Suppose we have a matrix where each cell represents a distinct symbol or integer. Our task is to decode this matrix by reading the cells in a particular order.
The decoding begins from the top-left cell of the matrix. We move in a bottom-left downward diagonal direction until we hit the left boundary. Upon hitting the left boundary, we move one cell down (unless we're at the bottom-left corner already, in which case we move one cell to the right) and start moving in an upward diagonal direction toward the upper-right.
While moving diagonally up-right, if we hit the top boundary, we move one cell to the right and start moving in a downward diagonal direction towards the bottom-left. However, if we hit the right boundary while moving diagonally upwards, we move one cell down and start moving in a bottom-left direction. In other words, we keep zigzagging diagonally across the matrix until every cell in the matrix is visited.
Upon completing this zigzag traversal, we will have a list of traversed cell values. Next, we process this list to uncover the indices of the perfect square numbers. The function diagonalTraverseAndSquares($matrix) should implement this traversal and return an array containing the positions of perfect square numbers in the traversed sequence.
Take a 3×4 matrix, for instance:
$matrix = [
[1, 2, 3, 4],
[5, 6, 7, 8],
[9, 10, 11, 12]
];
Upon completing the diagonal traversal, we'll get the list: {1, 5, 2, 3, 6, 9, 10, 7, 4, 8, 11, 12}. From this list, we see that 1, 9, and 4 are perfect squares and are located at the 1st, 6th, and 9th positions in the list. Thus, our function returns: [0, 5, 8].
First, let's examine the dimensions of the matrix. To map the landscape of our matrix, we'll use count($matrix) to determine the number of rows and count($matrix[0]) to determine the number of columns. Next, we initialize two arrays: $traversal and $results. The $traversal array will keep the cell values that we will obtain from the matrix based on our unique diagonal zigzag traversal. The $results array will later be populated with the positions of perfect square numbers that can be found in the $traversal array.
<?php
function diagonalTraverseAndSquares($matrix) {
$rows = count($matrix);
$cols = count($matrix[0]);
$traversal = [];
$results = [];
The next step involves the actual traversal of the matrix. This process is done diagonally in a zigzag pattern. We begin from the top-left corner (cell [0][0]) and make our journey through the matrix using two variables, $row and $col, to track the cell indices. We also initialize $dir with 1, which dictates that the starting direction is the down-left direction.
However, it's not just a simple left-right or up-down movement; as per the rules, we need to ensure we change our direction whenever we hit an edge. Let $dir = -1 dictate the up-right direction. To ensure we continue the correct diagonal movement and don't exceed the matrix boundaries, we use conditional checks within the loop.
$row = 0;
$col = 0;
$dir = 1;
for ($i = 0; $i < $rows * $cols; ++$i) { // Loop runs for the total number of cells in the matrix.
$traversal[] = $matrix[$row][$col]; // Append the current cell value to traversal.
// Logic to control direction based on edges:
if ($dir == 1) { // Moving down-left
if ($row == $rows - 1) {
$col += 1;
$dir = -1;
} elseif ($col == 0) {
$row += 1;
$dir = -1;
} else {
$row += 1;
$col -= 1;
}
} else { // Moving up-right
if ($col == $cols - 1) {
$row += 1;
$dir = 1;
} elseif ($row == 0) {
$col += 1;
$dir = 1;
} else {
$row -= 1;
$col += 1;
}
}
}
