# Finding Maximum Disk Space: Time and Space Complexity Analysis

Time and Space complexity are essential parameters of any algorithm. It teaches us to measure the performance of algorithms and helps us choose the most efficient approach for solving any problem. Here we will learn about the maximum disk space in Python.

Maximum Disk Space refers to the largest continuous block of free space available on a computer’s disk. Finding the maximum disk space involves analyzing disk partitions and identifying the combination that yields the largest contiguous free space within a target size. Two main approaches are Brute Force and Dynamic Programming, with the latter being more efficient in time complexity but requiring additional space for memoization.

Also Read: How to Work with Python TimeDelta?

## What is Maximum Disk Space?

Before going into complexities, we will first understand Maximum Disk Space. Maximum Disk Space is the largest contiguous block of unallocated or free space across the disk partitions on your computer’s storage drives. You must have heard this word while considering file allocation or enhancing the system’s performance. Thus, we will look at different approaches to finding Maximum Disk Space and analyse which suits the best.

## Time Complexity Analysis

• Brute Force Approach for Maximum Disk Space

We will go with the most common approach. To find maximum disk space, you can iterate through each disk block and keep track of free blocks (longest and consecutive). We can write a logic to find the maximum disk space in Python as follows:

```from itertools import combinations

def max_disk_space_bf(partitions, target_space):
max_space = 0
for r in range(1, len(partitions) + 1):
for combo in combinations(partitions, r):
if sum(combo) <= target_space:
max_space = max(max_space, sum(combo))
return max_space

partitions = [10, 20, 5, 30, 15]
target_space = 25
print("Maximized disk space (Brute Force):", max_disk_space_brute_force(partitions, target_space))
```

Here we have a function that takes two parameters: ‘partitions’ and ‘target_space.’ The partitions refer to the list containing the size of disk partitions, and target space is used to track the desired amount of space.

Inside the function, ‘max_space’ is initialized to zero. This will store the maximum free space found so far. The function has two loops that generate all kinds of combinations of partitions.

With each combination, we check if the sum of the partition sizes is less than or equal to the target size. If this is true, the max space has been updated. Thus, the maximum disk space is returned after going through all the combinations.

The time complexity of the brute force algorithm is O(2^n), where n is the number of disk partitions. This exponential time complexity arises because the algorithm iterates through all possible combinations of partitions, which grows exponentially as the number of partitions increases. As we spend time in iterating each block, the time complexity becomes exponential.

This approach is not suitable because the disk space is enormous. Hence, we will consider the second approach in terms of time complexity.

• Dynamic Programming Approach for Maximum Disk Space

The dynamic programming approach breaks down the problem into subproblems and stores the result in a memorization table. Let’s understand from the code:

```def max_disk_space_dp(partitions, target_space):
n = len(partitions)
dp = [[0] * (target_space + 1) for _ in range(n + 1)]

for i in range(1, n + 1):
for j in range(1, target_space + 1):
if partitions[i - 1] <= j:
dp[i][j] = max(dp[i - 1][j], dp[i - 1][j - partitions[i - 1]] + partitions[i - 1])
else:
dp[i][j] = dp[i - 1][j]

return dp[n][target_space]

partitions = [10, 20, 5, 30, 15]
target_space = 25
print("Maximized disk space (Dynamic Programming):", max_disk_space_dp(partitions, target_space))

```

The dynamic programming function `max_disk_space_dp` takes two inputs: `partitions` (a list of disk partition sizes) and `target_space` (the desired amount of free space).

First, it creates a 2D table `dp` with dimensions `(n+1) x (target_space+1)`, where `n` is the number of partitions. Each cell in the table is initially set to zero.

The function then iterates through each partition (`i`) and each possible target space value (`j`). For each combination of `i` and `j`, it checks if the current partition size `partitions[i-1]` is less than or equal to `j`.

If the current partition fits within the target space `j`, the function calculates the maximum disk space that can be obtained by either including or excluding the current partition. This is done by taking the maximum value between:

1. `dp[i-1][j]` (excluding the current partition), and
2. `dp[i-1][j-partitions[i-1]] + partitions[i-1]` (including the current partition).

The maximum value is stored in `dp[i][j]`.

However, if the current partition size exceeds the target space `j`, the function simply carries forward the maximum disk space from the previous row (`dp[i-1][j]`) since the current partition cannot be included for that target space.

This process continues until the entire table is filled. The final answer, which is the maximum disk space for the given `target_space`, is stored in the last cell `dp[n][target_space]`.

This approach’s time complexity is O(NM), where N is the number of partitions and M is the target space. The algorithm fills a 2D table with NM cells, making it more efficient than the brute force approach for large disk spaces, albeit with increased space usage for memoization.

## Space Complexity Analysis

To find the maximum disk space algorithm’s complexity, we will also include the space taken by the algorithm as a function of the input size.

In the Brute Force Approach, we have taken only a few parameters to track current and maximum space. Thus the space complexity is O(1)., indicating constant space usage.

In Dynamic Programming, we have taken a size table (N+1)(M+1) to store the data. Thus the space complexity for this approach is O(N*M).

### Which is the most suitable method to find maximum disk space?

The most suitable method depends on the user’s storage space and applicability. Every approach has its own pros and cons; thus, one should find the one suitable for the space and time constraints.

### Which approach takes the least time to find maximum disk space?

The dynamic programming approach takes the least time to find maximum disk space. However, as it requires the matrix to store the solutions of subproblems, it is not good in scenarios with limited space.

## Summary

Thus depending on your preferences and constraints, we should choose the algorithm for maximum disk space. As the Brute Force Approach takes the least space, it consumes the exponential time. Thus, if the disk space is larger, it will take time to find the result. Similarly, the dynamic programming approach takes the least time but takes a matrix space to store the solution to subproblems.

The dynamic programming approach is more efficient than the brute force approach for finding the maximum disk space, as it reduces the exponential time complexity to a polynomial time complexity, albeit at the cost of increased space usage for memoization.

## References

https://discuss.python.org/t/disk-space-used-by-a-file/45205

https://wiki.python.org/moin/TimeComplexity