Last Updated on November 23, 2023 by Ankit Kochar

Understanding the efficiency and resource utilization of algorithms is crucial in computer science. One fundamental aspect of analyzing algorithms is evaluating their memory usage, which is encapsulated in the concept of space complexity. Space complexity refers to the amount of memory space an algorithm requires to solve a computational problem concerning the input size. It plays a pivotal role in determining the practicality and scalability of algorithms, especially in scenarios where memory constraints are significant. This article delves into the intricacies of space complexity, its importance in algorithm analysis, and how it influences the design and evaluation of efficient algorithms.

## What is Space Complexity?

When we state that our algorithm is sufficient, we are referring to the fact that the algorithm solves the issue quickly and efficiently. As a result, in order to improve an algorithm’s effectiveness, it is crucial to analyze the algorithm after it has been designed.

You have all probably run into time complexity and space complexity while trying to analyze an issue or method.

When analyzing the effectiveness of an algorithm or an issue, space complexity is often overlooked, despite the fact that it is just as essential as time complexity.

Space complexity refers to the amount of memory required by an algorithm to solve a problem. It includes all the memory used by an algorithm, such as the space required for variables, data structures, function calls, and other temporary storage. The space complexity of an algorithm is usually measured in terms of the amount of memory it requires to solve a problem as a function of the input size.

In simple words, space complexity is nothing but the summation of all the memory space that an algorithm takes while executing. The space complexity will include the space occupied by both the variables and the input values with them.

Many people normally confuse space complexity and auxiliary space and consider both of them the same. But let’s make the confusion clear here as auxiliary space includes the total space taken by the algorithm during execution whereas the space complexity also includes the space of input variables with it.

By summarising we can say that space complexity is the summation of space used by input variables and the auxiliary space.

### Why is Space Complexity Important?

When analyzing the efficiency of an algorithm the space complexity of an algorithm plays an important part. The memory or space required by the program sometimes plays an important part in the performance of the algorithm and especially in cases when the memory is limited. Therefore minimizing the space complexity of the algorithm will be as important as reducing the time complexity.

### Calculating Space Complexity

To calculate the space complexity of the algorithm we need to know the memory taken or used by different data types, functions, constant values, etc. Below is the list of various data types corresponding to their memory requirement.

Type | Size |
---|---|

char, bool, unsigned char, _ _ int8, signed char | 1 byte |

short, __int16, wchar_t, unsigned short, __wchat_t | 2 byte |

__int32, long, unsigned int, unsigned long, int | 4 byte |

__int64, long long, long double, double | 8 byte |

### Memory Usage While Evaluation

While executing the program the computer needs more memory than required or consumed by the program. The extra space taken by computers is because of the reasons explained below:

**Instruction Space:** The RAM area used to hold a program’s instructions or code is known as the instruction space. The collection of commands used by the program to carry out its tasks are these directions. The executable code, libraries, and any other tools necessary for the application to execute is usually found in the instruction area. The complexity of the software and the number of instructions needed to complete its duties determine how large the instruction space will be.

**Environmental Space:** As we know, sometimes a function calls itself (recursively) or another function, storing or pushing the data of the previous function onto the stack until further execution is required, at which point the inner function is called. This space is used to store the address of the partially executed functions (a partially executed function is one that calls another function without fully executing itself).

**Data Space:** The memory space a software uses to keep its data is known as data space. Data used by the software includes variables, arrays, structures, and other kinds of data. The quantity of data the software uses, as well as the kinds of data used, determine the size of the data space. For instance, a program that uses complicated data structures or big groups will need more storage space than one that only uses straightforward variables.

### Examples of Space Complexity

In this section, we will discuss the space complexity of various examples.

**Example 1: Sum of all the elements in an array**

In this example, we will find the space complexity of the code where we are finding the sum of all the elements in an array.

**Code Implementation**

```
function sum(arr[],N){
ans=0
for(i = 0 to N){
ans=ans+arr[i]
}
print(ans)
}
```

**Calculating Space Complexity**

In the above example we are finding the sum of all the elements of an array of size N.

- The array is of type integer and each integer contains 4 bytes of memory hence the total memory consumed by the array is N*4
- The ans is an integer type variable and take 4 bytes of memory.
- The i variable will also take 4 bytes and will be used to iterate over the array.
- Now the initialization of for loop, function call, and print function will take the combined memory of 4 bytes(Assumed).
- Hence the total space complexity is (4*N+12) bytes. Hence O(N).

**Example 2: Factorial of a Number**

In this section of the blog, we will find the space complexity of the code of factorial of a number.

**Code Implementation**

```
factorial(N){
int fact=1;
for (int i=1; i<=N; i++)
{
fact*=i;
}
return fact;
}
```

**Calculating the Space Complexity**

In the above example we are finding the space complexity of the code of factorial of a number.

- The fact is int type so will take 4 bytes.
- The N is a number that will also take 4 bytes.
- The i is an iterator of int type which also takes 4 bytes.
- The return statement, for loop, and the function call will take 4 bytes of memory (Assumed).
- Hence the total complexity is 16 bytes. Hence O(1).

**Tips for Reducing Space Complexity**

Here are some tips that can help in reducing space complexity.That will improve the performance.

- We can use in-place algorithms as they do not require any additional space.
- We can also use dynamic programming as it saves space for redundant operations.
- We have to use efficient data structures that can manage the memory efficiently.

**Conclusion**

Space complexity serves as a crucial metric in algorithm analysis, offering insights into an algorithm's memory usage and scalability. Understanding an algorithm's space requirements helps in designing optimized solutions, especially in resource-constrained environments. By considering space complexity alongside time complexity, developers can make informed decisions in choosing the most efficient algorithms for various computational problems, ensuring optimal resource utilization and improved overall performance.

## Frequently Asked Questions Related to Space Complexity

Here is a list of some of the frequently asked questions about space complexity.

**1. What is a hash table?**

A hash table is a data structure that uses a hash function to map keys to values, providing constant-time lookup, insertion, and deletion operations on average.

**2. How is space complexity different from time complexity?**

While time complexity focuses on analyzing the computational time an algorithm requires to solve a problem concerning input size, space complexity deals with the amount of memory space needed by the algorithm.

**3. How is space complexity evaluated?**

Space complexity is typically evaluated in terms of auxiliary space and space used by input. Auxiliary space refers to additional space other than the input space that an algorithm uses, while space used by input refers to the memory space required to store the input itself.

**4. Why is space complexity important?**

Space complexity is essential as it helps in understanding how much memory an algorithm consumes based on the input size. It aids in optimizing memory usage, especially in constrained environments, and contributes to designing efficient algorithms.

**5. What are some common examples of algorithms with different space complexities?**

Algorithms can exhibit various space complexities. For instance, algorithms with constant space complexity (O(1)) use a fixed amount of memory regardless of input size. On the other hand, algorithms with linear space complexity (O(n)) increase memory usage proportionally with the input size.