Meaning of the constants that appear in the Master Theorem

Question

The general formula for time complexity is $T(n) = aT(n/c) + bn^k$.

• If $a> c^k$, the complexity is $O(n^{\log_c a})$.
• If $a = c^k$, it is $O(n^k \log n)$.
• If $a < c^k$, it is $O(n^k)$.

$a$ is the amount of times the recursive function is called, but what do $b$, $c$, and $k$ represent?

Answer 1

That is not the general formula for time complexity. There is no "general formula for time complexity", any more than there is a "general formula for the answer."

Rather, the formula you give is a recurrence relation that can be used to compute the running time of certain divide-and-conquer–style recursive algorithms. Specifically, it corresponds to a recursive function which, when given an input of size $n$, makes $a$ recursive calls, each on inputs of size $n/c$. If you pretended that the recursive calls were "free" (i.e., that they returned their answer in one computation step), then the algorithm would take $bn^k$ steps.

Answer 2

The Master theorem applies to recurrences of a certain form; in algorithm analysis, usually such for running-time functions of divide & conquer algorithms. If you look at some examples, you'll note that the common scheme is that

1. the input is divided into parts of size $n/c \pm 1$,
2. the algorithm is executed recursively on $a$ of these parts, and
3. the results of the recursive calls are then combined into the final result.

Now, steps 1 and 3 take some time, which the authors of that "general formula" of yours (which is anything but) assume to be of the form $bn^k$ (they probably mean $\Theta(n^k)$, or even $O(n^k)$, but simplify to this explicit function so the calculations work out more neatly).

See here for a longer explanation of the Master theorem with applications. There are some other questions on the Master theorem you may want to read through.

Answer 3

I think you must mean solving recurrences of the form T(n) = aT(n/b) + f(n).

From the relations that you have written I understand that you must be trying to learn the Master Method for solving recurrences. This is used where a divide and conquer strategy is used by the algorithm to solve problems. The D&C strategy divides a given problem into specific parts in order to solve it effectively.

Here a implies the number of parts a single problem is divided into.

(n/b) is the size of each sub-problem and

f(n) is the cost of solving(combining and splitting) each sub problem.

More details of the master method can be understood from here as suggested earlier by Raphael