The substitution method is a technique used in the analysis of recursive algorithms to determine their time complexity. It involves making an educated guess about the time complexity and then using mathematical induction to prove the correctness of the guess.

Here's a step-by-step explanation of the substitution method:

1. Guess the time complexity: Make an educated guess about the time complexity of the algorithm. This guess could be based on observations from the algorithm's structure or similarity to known algorithms.

2. Verify the base case: Prove that the guess holds for the smallest input size(s) of the algorithm. This step establishes the base case for the mathematical induction.

3. Assume correctness: Assume that the guess is correct for input sizes up to a certain value, which serves as the induction hypothesis.

4. Prove the induction step: Show that if the guess holds for input sizes up to a certain value, it also holds for the next larger input size. This step involves replacing the recurrence relation with the assumed time complexity and then proving the correctness of the new relation.

5. Conclude the time complexity: Based on the induction step, deduce that the guess is correct for all input sizes.

The substitution method can be used to analyze the time complexity of recursive algorithms by providing an upper bound on their time complexity. It is especially useful when the recurrence relation does not fit into the cases covered by the master theorem or when a more detailed analysis is needed.

It's important to note that the substitution method requires careful mathematical reasoning and may not always yield a straightforward solution. In some cases, the guess may need to be refined or adjusted based on the specific algorithm and its behavior.

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