Algorithm Space Complexity

Space complexity refers to the amount of memory space required by an algorithm, including the space of input values, for its execution. Just like time complexity, we use Big O notation to express space complexity.

Here's how space complexity relates to time complexity:

Trade-off

Often, there is a trade-off between time and space complexity. We can often reduce time complexity by using more memory, and vice versa. For example:

1int fibonacci(int n) {
2  if (n <= 1) return n;
3  return fibonacci(n-1) + fibonacci(n-2);
4}

This recursive Fibonacci function has $O(2^n)$ time complexity but $O(n)$ space complexity due to the recursive call stack.

We can optimize time complexity to $O(n)$ by using an array to store results, but this increases space complexity to $O(n)$:

1int fibonacci(int n) {
2  if (n <= 1) return n;
3  int fib[n+1];
4  fib[0] = 0; fib[1] = 1;
5  for (int i = 2; i <= n; i++) {
6    fib[i] = fib[i-1] + fib[i-2];
7  }
8  return fib[n];
9}

Auxiliary Space

Space complexity includes both the space of input values and the auxiliary space used by the algorithm. Auxiliary space refers to the extra space or temporary space used by an algorithm, not including the space used for input values.

Importance

In many cases, time complexity is more critical than space complexity because memory is often less costly than computational time. However, for systems with limited memory or very large datasets, space complexity can be a major constraint.

Big O Notation

Just like time complexity, we drop constants and lower order terms for space complexity. For example, an algorithm that uses a single integer and an array of size n would be considered to have $O(n)$ space complexity.

When designing algorithms, it's important to consider both time and space complexity and choose the appropriate trade-off for your specific problem and constraints.