Algorithm Analysis and Big O Notation

# Algorithm Space Complexity

## What is space complexity and how does it relate to time 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 spaceÂ complexity.

Here's how space complexity relates to timeÂ complexity:

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

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

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

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

int fibonacci(int n) {
if (n <= 1) return n;
int fib[n+1];
fib[0] = 0; fib[1] = 1;
for (int i = 2; i <= n; i++) {
fib[i] = fib[i-1] + fib[i-2];
}
return fib[n];
}

### 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 inputÂ 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 majorÂ 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 $O(n)$ spaceÂ complexity.

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

This Question is from the Lesson:

### Algorithm Analysis and Big O Notation

An introduction to algorithms - the foundations of computer science. Learn how to design, analyze, and compare them.

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This Question is from the Lesson:

### Algorithm Analysis and Big O Notation

An introduction to algorithms - the foundations of computer science. Learn how to design, analyze, and compare them.

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