Matrix

# Matrix Data Structure

# Overview

Matrix is a 2-dimensional array data structure that stores elements in rows and columns. It’s fundamental for solving problems involving grids, images, game boards, and mathematical computations.

# Key Properties

Time Complexity:
- Access: O(1)
- Traversal: O(m*n) where m=rows, n=columns
- Search: O(mn) for unsorted, O(log(mn)) for sorted matrices
Space Complexity: O(m*n) for storage
Core Idea: Elements accessed via [row][column] indexing
When to Use: Grid-based problems, 2D transformations, path finding, dynamic programming on grids

# Problem Categories

# Pattern 1: Matrix Traversal

Description: Navigate through matrix elements using specific patterns (spiral, diagonal, zigzag)
Examples: LC 54, 59, 498, 885
Key Pattern: Boundary-based movement with direction changes

# Pattern 2: Matrix Transformation

Description: Rotate, transpose, or flip matrices in-place or create new ones
Examples: LC 48, 867, 189, 1886
Key Pattern: Mathematical coordinate mapping

# Pattern 3: Matrix Search

Description: Find elements in sorted or partially sorted 2D matrices
Examples: LC 74, 240, 378, 668
Key Pattern: Binary search variants or elimination-based search

# Pattern 4: Matrix Modification

Description: Update matrix elements based on conditions (set zeros, smooth values)
Examples: LC 73, 661, 289, 1314
Key Pattern: Two-pass or auxiliary space approaches

# Pattern 5: Matrix Multiplication & Operations

Description: Mathematical operations between matrices or matrix computations
Examples: LC 311, 348, 1572, 1351
Key Pattern: Triple nested loops for multiplication, optimization for sparse matrices

# Pattern 6: Matrix Path & Dynamic Programming

Description: Find paths, count paths, or optimize values through matrix traversal
Examples: LC 62, 63, 64, 120, 931
Key Pattern: DP state transitions based on adjacent cells

# Templates & Algorithms

# Template Comparison Table

Template Type	Use Case	Key Structure	When to Use
Matrix Traversal	Spiral, diagonal movement	Boundary tracking + direction vectors	Ordered element processing
Matrix Transformation	Rotate, transpose, flip	Mathematical coordinate mapping	In-place modifications
Matrix Search	Find target in sorted matrix	Binary search or elimination	Sorted/partially sorted matrices
Matrix Modification	Set zeros, smooth values	Two-pass or auxiliary tracking	Conditional element updates
Matrix Multiplication	Dot product operations	Triple nested loops	Mathematical computations
Matrix Path DP	Path counting, min/max path	DP state transitions	Optimization problems on grids

# Universal Matrix Template

def solve_matrix_problem(matrix):
    if not matrix or not matrix[0]:
        return default_result
    
    rows, cols = len(matrix), len(matrix[0])
    
    # Initialize result structure
    result = initialize_result(rows, cols)
    
    # Main processing loop
    for i in range(rows):
        for j in range(cols):
            # Process current cell
            process_cell(matrix, i, j, result)
    
    return result

def process_cell(matrix, row, col, result):
    # Template for cell processing
    # - Check boundaries
    # - Apply logic
    # - Update result
    pass

# Pattern-Specific Templates

# Template 1: Matrix Traversal (Spiral/Diagonal)

def spiral_traversal(matrix):
    if not matrix or not matrix[0]:
        return []
    
    result = []
    rows, cols = len(matrix), len(matrix[0])
    
    # Define boundaries
    top, bottom = 0, rows - 1
    left, right = 0, cols - 1
    
    while top <= bottom and left <= right:
        # Right movement
        for j in range(left, right + 1):
            result.append(matrix[top][j])
        top += 1
        
        # Down movement  
        for i in range(top, bottom + 1):
            result.append(matrix[i][right])
        right -= 1
        
        # Left movement (if still valid row)
        if top <= bottom:
            for j in range(right, left - 1, -1):
                result.append(matrix[bottom][j])
            bottom -= 1
        
        # Up movement (if still valid column)
        if left <= right:
            for i in range(bottom, top - 1, -1):
                result.append(matrix[i][left])
            left += 1
    
    return result

# Template 2: Matrix Transformation (Rotate/Transpose)

def rotate_matrix_90_clockwise(matrix):
    """
    Two-step approach: Transpose + Reverse rows
    """
    n = len(matrix)
    
    # Step 1: Transpose matrix (swap matrix[i][j] with matrix[j][i])
    for i in range(n):
        for j in range(i + 1, n):  # Start from i+1 to avoid double swap
            matrix[i][j], matrix[j][i] = matrix[j][i], matrix[i][j]
    
    # Step 2: Reverse each row
    for i in range(n):
        matrix[i].reverse()
    
    return matrix

def transpose_matrix(matrix):
    """
    Create new matrix with swapped dimensions
    """
    if not matrix or not matrix[0]:
        return []
    
    rows, cols = len(matrix), len(matrix[0])
    result = [[0] * rows for _ in range(cols)]
    
    for i in range(rows):
        for j in range(cols):
            result[j][i] = matrix[i][j]
    
    return result

# Template 3: Matrix Search

def search_matrix_binary(matrix, target):
    """
    Binary search on sorted matrix (treat as 1D array)
    """
    if not matrix or not matrix[0]:
        return False
    
    rows, cols = len(matrix), len(matrix[0])
    left, right = 0, rows * cols - 1
    
    while left <= right:
        mid = (left + right) // 2
        mid_row, mid_col = mid // cols, mid % cols
        mid_val = matrix[mid_row][mid_col]
        
        if mid_val == target:
            return True
        elif mid_val < target:
            left = mid + 1
        else:
            right = mid - 1
    
    return False

def search_matrix_elimination(matrix, target):
    """
    Search in row-wise and column-wise sorted matrix
    Start from top-right or bottom-left corner
    """
    if not matrix or not matrix[0]:
        return False
    
    rows, cols = len(matrix), len(matrix[0])
    row, col = 0, cols - 1  # Start from top-right
    
    while row < rows and col >= 0:
        if matrix[row][col] == target:
            return True
        elif matrix[row][col] > target:
            col -= 1  # Move left
        else:
            row += 1  # Move down
    
    return False

# Template 4: Matrix Modification

def set_matrix_zeros(matrix):
    """
    Set entire row and column to zero if any element is zero
    """
    if not matrix or not matrix[0]:
        return
    
    rows, cols = len(matrix), len(matrix[0])
    
    # Use first row and column as markers
    first_row_zero = any(matrix[0][j] == 0 for j in range(cols))
    first_col_zero = any(matrix[i][0] == 0 for i in range(rows))
    
    # Mark zeros in first row and column
    for i in range(1, rows):
        for j in range(1, cols):
            if matrix[i][j] == 0:
                matrix[0][j] = 0  # Mark column
                matrix[i][0] = 0  # Mark row
    
    # Set zeros based on markers
    for i in range(1, rows):
        for j in range(1, cols):
            if matrix[0][j] == 0 or matrix[i][0] == 0:
                matrix[i][j] = 0
    
    # Handle first row and column
    if first_row_zero:
        for j in range(cols):
            matrix[0][j] = 0
    if first_col_zero:
        for i in range(rows):
            matrix[i][0] = 0

# Template 5: Matrix Multiplication

def multiply_matrices(A, B):
    """
    Standard matrix multiplication: C[i][j] = sum(A[i][k] * B[k][j])
    """
    if not A or not A[0] or not B or not B[0]:
        return []
    
    rows_A, cols_A = len(A), len(A[0])
    rows_B, cols_B = len(B), len(B[0])
    
    if cols_A != rows_B:
        return []  # Invalid dimensions
    
    result = [[0] * cols_B for _ in range(rows_A)]
    
    for i in range(rows_A):
        for j in range(cols_B):
            for k in range(cols_A):
                result[i][j] += A[i][k] * B[k][j]
    
    return result

def multiply_sparse_matrices(A, B):
    """
    Optimized multiplication for sparse matrices
    """
    if not A or not A[0] or not B or not B[0]:
        return []
    
    rows_A, cols_A = len(A), len(A[0])
    rows_B, cols_B = len(B), len(B[0])
    result = [[0] * cols_B for _ in range(rows_A)]
    
    for i in range(rows_A):
        for k in range(cols_A):
            if A[i][k] != 0:  # Skip if zero
                for j in range(cols_B):
                    result[i][j] += A[i][k] * B[k][j]
    
    return result

# Template 6: Matrix Path DP

def min_path_sum(grid):
    """
    Find minimum path sum from top-left to bottom-right
    """
    if not grid or not grid[0]:
        return 0
    
    rows, cols = len(grid), len(grid[0])
    
    # Initialize DP table (can modify grid in-place to save space)
    dp = [[0] * cols for _ in range(rows)]
    dp[0][0] = grid[0][0]
    
    # Fill first row
    for j in range(1, cols):
        dp[0][j] = dp[0][j-1] + grid[0][j]
    
    # Fill first column
    for i in range(1, rows):
        dp[i][0] = dp[i-1][0] + grid[i][0]
    
    # Fill rest of the table
    for i in range(1, rows):
        for j in range(1, cols):
            dp[i][j] = grid[i][j] + min(dp[i-1][j], dp[i][j-1])
    
    return dp[rows-1][cols-1]

def unique_paths(m, n):
    """
    Count unique paths from top-left to bottom-right
    """
    dp = [[1] * n for _ in range(m)]
    
    for i in range(1, m):
        for j in range(1, n):
            dp[i][j] = dp[i-1][j] + dp[i][j-1]
    
    return dp[m-1][n-1]

# Essential Matrix Properties

# Diagonal Properties

Main Diagonal: For position (i,j), i - j = constant
- Elements where row - col = 0: (0,0), (1,1), (2,2)…
Anti-Diagonal: For position (i,j), i + j = constant
- Elements where row + col = n-1: (0,n-1), (1,n-2)…

# Coordinate System

# Standard matrix indexing: matrix[row][col]
# For matrix[i][j]:
# - i represents row (vertical position)
# - j represents column (horizontal position)

# Direction vectors for 4-directional movement
directions = [(0,1), (1,0), (0,-1), (-1,0)]  # right, down, left, up

# Direction vectors for 8-directional movement  
directions_8 = [(0,1), (1,0), (0,-1), (-1,0), (1,1), (1,-1), (-1,1), (-1,-1)]

# Boundary checking
def is_valid(row, col, rows, cols):
    return 0 <= row < rows and 0 <= col < cols

# Problems by Pattern

# Pattern-Based Problem Classification

# Pattern 1: Matrix Traversal Problems

Problem	LC #	Key Technique	Difficulty	Template Used
Spiral Matrix	54	Boundary tracking with 4 directions	Medium	Traversal Template
Spiral Matrix II	59	Fill matrix in spiral order	Medium	Traversal Template
Diagonal Traverse	498	Direction alternation with boundary handling	Medium	Traversal Template
Walking Robot Simulation	885	Direction vectors + obstacle checking	Easy	Traversal Template
Spiral Matrix III	885	Expanding spiral with bounds checking	Medium	Traversal Template
Matrix Cells in Distance Order	1030	Manhattan distance sorting	Easy	Traversal Template
Shift 2D Grid	1260	Circular array shifting in 2D	Easy	Traversal Template

# Pattern 2: Matrix Transformation Problems

Problem	LC #	Key Technique	Difficulty	Template Used
Rotate Image	48	Transpose + reverse rows	Medium	Transformation Template
Transpose Matrix	867	Swap rows and columns	Easy	Transformation Template
Flip Image	832	Horizontal flip + bit inversion	Easy	Transformation Template
Flipping an Image	832	Row reversal with bit operations	Easy	Transformation Template
Rotate Array	189	Array rotation using reversal	Medium	Transformation Template
Determine Whether Matrix Can Be Obtained by Rotation	1886	Multiple 90° rotations	Easy	Transformation Template

# Pattern 3: Matrix Search Problems

Problem	LC #	Key Technique	Difficulty	Template Used
Search a 2D Matrix	74	Binary search on flattened matrix	Medium	Search Template (Binary)
Search a 2D Matrix II	240	Elimination from top-right corner	Medium	Search Template (Elimination)
Kth Smallest Element in Sorted Matrix	378	Binary search on value range	Medium	Search Template (Binary)
Find K Pairs with Smallest Sums	373	Priority queue with matrix properties	Medium	Search Template
Shortest Distance from All Buildings	317	BFS from each building	Hard	Search Template
Count Negative Numbers in Sorted Matrix	1351	Binary search or elimination	Easy	Search Template
Find a Peak Element II	1901	Binary search on 2D peak	Medium	Search Template
Median in a Row-Wise Sorted Matrix	-	Binary search on median value	Medium	Search Template

# Pattern 4: Matrix Modification Problems

Problem	LC #	Key Technique	Difficulty	Template Used
Set Matrix Zeroes	73	In-place marking using first row/column	Medium	Modification Template
Image Smoother	661	8-directional averaging	Easy	Modification Template
Game of Life	289	In-place state transitions	Medium	Modification Template
Range Sum Query 2D - Mutable	308	Segment tree or binary indexed tree	Hard	Modification Template
Bomb Enemy	361	DP with obstacle handling	Medium	Modification Template
Shortest Distance from All Buildings	317	BFS with distance accumulation	Hard	Modification Template
Max Area of Island	695	DFS with visited marking	Medium	Modification Template
Number of Islands	200	DFS/BFS with grid modification	Medium	Modification Template

# Pattern 5: Matrix Multiplication & Operations

Problem	LC #	Key Technique	Difficulty	Template Used
Sparse Matrix Multiplication	311	Skip zeros optimization	Medium	Multiplication Template
Design Tic-Tac-Toe	348	Row/column/diagonal sum tracking	Medium	Operations Template
Matrix Diagonal Sum	1572	Main + anti-diagonal sum	Easy	Operations Template
Count Negative Numbers in Sorted Matrix	1351	Efficient counting in sorted matrix	Easy	Operations Template
Lucky Numbers in a Matrix	1380	Row min + column max	Easy	Operations Template
Maximum Side Length of Square	1292	2D prefix sum + binary search	Medium	Operations Template
Range Sum Query 2D Immutable	304	2D prefix sum	Medium	Operations Template
Minimum Falling Path Sum	931	DP with adjacent cell transitions	Medium	Operations Template

# Pattern 6: Matrix Path & Dynamic Programming

Problem	LC #	Key Technique	Difficulty	Template Used
Unique Paths	62	DP counting paths	Medium	Path DP Template
Unique Paths II	63	DP with obstacles	Medium	Path DP Template
Minimum Path Sum	64	DP with cost optimization	Medium	Path DP Template
Triangle	120	DP on triangular matrix	Medium	Path DP Template
Minimum Falling Path Sum	931	DP with adjacent constraints	Medium	Path DP Template
Cherry Pickup	741	3D DP for round trip	Hard	Path DP Template
Dungeon Game	174	Reverse DP from destination	Hard	Path DP Template
Minimum Path Sum	64	Basic grid DP	Medium	Path DP Template
Maximum Path Sum	124	Tree DP adapted to grid	Hard	Path DP Template
Path with Maximum Gold	1219	DFS with backtracking	Medium	Path DP Template

# Additional Matrix Problems by Difficulty

# Easy Problems (Foundation)

Problem	LC #	Pattern	Key Learning
Reshape the Matrix	566	Transformation	1D to 2D conversion
Toeplitz Matrix	766	Pattern Recognition	Diagonal property checking
Available Captures for Rook	999	Traversal	Direction-based movement
Find Winner on a Tic Tac Toe Game	1275	Operations	Game state evaluation
Cells with Odd Values in Matrix	1252	Modification	Index-based updates
Matrix Block Sum	1314	Operations	2D range sum
Sum of All Odd Length Subarrays	1588	Operations	Subarray contribution

# Medium Problems (Core Skills)

Problem	LC #	Pattern	Key Learning
Valid Sudoku	36	Validation	Set operations for uniqueness
Word Search	79	DFS/Backtracking	Path exploration with backtracking
Surrounded Regions	130	DFS/BFS	Boundary-based region identification
Rotate Array	189	Transformation	Multiple rotation techniques
Maximal Square	221	DP	2D DP for shape optimization
Longest Increasing Path in Matrix	329	DFS + Memoization	DAG longest path
Island Perimeter	463	Traversal	Boundary counting
Pacific Atlantic Water Flow	417	DFS	Multi-source reachability

# Hard Problems (Advanced Techniques)

Problem	LC #	Pattern	Key Learning
Sudoku Solver	37	Backtracking	Constraint satisfaction
N-Queens	51	Backtracking	Complex constraint checking
The Maze III	499	Dijkstra/BFS	Shortest path with direction
Robot Room Cleaner	489	DFS	Unknown grid exploration
Minimum Number of Taps	1326	Greedy/DP	Interval covering optimization
Cherry Pickup II	1463	3D DP	Multi-agent path optimization
Largest Rectangle in Histogram	84	Stack	Histogram-based optimization

# Pattern Selection Strategy

Matrix Problem Analysis Flowchart:

1. Is the problem about traversing matrix in a specific order?
   ├── YES → Use Matrix Traversal Template
   │   ├── Spiral order? → Boundary tracking approach
   │   ├── Diagonal order? → Direction alternation approach
   │   └── Custom order? → Direction vectors approach
   └── NO → Continue to 2

2. Does the problem require matrix transformation (rotate, flip, transpose)?
   ├── YES → Use Matrix Transformation Template
   │   ├── Rotate 90°? → Transpose + reverse rows
   │   ├── General rotation? → Mathematical coordinate mapping
   │   └── Transpose? → Swap (i,j) with (j,i)
   └── NO → Continue to 3

3. Is it a search problem in a sorted/partially sorted matrix?
   ├── YES → Use Matrix Search Template
   │   ├── Fully sorted (row-wise + col-wise)? → Binary search as 1D array
   │   ├── Row-wise and column-wise sorted? → Elimination approach
   │   └── Partially sorted? → Modified binary search
   └── NO → Continue to 4

4. Does the problem modify matrix elements based on conditions?
   ├── YES → Use Matrix Modification Template
   │   ├── Set zeros? → Use first row/column as markers
   │   ├── Smooth/average? → 8-directional neighbor processing
   │   └── State transitions? → Two-pass or auxiliary space
   └── NO → Continue to 5

5. Is it a mathematical operation between matrices?
   ├── YES → Use Matrix Multiplication Template
   │   ├── Standard multiplication? → Triple nested loop
   │   ├── Sparse matrices? → Skip-zero optimization
   │   └── Special operations? → Customize based on operation
   └── NO → Continue to 6

6. Is it about finding paths or optimizing values through the matrix?
   ├── YES → Use Matrix Path DP Template
   │   ├── Count paths? → DP with path counting
   │   ├── Min/Max path cost? → DP with optimization
   │   └── Complex constraints? → DFS + memoization
   └── NO → Use Universal Matrix Template

# Implementation Decision Tree

# Step 1: Problem Classification

Read the problem carefully and identify key requirements
Determine the input constraints (matrix size, value ranges)
Identify the expected output (modified matrix, single value, list)
Look for keywords: traverse, rotate, search, modify, multiply, path

# Step 2: Pattern Recognition

Traversal indicators: “spiral”, “diagonal”, “clockwise”, “order”
Transformation indicators: “rotate”, “transpose”, “flip”, “mirror”
Search indicators: “find”, “search”, “locate”, “sorted matrix”
Modification indicators: “set”, “update”, “smooth”, “change”
Math operation indicators: “multiply”, “sum”, “product”, “diagonal”
Path/DP indicators: “path”, “minimum”, “maximum”, “count”, “ways”

# Step 3: Template Selection

Choose the most specific template that matches the pattern
Adapt the template to problem-specific requirements
Consider edge cases: empty matrix, single element, rectangular vs square
Optimize for constraints: in-place vs extra space, time complexity

# Code Examples & Implementations

# Advanced Example 1: Set Matrix Zeroes (LC 73)

def setZeroes(matrix):
    """
    Set entire row and column to zero if element is zero
    Time: O(m*n), Space: O(1)
    """
    if not matrix or not matrix[0]:
        return
    
    rows, cols = len(matrix), len(matrix[0])
    
    # Check if first row and column need to be zero
    first_row_zero = any(matrix[0][j] == 0 for j in range(cols))
    first_col_zero = any(matrix[i][0] == 0 for i in range(rows))
    
    # Use first row and column as markers
    for i in range(1, rows):
        for j in range(1, cols):
            if matrix[i][j] == 0:
                matrix[0][j] = 0
                matrix[i][0] = 0
    
    # Set zeros based on markers
    for i in range(1, rows):
        for j in range(1, cols):
            if matrix[0][j] == 0 or matrix[i][0] == 0:
                matrix[i][j] = 0
    
    # Handle first row and column
    if first_row_zero:
        for j in range(cols):
            matrix[0][j] = 0
    if first_col_zero:
        for i in range(rows):
            matrix[i][0] = 0

# Advanced Example 2: Search 2D Matrix II (LC 240)

def searchMatrix(matrix, target):
    """
    Search in row-wise and column-wise sorted matrix
    Time: O(m+n), Space: O(1)
    """
    if not matrix or not matrix[0]:
        return False
    
    rows, cols = len(matrix), len(matrix[0])
    row, col = 0, cols - 1  # Start from top-right
    
    while row < rows and col >= 0:
        current = matrix[row][col]
        if current == target:
            return True
        elif current > target:
            col -= 1  # Eliminate current column
        else:
            row += 1  # Eliminate current row
    
    return False

# Advanced Example 3: Image Smoother (LC 661)

def imageSmoother(M):
    """
    Smooth image by averaging 8-connected neighbors
    Time: O(m*n), Space: O(m*n)
    """
    if not M or not M[0]:
        return []
    
    rows, cols = len(M), len(M[0])
    result = [[0] * cols for _ in range(rows)]
    
    # 8-directional + current cell
    directions = [(di, dj) for di in [-1, 0, 1] for dj in [-1, 0, 1]]
    
    for i in range(rows):
        for j in range(cols):
            total = 0
            count = 0
            
            for di, dj in directions:
                ni, nj = i + di, j + dj
                if 0 <= ni < rows and 0 <= nj < cols:
                    total += M[ni][nj]
                    count += 1
            
            result[i][j] = total // count
    
    return result

# Summary & Quick Reference

# Complexity Quick Reference

Operation	Time Complexity	Space Complexity	Notes
Matrix Access	O(1)	O(1)	Direct indexing
Full Traversal	O(m*n)	O(1)	Visit every element
Spiral Traversal	O(m*n)	O(1)	Boundary tracking
Binary Search (Sorted)	O(log(m*n))	O(1)	Treat as 1D array
Elimination Search	O(m+n)	O(1)	Start from corner
Matrix Rotation	O(m*n)	O(1)	Transpose + reverse
Matrix Multiplication	O(mnp)	O(m*p)	Standard algorithm
Sparse Multiplication	O(mnk)	O(m*p)	k = average non-zeros per row
DP Path Problems	O(m*n)	O(m*n) or O(n)	Can optimize space

# Template Quick Reference

Template	Pattern	Key Code Structure
Universal	General processing	`for i in range(rows): for j in range(cols):`
Traversal	Spiral/Diagonal	Boundary tracking with direction vectors
Transformation	Rotate/Transpose	Mathematical coordinate mapping
Search	Find elements	Binary search or elimination approach
Modification	Update elements	Two-pass or auxiliary space techniques
Multiplication	Math operations	Triple nested loop with optimizations
Path DP	Optimization	DP state transitions between adjacent cells

# Common Patterns & Tricks

# Boundary Tracking (Spiral)

top, bottom = 0, rows - 1
left, right = 0, cols - 1
while top <= bottom and left <= right:
    # Process boundaries and update them

# Direction Vectors

# 4-directional movement
directions = [(0,1), (1,0), (0,-1), (-1,0)]
# 8-directional movement  
directions = [(di,dj) for di in [-1,0,1] for dj in [-1,0,1]]

# In-Place Modifications

# Use first row/column as markers
first_row_zero = any(matrix[0][j] == 0 for j in range(cols))
first_col_zero = any(matrix[i][0] == 0 for i in range(rows))

# Coordinate Transformation

# 90° clockwise rotation: (i,j) → (j, n-1-i)
# Transpose: (i,j) → (j,i)
# Flip horizontal: (i,j) → (i, n-1-j)

# Matrix to 1D Index Conversion

# Matrix[row][col] → index = row * cols + col
# Index → row = index // cols, col = index % cols

# Problem-Solving Steps

Understand the Problem
- Identify input format (matrix dimensions, constraints)
- Determine output requirements (modified matrix, values, coordinates)
- Look for special properties (sorted, sparse, square vs rectangular)
Choose the Right Pattern
- Use the decision flowchart to identify the pattern
- Select the most specific template that matches
- Consider time/space complexity requirements
Handle Edge Cases
- Empty matrix: if not matrix or not matrix[0]:
- Single element: special handling for 1x1 matrices
- Rectangular matrices: different row and column counts
Optimize for Constraints
- In-place vs extra space based on requirements
- Choose appropriate algorithm based on matrix size
- Consider sparse matrix optimizations when applicable

# Common Mistakes & Tips

🚫 Common Mistakes:

Index confusion: Mixing up matrix[row][col] vs matrix[col][row]
Boundary errors: Off-by-one errors in range calculations
Direction mistakes: Incorrect direction vector calculations
In-place modification: Modifying matrix while reading (use markers)
Edge case neglect: Not handling empty or single-element matrices
Coordinate transformation errors: Incorrect rotation/transpose formulas

✅ Best Practices:

Always check bounds: Use 0 <= i < rows and 0 <= j < cols
Use meaningful names: rows, cols instead of m, n
Handle edge cases first: Check for empty/invalid input
Draw examples: Visualize transformations on small matrices
Use direction vectors: More readable than hardcoded movements
Consider space optimization: In-place modifications when possible
Test with different shapes: Square, rectangular, single row/column

# Interview Tips

Clarify Matrix Properties
- Ask about matrix dimensions and constraints
- Confirm if modification in-place is allowed
- Check if matrix is guaranteed to be non-empty
Start with Brute Force
- Implement the straightforward O(m*n) solution first
- Then optimize based on specific problem constraints
- Explain your approach clearly before coding
Trace Through Examples
- Use small matrices (2x2, 3x3) to verify logic
- Walk through boundary conditions step-by-step
- Check your algorithm with edge cases
Optimize Systematically
- Identify bottlenecks in your initial solution
- Consider mathematical properties for optimizations
- Discuss space-time tradeoffs with the interviewer
Common Interview Patterns
- Matrix traversal: Focus on boundary management
- Matrix transformation: Know rotation/transpose patterns
- Matrix search: Master binary search and elimination approaches
- Matrix DP: Understand state transitions and space optimization

Arrays: Matrix is a 2D extension of array concepts
Dynamic Programming: Many matrix problems use DP patterns
Graph Algorithms: Matrix can represent graphs (adjacency matrix)
Binary Search: Essential for sorted matrix search problems
Backtracking: Used in matrix exploration problems (N-Queens, Sudoku)
String Processing: Matrix problems often involve pattern matching
Greedy Algorithms: Some matrix optimization problems use greedy approach

# Additional Resources

Visualization Tools: Draw.io for matrix transformation visualization
Practice Platforms: LeetCode matrix problems by difficulty
Mathematical Background: Linear algebra for advanced matrix operations
Algorithm Analysis: Big-O notation for matrix algorithm complexity

Matrix

Table of Contents

# Matrix Data Structure

# Overview

# Key Properties

# Problem Categories

# Pattern 1: Matrix Traversal

# Pattern 2: Matrix Transformation

# Pattern 3: Matrix Search

# Pattern 4: Matrix Modification

# Pattern 5: Matrix Multiplication & Operations

# Pattern 6: Matrix Path & Dynamic Programming

# Templates & Algorithms

# Template Comparison Table

# Universal Matrix Template

# Pattern-Specific Templates

# Template 1: Matrix Traversal (Spiral/Diagonal)

# Template 2: Matrix Transformation (Rotate/Transpose)

# Template 3: Matrix Search

# Template 4: Matrix Modification

# Template 5: Matrix Multiplication

# Template 6: Matrix Path DP

# Essential Matrix Properties

# Diagonal Properties

# Coordinate System

# Problems by Pattern

# Pattern-Based Problem Classification

# Pattern 1: Matrix Traversal Problems

# Pattern 2: Matrix Transformation Problems

# Pattern 3: Matrix Search Problems

# Pattern 4: Matrix Modification Problems

# Pattern 5: Matrix Multiplication & Operations

# Pattern 6: Matrix Path & Dynamic Programming

# Additional Matrix Problems by Difficulty

# Easy Problems (Foundation)

# Medium Problems (Core Skills)

# Hard Problems (Advanced Techniques)

# Pattern Selection Strategy

# Implementation Decision Tree

# Step 1: Problem Classification

# Step 2: Pattern Recognition

# Step 3: Template Selection

# Code Examples & Implementations

# Advanced Example 1: Set Matrix Zeroes (LC 73)

# Advanced Example 2: Search 2D Matrix II (LC 240)

# Advanced Example 3: Image Smoother (LC 661)

# Summary & Quick Reference

# Complexity Quick Reference

# Template Quick Reference

# Common Patterns & Tricks

# Boundary Tracking (Spiral)

# Direction Vectors

# In-Place Modifications

# Coordinate Transformation

# Matrix to 1D Index Conversion

# Problem-Solving Steps

# Common Mistakes & Tips

# Interview Tips

# Related Topics

# Additional Resources