# Space Complexity in Data Structures

The space complexity helps to determine the efficiency and scalability of a solution, and it is an important factor to consider when choosing a data structure or designing an algorithm.

This article aims to provide insight into the significance of space complexity. We will be covering the following sections:

- Introduction
- Definition
- Why do we need to Calculate Complexity?
- How to Calculate Complexity?
- Table for Common Algorithms

So, without further ado, let’s get started!

**What is Space Complexity in Data Structures?**

Space complexity in data structures refers to the amount of memory used by an algorithm to solve a problem. It measures the amount of memory space required to store the data and structures used by an algorithm. This complexity is important because it determines the scalability of a solution and the ability of a program to handle large amounts of data. The space complexity is often expressed in terms of the size of the input, and it is an important factor to consider when choosing a data structure for a specific problem.** **

**Definition**

Space complexity is a measure of the amount of memory an algorithm uses, expressed in terms of the size of the input. It refers to the amount of memory storage required to execute the algorithm and solve a problem. A low space complexity means that an algorithm requires relatively little memory to solve the problem, while a high space complexity means that it requires a large amount of memory, potentially leading to slow performance or memory limitations.

**Why do we need to Calculate Space Complexity?**

To ensure the efficiency and effectiveness of an algorithm or problem, it is necessary to calculate its space complexity. Despite the fact that modern systems usually have ample memory, analyzing the space complexity is still important to optimize the algorithm and make it run with a minimal amount of memory. This is particularly important for real-world applications, where developers must consider the memory limitations of the systems they are using and avoid processes that use more memory than is available. By considering the space complexity, developers can make informed decisions about which data structures and algorithms to use, and can ensure that their applications run smoothly and efficiently.

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**How to Calculate Space Complexity?**

Evaluating the space complexity of an algorithm involves determining the amount of memory used by various elements such as variables of different data types, program instructions, constant values, and in some cases, function calls and the recursion stack. The exact amount of memory used by different data types may vary depending on the operating system, but the method of calculating the space complexity remains constant. To determine the space-complexity, it is important to consider all of these factors and to add up the memory used by each element to get an overall measure of the algorithm’s memory usage. For example, here is a table summarizing the memory space taken by various data types in the C programming language:

Data Type |
Memory Space (in bytes) |

int | 4 |

float | 4 |

double | 8 |

char | 1 |

short int | 2 |

long int | 4 |

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**Note:** The above table is based on common memory configurations and may vary depending on the specific implementation and architecture of the system being used.

Consider the following example:

**Example 1:**

int main(){ int a = 10; float b = 20.5; char c = 'A'; int d[10]; return 0;}

To calculate the complexity of this algorithm, we need to determine the amount of memory used by each of the variables. In this case:

- a is an integer, which takes up 4 bytes of memory.
- b is a float, which takes up 4 bytes of memory.
- c is a character, which takes up 1 byte of memory.
- d is an array of 10 integers, which takes up 40 bytes of memory (10 x 4).

So, the total amount of memory used by this algorithm is 4 + 4 + 1 + 40 = 49 bytes.

These are just simple examples, and in a real-world scenarios, this complexity can be more complex and involve more variables, data structures, and functions. However, the process of calculating the space-complexity remains the same: we need to add up the memory used by each element to get an overall measure of the algorithm’s memory usage.

Let’s consider one more example:

**Example 2:**

int factorial(int n){ if (n == 0) return 1; else return n * factorial(n-1);}

To calculate the complexity of this algorithm, we need to determine the amount of memory used by the variables and functions. In this case:

**n**is an integer input parameter, which takes up 4 bytes of memory.- The function call
**factorial**takes up some memory for the function call stack, which is implementation-dependent.

In this case, the function **factorial** is recursive, so it makes multiple function calls and uses memory on the function call stack. The complexity of this algorithm is proportional to the number of function calls, which is directly proportional to the value of **n**. The more calls, the more memory will be used on the function call stack.

In the worst-case scenario, where **n** is very large, this algorithm can use a significant amount of memory on the function call stack, leading to a high space-complexity.

**Space Complexity Table for Common Algorithms**

Algorithm |
Space Complexity |

Linear Search | O(1) |

Binary Search | O(1) |

Bubble Sort | O(1) |

Insertion Sort | O(1) |

Selection Sort | O(1) |

Quick Sort | O(log n) |

Merge Sort | O(n) |

Depth First Search (DFS) | O(n) |

Breadth First Search (BFS) | O(n) |

Dynamic Programming | O(n^2) or O(n*m) |

Greedy Algorithm | O(n) |

Backtracking | O(2^n) |

Please note that the space-complexity of some algorithms may vary based on their implementation and the data structure used.

**Endnotes**

In conclusion, it’s crucial to keep the space usage as low as possible when writing a program or algorithm to minimize its space-complexity. By analyzing the worst-case scenario, you can ensure that the program can handle large inputs, increase its adaptability, and maintain stability. This will help keep the resources created using the algorithm in good condition, avoiding any potential heating issues. Explore more such articles to learn more about the data structures and consolidate your knowledge of the fundamentals.

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