Coding Patterns
Matrix Algorithms

Matrix Algorithms

Matrix algorithms involve efficient manipulation, traversal, and transformation of 2D arrays (grids). These problems are ubiquitous in technical interviews, appearing as image processing tasks, board games, or dynamic programming grids.

Real-World Analogy

Imagine a library reference section organized in a grid.

  • Rows represent different categories of books (History, Science, Art).
  • Columns represent the specific aisle number for that category.

To find a specific book, you need two pieces of information: the Category (Row) and the Aisle (Column). A Matrix Algorithm is simply a strategy for efficient navigation—whether you’re walking through every aisle (traversal), jumping to a specific section (search), or rearranging the shelves (rotation).

Visual Explanation

In memory, a matrix is often contiguous, but conceptually we treat it as a grid. Here’s how we visualize 2D traversal:

  graph TD
    subgraph Grid["2D Grid / Matrix"]
        direction TB
        R0["Row 0: 1, 2, 3"]
        R1["Row 1: 4, 5, 6"]
        R2["Row 2: 7, 8, 9"]
        R0 --- R1
        R1 --- R2
    end

    subgraph Access["Access Pattern"]
        P["Pointer @ grid[1][1]"] --> V["Value: 5"]
        P --> N["Neighbors"]
        N -- Up --> U["2"]
        N -- Down --> D["8"]
        N -- Left --> L["4"]
        N -- Right --> R["6"]
    end

    style P fill:#f9f,stroke:#333
    style V fill:#bbf,stroke:#333

When to Use Matrix Algorithms

Check for these signs in the problem statement:

  • The input is a 2D array, grid, or board.
  • You need to perform image processing (rotate, flip, scale).
  • The problem involves a game board (Sudoku, Chess, Tic-Tac-Toe, Game of Life).
  • You are calculating paths or costs in a grid-based map.
  • You need to search for a value in a sorted 2D structure.

Complexity Analysis

OperationTime ComplexitySpace ComplexityNotes
Access
O(1)
O(1)
Direct indexing matrix[r][c]
Traversal
O(M*N)
O(1)
For simple iteration; extra space if recursion used
Search (Unsorted)
O(M*N)
O(1)
Must check every element
Search (Sorted)
O(M + N)
O(1)
Staircase search or Binary Search variations
Rotation
O(M*N)
O(1)
In-place rotation possible

Variables:

  • M: Number of rows.
  • N: Number of columns.

Common Mistakes

  • Confusing Row/Column Indices: Remembering that matrix[y][x] corresponds to row y and column x, which is often visually opposite to (x, y) Cartesian coordinates.
  • Boundary Checks: Forgetting to check r >= 0, r < rows, c >= 0, and c < cols before accessing neighbors.
  • Empty Matrices: Failing to handle edge cases where the matrix is [] or [[]].
  • Jagged Arrays: Assuming all rows have the same length (though usually true in interview grid problems).

Next Steps

  1. Memorize the Templates: Learn the standard nested-loop and boundary-check patterns in the Code Templates section.
  2. Practice Problems: Apply these concepts to real interview questions in the Problems section.