Recurrences

There are several techniques to solve recurrence relations, each suited to different forms and complexities of the equation.

• The iteration method, also known as backward substitution expands the recurrence step-by-step, revealing a pattern that can be summed and then closed into a direct formula.
• The recursion tree method visualizes recursive calls as a tree and estimates the total work at each level, summing them to find a bound.
• The master theorem provides a shortcut for divide-and-conquer relations of the form \( T(n) = aT(n/b) + f(n) \) when \( a, b \) are constants and $f(n)$ is some expression of $n$.

All of these approaches will be introduced, exemplified, and practiced with during the Analysis of Algorithms course you'll take.

The following lecture slides from an Algorithms course in University of Tennessee will give you a taste (+ examples) of how these methods work: (1) slides (2) slides. [Our course's final exam won't ask questions about these linked slides.]