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For a rectangular matrix, traverse a diagonal by moving one row at a time and changing the column by one in the chosen direction. Check row and column bounds separately before accessing a cell. The title does not specify whether “detecting” means listing diagonal cells, finding a pattern, or selecting the main diagonals; this guide treats it primarily as diagonal-wise traversal, then explains how to adapt it to pattern matching and principal diagonals.
What counts as a diagonal?
Represent a matrix cell with a row and column, (r, c). Moving down and right gives (r + 1, c + 1); moving down and left gives (r + 1, c - 1). A diagonal is the sequence of cells reached by repeatedly taking one of those steps while the coordinates remain valid.
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For a matrix with R rows and C columns, valid row indices are 0 through R - 1, and valid column indices are 0 through C - 1. Keep these limits distinct: a rectangular matrix cannot safely use one shared n bound unless it is known to be square.
Choose what “detecting” means
Traverse every diagonal
If the goal is to output values in diagonal groups or a diagonal-wise order, first decide which slope and ordering are required. A straightforward approach starts each diagonal at a boundary cell, then walks along it until reaching an edge. For one slope, starting across the top edge and then down the left edge (excluding the already-used top-left corner) covers all R + C - 1 diagonals.
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Find a pattern on a diagonal
If the goal is to find a sequence or pattern, specify the pattern, slope, minimum run length, and whether a match may begin at any cell. Enumerate each diagonal as above and test each in-bounds run. Stop a candidate run at the diagonal’s endpoint rather than reading past it.
Select the two principal diagonals
For a square matrix of side N, the main diagonal contains (i, i) and the anti-diagonal contains (i, N - 1 - i) for i from 0 to N - 1. These are only the two principal diagonals; they are not all the diagonals of the matrix. For a non-square matrix, do not apply the square-matrix formula without defining which cells are intended.
Boundary-start traversal algorithm
The following language-neutral pseudocode enumerates each diagonal that slopes down and right. It visits every cell exactly once overall.
- For each column
startColumnfrom0toC - 1, begin at(0, startColumn). - While
row < Randcolumn < C, process the cell, then increment both coordinates. - For each row
startRowfrom1toR - 1, begin at(startRow, 0). - Walk and process cells using the same bounds check and increments.
For the opposite slope, use starts along the top edge and right edge, and decrement the column at each step while incrementing the row. Check column >= 0 as well as row < R.
Because the algorithm processes each of the R × C cells once, its running time is O(RC). If values are emitted as they are visited, the extra space is O(1), not counting the output destination. These are algorithmic bounds derived from the traversal, not benchmark results.
Why a zigzag traversal is a different choice
A direction-switching traversal alternates between moving diagonally up and diagonally down, taking a step right or down when it reaches a boundary. It produces one zigzag ordering rather than listing each diagonal as an independent group. The Indian Institute of Technology Kharagpur examination solution demonstrates a 5×3 example whose zigzag output is 1, 4, 2, 3, 5, 7, 10, 8, 6, 9, 11, 13, 14, 12, 15. Treat that as one defined ordering, not as the universal meaning of diagonal traversal.
Represent the matrix correctly in C++ and C#
| Language and representation | Access syntax | Shape and bounds to handle |
|---|---|---|
| C++ built-in two-dimensional array | a[row][column] |
Supply or otherwise know the row and column dimensions; do not assume they are equal. Microsoft documents successive subscripts for multidimensional arrays in its C++ array reference. |
C# rectangular array, T[,] |
a[row, column] |
Get dimensions with a.GetLength(0) and a.GetLength(1). Microsoft’s C# array reference documents the index order and rectangular-array form. |
C# jagged array, T[][] |
a[row][column] |
Rows may differ in length. Check the outer row index, whether the row is null if null rows are allowed, and the selected row’s own length before accessing the column. |
Microsoft’s CA1814 guidance recommends considering jagged arrays when multidimensional storage would waste space; it also allows multidimensional arrays when they do not waste space. This is a storage-shape trade-off, not a claim that one representation is always faster or better. Rectangular arrays suit uniform rows; jagged arrays can represent uneven rows.
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Implementation checks and edge cases
- Empty matrix: If either dimension is zero, return without indexing a cell.
- One row or one column: Boundary-start enumeration still works; some diagonals simply contain one cell.
- Rectangular dimensions: Check row and column bounds independently at every step.
- Jagged C# rows: Use each row’s actual length, and account for null row references if permitted by the program.
- Both slopes: Enumerate the matching top and side boundaries separately for each direction.
- C++ dimensions: Make dimensions available to the traversal, or use a container from which row and column sizes can be obtained.
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