Diagonal Traverse
The “Diagonal Traverse” problem asks you to return the elements of a matrix in a diagonal zigzag order. Starting from the top-left corner, you move along diagonals that run from top-right to bottom-left, alternating direction at each diagonal. One diagonal is traversed upward, the next downward, and this pattern continues until you’ve visited every element exactly once.
What makes this problem deceptively tricky is that the traversal pattern is not row-wise or column-wise. You must coordinate movement across both dimensions while also switching direction at the right moments. The goal is to produce a clean, predictable ordering without missing elements or stepping outside the matrix.
Why this isn’t a standard matrix scan
A typical matrix scan is easy because the next element is always adjacent in a consistent direction, such as moving left-to-right across rows. Diagonal traversal breaks that simplicity. The “next” element depends on your current direction, and the direction changes based on which diagonal you are on.
Because diagonals interact with boundaries differently, you cannot rely on one universal step rule. You need a strategy that explicitly manages direction changes and boundary behavior.
Want to explore more coding problem solutions? Check out the Minimum Cost for Tickets and Minimum Falling Path Sum.
Recognizing the diagonal index pattern
A useful observation is that every matrix cell belongs to exactly one diagonal based on the sum of its row and column indices. Cells with the same row-plus-column value sit on the same diagonal. As you traverse, you are effectively visiting diagonals in increasing order of that sum.
This diagonal index pattern gives the traversal a structured backbone. Instead of thinking of movement as a free-form zigzag, you can think of it as iterating through diagonals one by one, alternating the direction used to read each diagonal.
Two ways to solve the problem: simulate movement or group by diagonal
There are two common solution styles for diagonal traverse. One approach simulates the movement step by step, keeping track of the current row, column, and direction. The other group elements by their diagonal index and then outputs each diagonal in the correct direction.
Both approaches are valid. The movement simulation is closer to the actual traversal behavior, while diagonal grouping often feels cleaner conceptually because it leverages the row-plus-column structure directly.
Understanding direction changes and boundary behavior
The core challenge in a simulation-based solution is deciding what happens when you hit a boundary. When moving upward along a diagonal, you typically decrease the row and increase the column. When moving downward, you increase the row and decrease the column.
However, when you reach the top row, bottom row, leftmost column, or rightmost column, the next move cannot follow the usual diagonal step. Instead, you shift to the beginning of the next diagonal and flip direction. The correctness of the traversal depends on handling these transitions precisely.
Why alternating directions is essential
The zigzag effect comes from alternating the direction you use to read each diagonal. Without this alternation, you would still move diagonally, but the output order would not match the required pattern.
This alternation is not arbitrary. It ensures that the traversal stays continuous and visits adjacent diagonals in a way that covers the entire matrix without skipping. Each diagonal is processed completely, then the traversal transitions cleanly to the next one.
Why this approach guarantees visiting every cell exactly once
Diagonal traversal works because diagonals partition the matrix. Every cell belongs to exactly one diagonal, and the sequence of diagonal indices covers all possible sums from zero up to the maximum valid sum.
As long as you process each diagonal exactly once and ensure you include every cell on that diagonal, you will visit every element exactly once. The direction alternation changes only the order within each diagonal, not which cells are included.
Time and space complexity considerations
The traversal runs in linear time relative to the number of elements in the matrix because each cell is output exactly once. Space usage depends on the approach. A movement simulation uses constant extra space beyond the output. A diagonal grouping approach may use additional storage to temporarily hold diagonals, though it remains linear overall.
Both strategies are efficient for typical input sizes, and the choice often comes down to which reasoning style feels clearer.
Why this problem is common in interviews
Diagonal Traverse is frequently used in interviews because it tests attention to detail and controlled simulation. It requires careful handling of boundaries and direction toggling, which are common sources of off-by-one errors.
The problem also checks whether you can recognize hidden structure in a traversal pattern and turn it into a predictable algorithm rather than improvising movement rules.
What this problem teaches beyond the diagonal order
Beyond producing a traversal output, this problem teaches a broader lesson about pattern-based iteration. Many matrix problems become easier when you identify an invariant that groups cells, such as row-plus-column diagonals.
If you can clearly explain how diagonal indices structure the traversal, why direction alternates, and how boundary transitions work, you demonstrate strong algorithmic thinking. That depth of understanding makes “Diagonal Traverse” a valuable exercise in matrix traversal design and implementation discipline.
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