ", ");
                while ($j > 0 && $i < $n-1) {    //end at left or bottom edge
                    $i++; $j--;    //go down and left
                    echo($mat[$i][$j] . ", ");
                }
            }
        
            if ($i == $n-1 && $j == 0) {    //left-bottom corner
                $j++;    //go right once
                echo($mat[$i][$j] . ", ");
                while ($i > 0 && $j < $m - 1) {    //end at top or right edge
                    $i--; $j++;   //go right and up
                    echo($mat[$i][$j] . ", ");
                }
            }
        
            if ($i == 0 && $j == $m-1) {    //top-right corner
                $i++;    //go down once
                echo($mat[$i][$j] . ", ");
                while ($j > 0 && $i < $n-1) {    //end at left or bottom edge
                    $i++; $j--;   //go down and left
                    echo($mat[$i][$j] . ", ");
                }
            }
        
            if ($i > 0 && $i < $n-1 && $j == $m-1) {    //right edge and not top or bottom ends
                $i++;    //go down once
                echo($mat[$i][$j] . ", ");
                while ($j > 0 && $i < $n-1) {    //end at left or bottom edge
                    $i++; $j--;    //go down and left
                    echo($mat[$i][$j] . ", ");
                }
                //bottom-right end of matrix
                if ($i == $n-1 && $j == $m-1) {
                    $i++; $j++;    //increase i and j to stop the big while-loop
                }
            }
        
            if ($j > 0 && $j < $m-1 && $i == $n-1) {    //bottom edge and not left or right ends
                $j++;    //go right once
                
                echo($mat[$i][$j] . ", ");
                while ($i > 0 && $j < $m-1) {    //end at top or right edge
                    $i--; $j++;    //go right and up
                    echo($mat[$i][$j] . ", ");
                }
                //bottom-right end of matrix
                if ($i == $n-1 && $j == $m-1) {
                    $i++; $j++;    //increase i and j to stop the big while-loop
                }
            }    
        }
        echo PHP_EOL;
    }

  // main area
/*
    $mat = $mo->array([[1,  2,  3], 
                       [4,  5,  6],
                       [7,  8,  9]]);
                     
    zigZagMatrix($mat);
*/
/*
    $mat2 = $mo->array([[1,  2,  3,  4,  5], 
                        [6,  7,  8,  9,  10],
                        [11, 12, 13, 14, 15],
                        [16, 17, 18, 19, 20]]);
      
    zigZagMatrix($mat2);
*/

?>

Remove the first pair of the block comment markers (/* */). Make the first argument of the called function, "int mat[][3]". Run the program and the output should be:

    1, 2, 4, 7, 5, 3, 6, 8, 9, 

Put back the first pair of the block comment markers (/* */). Remove the second pair of the block comment markers (/* */). Make the first argument of the called function, "int mat[][5]". Run the program and the output should be:

1, 2, 6, 11, 7, 3, 4, 8, 12, 16, 17, 13, 9, 5, 10, 14, 18, 19, 15, 20,

The time complexity is actually O(2 n x m), with the first O(n x m) for populating the input matrix with all the elements, and the second O(n x m) for iterating over all the elements in the matrix in the called function. Since the coefficient (multiplicand of 2) is usually omitted, the time complexity is quoted as O(n x m).

The space complexity is O(n x m), for the input matrix, which is the same as the matrix in the called function (same reference).

Thanks for reading.