Master Theorem

Master Theorem

$$T(n)=aT(\frac{n}{b})+f(n)$$

where $a\ge 1,b\ge 1$ be constant and $f(n)$ be a function

Let $T(n)$ is defined on non-negative integers by the recurrence

• $n$ is the size of the problem
• $a$ is the number of sub problems in the recursion
• $\frac{n}{b}$ is the size of each sub problem (Here it is assumed that all sub problems are essentially the same size)
• $f(n)$ is the time to create the sub problems and combine their results in the above procedure

There are following three cases:

1. If $f(n)=\Theta (n^c)$ where $c<lo{g}_{b}a$ then $T(n)=\Theta (n^{lo{g}_{b}a})$
2. If $f(n)=\Theta (n^c)$ where $c=lo{g}_{b}a$ then $T(n)=\Theta (n^{c}logn)$
3. If $f(n)=\Theta (n^c)$ where $c>lo{g}_{b}a$ then $T(n)=\Theta (f(n))$

Indamissible equations

$$T(n)=2^{n}T(\frac{n}{2})+n$$

$a$ is not constant. The number of subproblems should be fixed

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$$T(n)=0.5T(\frac{n}{2})+n$$
$a<1$. Can’t have less than 1 subproblems

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$$T(n)=16T(\frac{n}{2})-n^2$$

$f(n)$ which is the combination time is not positive