How does memoization improve the performance of recursive algorithms?

Memoisation improves the performance of recursive algorithms by storing the results of expensive function calls and reusing them when needed.

In more detail, memoisation is a technique used in computer programming to optimise the execution time of a function by storing the results of expensive function calls and reusing them when the same inputs occur again. This is particularly useful in recursive algorithms, where the same function is called multiple times with the same arguments.

Recursive algorithms often involve repeated computation of the same subproblems. For example, in the calculation of Fibonacci numbers, the same Fibonacci number may be calculated multiple times in the process of calculating a larger Fibonacci number. This repeated computation can lead to an exponential increase in the time complexity of the algorithm.

Memoisation addresses this issue by storing the results of these repeated computations in a data structure, such as an array or a hash table. When the function is called with the same arguments, the algorithm first checks if the result is already stored in the data structure. If it is, the stored result is returned immediately, avoiding the need for repeated computation. If the result is not stored, the function is executed, and the result is stored in the data structure for future use.

This technique can significantly reduce the time complexity of recursive algorithms. For example, the time complexity of a naive recursive algorithm for calculating Fibonacci numbers is exponential, but with memoisation, it can be reduced to linear.

However, memoisation does come with a trade-off. While it can significantly improve the time efficiency of an algorithm, it does so at the expense of space efficiency. The storage of results in a data structure requires additional memory, which can be a concern in applications where memory is limited.

In conclusion, memoisation is a powerful technique for optimising recursive algorithms. It can significantly improve the time efficiency of these algorithms by avoiding repeated computation, but it does so at the expense of space efficiency.