Implementing
2D Array Algorithms

Master row-major traversal, nested loops, and common patterns for processing two-dimensional arrays in Java.

AP Computer Science A

What Is a 2D Array?

A 2D array in Java is an array of arrays: each element of the outer array is itself an array. You can think of it as a grid with rows and columns, where every cell holds a single value.

Declaring & Initializing

Java provides two common ways to create a 2D array: specifying dimensions only (filled with defaults), or providing an initializer list with explicit values.

// Method 1: Dimensions only (3 rows, 4 cols of 0s)
int[][] grid = new int[3][4];

// Method 2: Initializer list
int[][] grid = {
    {3, 7, 1, 9},
    {4, 2, 8, 5},
    {6, 0, 3, 7}
};

Why 2D Arrays Matter

Two-dimensional arrays model countless real-world structures. Understanding them is essential for the AP CSA exam and for building real software.

Game Boards

Chess, tic-tac-toe, minesweeper: every cell maps to a grid position.

Images

Pixel data is stored as rows and columns of color values.

Spreadsheets

Tables of data, gradebooks, and matrices are naturally 2D.

The Row-Major Traversal Pattern

Row-major order visits every element by going across each row from left to right, top to bottom. This is the standard traversal on the AP exam.

for (int r = 0; r < arr.length; r++) {
    for (int c = 0; c < arr[r].length; c++) {
        // process arr[r][c]
    }
}

Column-Major Traversal

In column-major order, the outer loop iterates over columns and the inner loop iterates over rows: you visit every element in column 0 first, then column 1, and so on.

for (int c = 0; c < arr[0].length; c++) {
    for (int r = 0; r < arr.length; r++) {
        // process arr[r][c]
    }
}

Row-Major vs Column-Major

	Row-Major	Column-Major
Outer loop	rows (r)	columns (c)
Inner loop	columns (c)	rows (r)
Visit pattern	Left to right, top to bottom	Top to bottom, left to right
AP default?	✓ Yes	Sometimes tested
Use case	Summing all elements, printing	Column averages, transposing

Enhanced For-Each Loop

When you only need to read values (no index needed), the enhanced for loop is cleaner. The outer loop iterates over rows (each an int[]), and the inner loop iterates over values in that row.

for (int[] row : arr) {
    for (int val : row) {
        System.out.print(val + " ");
    }
}

Step-by-Step Algorithm Design

1

Identify what you need: Are you summing, searching, counting, or modifying? This determines your accumulator variable.

2

Choose traversal order: Row-major for general processing; column-major if you need column-wise results.

3

Set loop bounds carefully: Outer uses arr.length (rows), inner uses arr[r].length (cols). Never hard-code sizes.

4

Process each cell: Apply your logic inside the inner loop using arr[r][c].

5

Return or output: After the loops complete, return your accumulated result.

A restorer repairs a luminous pixel mosaic

A restorer lifts a tile from a luminous rectangular mosaic.

Grid algorithms turn a visual pattern into explicit row and column operations. Bounds checks determine which neighboring cells can be accessed.

Check neighboring cells

A 3-by-4 grid marks corner, edge and interior neighbors and shows row-before-column bounds checks.

In this rectangular grid, an interior cell has four orthogonal neighbors, a noncorner edge cell has three, and a corner has two. Validate the target row before accessing that row’s length.

Example 1: Sum All Elements

Write a method that returns the sum of every element in a 2D integer array.

public static int totalSum(int[][] arr) {
    int sum = 0;
    for (int r = 0; r < arr.length; r++) {
        for (int c = 0; c < arr[r].length; c++) {
            sum += arr[r][c];
        }
    }
    return sum;
}

Example 2: Count Matches

Count how many times a target value appears in the 2D array.

public static int countOccurrences(int[][] arr, int target) {
    int count = 0;
    for (int[] row : arr) {
        for (int val : row) {
            if (val == target) {
                count++;
            }
        }
    }
    return count;
}

Example 3: Row Sums Array

Return a 1D array where each element is the sum of the corresponding row.

public static int[] rowSums(int[][] arr) {
    int[] sums = new int[arr.length];
    for (int r = 0; r < arr.length; r++) {
        for (int c = 0; c < arr[r].length; c++) {
            sums[r] += arr[r][c];
        }
    }
    return sums;
}

Example 4: Find the Maximum

Locate the largest value in a 2D array. Initialize the max with the first element, then compare every other cell.

public static int findMax(int[][] arr) {
    int max = arr[0][0];
    for (int r = 0; r < arr.length; r++) {
        for (int c = 0; c < arr[r].length; c++) {
            if (arr[r][c] > max) {
                max = arr[r][c];
            }
        }
    }
    return max;
}

Example 5: Modify In Place

Replace every negative value in the array with 0. This requires the standard indexed loop because we need to write to the array.

public static void replaceNegatives(int[][] arr) {
    for (int r = 0; r < arr.length; r++) {
        for (int c = 0; c < arr[r].length; c++) {
            if (arr[r][c] < 0) {
                arr[r][c] = 0;
            }
        }
    }
}

Common Mistakes

✕ Swapped Indices

Using arr[c][r] instead of arr[r][c] causes an ArrayIndexOutOfBoundsException when rows and columns differ.

✕ Hard-coded Bounds

Writing c < 4 instead of c < arr[r].length breaks when the array size changes.

✕ Modifying with For-Each

Assigning val = 0 in an enhanced for loop changes only the local copy, not the array itself.

Algorithm Summary

Algorithm	Pattern	Loop Type
Sum all elements	Accumulator += each cell	Either
Count matches	if (val == target) count++	Either
Row/column sums	Reset accumulator per row/col	Indexed
Find max/min	Compare and update tracker	Either
Modify in place	arr[r][c] = newValue	Indexed only
Search (find location)	Return r, c when found	Indexed only

Question 1 of 5

Given int[][] m = new int[3][5];, what does m.length return?

m.length returns the number of rows (the outer array's length), which is 3. To get the number of columns, use m[0].length, which returns 5.

Question 2 of 5

int count = 0;
for (int[] row : arr) {
    for (int val : row) {
        if (val % 2 == 0) {
            count++;
        }
    }
}

▼ Tap to read code

What does this code compute?

The condition val % 2 == 0 checks whether a value is even, and count++ increments a counter for each even value found. The code counts how many even elements exist.

Question 3 of 5

You want to replace every negative value in a 2D array with 0. Which loop type is required?

Modifying array elements requires direct access via arr[r][c] = 0. The enhanced for loop only gives you a copy of each value, so assigning to the loop variable does not change the original array.

Question 4 of 5

int result = 0;
for (int c = 0; c < arr[0].length; c++) {
    for (int r = 0; r < arr.length; r++) {
        result += arr[r][c];
    }
}

▼ Tap to read code

This code traverses the array in which order?

The outer loop iterates over columns (c) and the inner loop iterates over rows (r). This means all elements in column 0 are visited first, then column 1, etc.: this is column-major order.

Question 5 of 5

int max = arr[0][0];
for (int r = 0; r < arr.length; r++) {
    for (int c = 0; c < arr[r].length; c++) {
        if (arr[r][c] > max) {
            max = arr[r][c];
        }
    }
}
return max;

▼ Tap to read code

If the array is {{-5, -2}, {-8, -1}}, what does this method return?

Since max is initialized to arr[0][0] which is \( -5 \), the algorithm correctly compares all values. The largest value is \( -1 \) at position [1][1]. This is why initializing to the first element (not 0) is critical for arrays that may contain only negative values.

Implementing2D Array Algorithms

What Is a 2D Array?

Declaring & Initializing

Why 2D Arrays Matter

The Row-Major Traversal Pattern

Column-Major Traversal

Row-Major vs Column-Major

Enhanced For-Each Loop

Step-by-Step Algorithm Design

A restorer repairs a luminous pixel mosaic

Check neighboring cells

Example 1: Sum All Elements

Example 2: Count Matches

Example 3: Row Sums Array

Example 4: Find the Maximum

Example 5: Modify In Place

Common Mistakes

Algorithm Summary

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Implementing
2D Array Algorithms