Time complexity and space complexity are measures used in algorithm analysis to evaluate the efficiency of an algorithm.

They provide insights into how the algorithm’s performance scales with increasing input size.

Let’s explore each of these concepts:

## 1. Time Complexity:

- Time complexity measures the amount of time an algorithm takes to run as a function of the input size.
- It provides an estimation of the worst-case, best-case, or average-case running time of an algorithm.
- Time complexity is commonly expressed using Big O notation, which gives an upper bound on the growth rate of the algorithm’s running time.
- It helps in understanding how the algorithm’s performance will change as the input size increases.
- Common time complexity classes include O(1) (constant time), O(log n) (logarithmic time), O(n) (linear time), O(n log n) (linearithmic time), O(n^2) (quadratic time), and O(2^n) (exponential time), among others.
- The lower the time complexity, the more efficient the algorithm.

## 2. Space Complexity:

- Space complexity measures the amount of memory an algorithm requires as a function of the input size.
- It provides an estimation of the worst-case, best-case, or average-case memory usage of an algorithm.
- Space complexity is commonly expressed using Big O notation, which gives an upper bound on the growth rate of the algorithm’s memory usage.
- It helps in understanding how the algorithm’s memory requirements will change as the input size increases.
- Common space complexity classes include O(1) (constant space), O(log n) (logarithmic space), O(n) (linear space), O(n log n) (linearithmic space), O(n^2) (quadratic space), and O(2^n) (exponential space), among others.
- The lower the space complexity, the more memory-efficient the algorithm.

## Difference between time complexity and space complexity:

Time Complexity | Space Complexity | |

Definition | Measures the amount of time an algorithm takes to run. | Measures the amount of memory an algorithm requires to execute. |

Focus | Emphasizes the running time or execution speed of an algorithm. | Emphasizes the memory usage or storage requirements of an algorithm. |

Evaluation | Expressed as a function of the input size (n). | Expressed as a function of the input size (n). |

Analysis | Provides insights into the algorithm’s efficiency and scalability with respect to time. | Provides insights into the algorithm’s efficiency and scalability with respect to memory. |

Notation | Typically expressed using Big O notation. | Typically expressed using Big O notation. |

Relationship | Time complexity and space complexity can have different values and behaviors independently of each other. | Time complexity and space complexity can have different values and behaviors independently of each other. |

Importance | A lower time complexity indicates a more efficient algorithm in terms of execution speed. | A lower space complexity indicates a more memory-efficient algorithm. |

Trade-Offs | Optimizing for better time complexity might lead to increased space complexity and vice versa. | Optimizing for better space complexity might impact the time complexity and vice versa. |

Analysis Factors | Dominant operations, loops, recursion, and iterations in the algorithm. | Memory allocation, data structures, and auxiliary space used by the algorithm. |