### The master theorem

```Master Theorem
Chen Dan Dong
Feb. 19, 2013
Outline




Review of asymptotic notations
Understand the Master Theorem
Prove the theorem
Examples and applications
Review of Asymptotic Notation



Θ notation: asymptotic tight bound
Θ(g(n)) = { f(n): there exist positive constants c1, c2,
and n0 such that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for all n ≥
n0}.
O notation: asymptotic upper bound
O(g(n)) = { f(n): there exist positive constants c, and
n0 such that 0≤ f(n) ≤ cg(n) for all n ≥ n0}.
Ω notation: asymptotic lower bound
Ω(g(n)) = { f(n): there exist positive constants c, and
n0 such that 0≤ cg(n) ≤ f(n) for all n ≥ n0}.
Review of Asymptotic Notation (Con.)


Asymptotic notation in equations
Theorem:
For any two functions f(n) and g(n), we have
f(n) = Θ(g(n))
if and only if f(n) = O(g(n)) and f(n) = Ω(g(n)).
Master Theorem
Theorem: Let a ≥ 1 and b > 1 be constants, let f(n) be a
function and let T(n) be defined on the nonnegative
integers by recurrence T(n) = aT(n/b) + f(n),
Then T(n) has the following asymptotic bounds.
Case 1. If
for some constant ϵ > 0,
then
Case 2. If
then
Case 3. If
for some constant ϵ > 0,
and if a f(n/b) ≤ c f(n) for some constant c < 1 and all
sufficiently large n, then T(n) = Θ(f(n)).
Proof of Master Theorem

The proof consists of two parts
 The
first part analyzes the recurrence under the
simplifying assumption that T(n) is defined only on
exact powers of b, for n = 1, b, b2, ….
 The second part extends the analysis to all
positive integers n with handling floors and
ceilings.

Due to time limit, I will only show the proof for
the first part for n as exact powers of b.
Proof – Lemma 1
Lemma 1. Let a ≥ 1 and b > 1 be constants,
let f(n) be a nonnegative function defined
on exact powers of b. Define T(n) exact
powers of b by the recurrence
where j is a positive integer. Then
Proof – Lemma 2
Lemma 2. Let a ≥ 1 and b > 1 be constants, let f(n) be a
nonnegative function defined on exact powers of b. A
function g(n) defined over exact power of b by
has the following asymptotic bounds:
Case 1. If
for some constant ϵ > 0,
then
Case 2. If
then
Case 3. if a f(n/b) ≤ c f(n) for some constant c < 1 and all
sufficiently large n, then g(n) = Θ(f(n)).
Combining Lemma 1 and Lemma 2
Lemma 3. Let a ≥ 1 and b > 1 be constants, let f(n) be a
nonnegative function defined on exact powers of b.
Define T(n) exact powers of b by the recurrence
Then T(n) has the following asymptotic bounds:
Case 1. If
for some constant ϵ > 0,
then
Case 2. If
then
Case 3. if
for some constant ϵ > 0,
and if a f(n/b) ≤ c f(n) for some constant c < 1 and all
sufficiently large n, then g(n) = Θ(f(n)).
If n is not exact powers of b…



If n is not exact powers of b, then n/b is an not
integer.
We need to obtain a lower bound on
T(n) = a T(⌈n/b⌉) + f(n)
and an upper bound on
T(n) = a T(⌊n/b⌋) + f(n).
If intersted, check out here.
Master Theorem
Theorem: Let a ≥ 1 and b > 1 be constants, let f(n) be a
function and let T(n) be defined on the nonnegative
integers by recurrence T(n) = aT(n/b) + f(n),
Then T(n) has the following asymptotic bounds.
Case 1. If
for some constant ϵ > 0,
then
Case 2. If
then
Case 3. If
for some constant ϵ > 0,
and if a f(n/b) ≤ c f(n) for some constant c < 1 and all
sufficiently large n, then T(n) = Θ(f(n)).
Some examples

T(n) = 9 T(n/3) + n
Case 1.

T(n) = T(2n/3) +1
Case 2.

T(n) = 2T(n/2) + n lgn
Can’t apply Master Theorem.
Binary Search

Binary search finds the position of a specified
value within a sorted array.

Finding the recurrence relationship

Applying Master Theorem.
Case 2. T(n) = O(lg n).
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