The substitution method
The substitution method encompasses a variety of techniques, including induction, to provide proofs for recurrence functions. Often, we need to be innovative in making variable substitutions to transform the recurrence function into a form for which we already know the solution. One key characteristic of this method is its flexibility – there can be multiple approaches to solving the same recurrence. Although it does not offer a uniform solution for all problems, the substitution method is a powerful tool. In fact, even the master theorem (discussed in the next section) is proven using this method.
By employing the substitution method, we can address a wide range of recurrence functions. The process typically involves hypothesizing a solution, substituting it into the original recurrence, and then using induction to prove that the hypothesis is correct. This approach allows for creativity and adaptability, making it a valuable technique in algorithm...
