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Analyzing Space Complexity of Algorithms

In this lesson, we’ll explore how to analyze the space requirements of algorithms by looking at two examples. We’ll start with an algorithm that uses constant space and then move to one that requires space. Through these examples, we’ll learn how to calculate space complexity by examining variables, data structures, and other memory requirements.

Example 1: Finding the Maximum Element (Constant Space – )

Let’s begin with a simple algorithm to find the maximum element in a list. This algorithm only needs a small, fixed amount of memory, regardless of the input size.

Algorithm Code

java

class Solution {

  public int find_max(int[] arr) {
    int max_val = arr[0]; // Assume first element is max

    for (int num : arr) {
      if (num > max_val) {
        max_val = num; // Update max if a larger value is found
      }
    }
    return max_val; // Return the maximum value found
  }

  public static void main(String[] args) {
    int[] arr = { 12, 5, 8, 20, 3 }; // Example array

    Solution solution = new Solution();
    int maxValue = solution.find_max(arr);

    System.out.println("Maximum value: " + maxValue);
  }
}

Step-by-Step Space Complexity Analysis

Input Storage:
- The input array arr has elements, where is the length of the list. However, input space is not counted in auxiliary space complexity, so we focus on the additional memory used by the algorithm.
Auxiliary Space:
- Variable max_val: We use this variable to store the current maximum value. No matter how large n is, this variable only takes up one unit of memory.
- Loop Variable num: This variable stores each element in arr as we iterate through it. Again, this requires only one unit of memory, as num is overwritten in each iteration.
Total Space Complexity:
- Since we’re only storing a fixed number of variables (max_val and num), the memory required does not increase with n. Therefore, the space complexity is .

Example 2: Matrix Multiplication (Quadratic Space – )

Now, let’s look at an algorithm for multiplying two matrices. Here, we’ll require memory that grows with the size of the input, leading to a quadratic space complexity.

Algorithm Code

java

class Solution {

  public int[][] matrix_multiply(int[][] A, int[][] B) {
    int n = A.length;
    int[][] result = new int[n][n];

    // Perform matrix multiplication
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < n; j++) {
        for (int k = 0; k < n; k++) {
          result[i][j] += A[i][k] * B[k][j];
        }
      }
    }
    return result;
  }

  public static void main(String[] args) {
    int[][] A = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };

    int[][] B = { { 9, 8, 7 }, { 6, 5, 4 }, { 3, 2, 1 } };

    Solution solution = new Solution();
    int[][] result = solution.matrix_multiply(A, B);

    // Print the result matrix
    System.out.println("Resultant Matrix:");
    for (int[] row : result) {
      for (int val : row) {
        System.out.print(val + " ");
      }
      System.out.println();
    }
  }
}

Step-by-Step Space Complexity Analysis

Input Storage:
- The matrices A and B are both matrices, so each one requires memory units. But as with the first example, we focus on auxiliary space and ignore the memory used to store inputs.
Auxiliary Space:
- Output Matrix result: The result of multiplying two matrices is another matrix. We need to create this matrix to store the final product. Since result has elements, it takes up memory.
- Loop Variables i, j, and k: These variables are used for iteration and only occupy a fixed amount of memory, which is .
Total Space Complexity:
- The dominant space usage here is the result matrix, which requires memory units. Therefore, the space complexity is .

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