Hello, Space Explorer! Are you ready to embark on a journey into the land of matrices? In this unit, we're in for a thrilling ride into the world of unique matrix traversal. We'll be navigating through the realm of 2D arrays, following a distinctive order that oozes intrigue. Stay seated, and let's dive right in!
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($matrix) should implement this traversal and return an array containing the positions of perfect square numbers in the traversed sequence.
Take a matrix, for instance:
Upon completing the diagonal traversal, we'll get the list: {1, 5, 2, 3, 6, 9, 10, 7, 4, 8, 11, 12}. From this list, we see that 1, 9, and 4 are perfect squares and are located at the 1st, 6th, and 9th positions in the list. Thus, our function returns: [0, 5, 8].
First, let's examine the dimensions of the matrix. To map the landscape of our matrix, we'll use count($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.
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.
With a completed traversal, we have obtained a list of integers. Next, we evaluate this list to find perfect squares, i.e., numbers that are squares of other integers. For every perfect square we encounter, we add its position in the $traversal array to the $results array. In PHP, we can use the sqrt function. If the integer square root, when squared, equals the initial number, we know the initial number is a perfect square, so we add its position to our results.
Bravo! You've successfully navigated a challenging task involving a unique matrix traversal pattern. You've demonstrated solid skills in PHP programming, specifically array manipulation, and brilliantly tackled the challenges of moving around multidimensional arrays.
Now, it's time to embark on your journey and put your newly learned skills to the test! Try out more complex matrices and different values to truly master the concept. Keep experimenting, and you'll soon become a wizard at matrix traversals. Happy coding!
