How do you determine the complexity of a recursive algorithm?

The complexity of a recursive algorithm is determined by analysing its time and space complexity using recurrence relations.

To understand the complexity of a recursive algorithm, we need to delve into the concept of time and space complexity. Time complexity refers to the computational complexity that describes the amount of computational time taken by an algorithm to run, as a function of the size of the input to the program. Space complexity, on the other hand, is a measure of the amount of memory an algorithm needs in relation to the size of the input.

The complexity of a recursive algorithm is often expressed using recurrence relations. A recurrence relation is an equation that recursively defines a sequence; each term of the sequence is defined as a function of the preceding terms. In the context of a recursive algorithm, the recurrence relation often gives the number of operations needed to solve a problem of size n, in terms of the number of operations needed to solve smaller problems.

For example, consider a simple recursive algorithm for calculating the factorial of a number. The time complexity of this algorithm can be expressed with the recurrence relation T(n) = T(n-1) + c, where c is a constant representing the time taken for the non-recursive part of the algorithm. This relation tells us that the time taken to calculate the factorial of n is equal to the time taken to calculate the factorial of n-1, plus some constant time.

To solve this recurrence relation and find the time complexity of the algorithm, we can use a method called 'recursion tree method' or 'substitution method'. For the factorial algorithm, if we expand the recurrence relation, we get T(n) = T(n-1) + c = T(n-2) + 2c = ... = T(1) + (n-1)c. This simplifies to T(n) = nc - (n-1)c, which is O(n), meaning the time complexity of the factorial algorithm is linear.

Similarly, the space complexity of a recursive algorithm can be determined by analysing the maximum depth of the recursion tree, which gives the maximum amount of memory needed by the algorithm at any point in time.

In conclusion, determining the complexity of a recursive algorithm involves analysing its time and space complexity, often using recurrence relations to express the number of operations needed to solve the problem.