The document discusses the Master theorem, which provides a way to analyze the time complexity of divide-and-conquer algorithms. The theorem applies to algorithms represented by the recurrence relation T(n) = aT(n/b) + f(n). It describes three cases to determine the complexity based on comparing values of a, b, and f(n). An example problem is provided that uses the Master theorem to solve the recurrence relation T(n) = 2T(n/2) + n, finding the complexity is θ(n log n).

1. MASTER THEOREM
FOR SOLVING RECURRENCE RELATIONS
BIRLA INSTITUTE OF TECHNOLOGY, MESRA
JAIPUR CAMPUS
Submitted By:
DR. MADHAVI SINHA
Submitted To:
RIDYA GUPTA MCA/25004/22

2. Introduction to the Master theorem
• The Master theorem is a mathematical tool used to analyse the complexity of
divide-and-conquer algorithms.
• It provides a way to express the time complexity of a recursive algorithm in terms
of its subproblems and their sizes.
• The theorem applies to algorithms that can be represented by the following
recurrence relation:
)
(
)
( n
f
b
n
aT
n
T 







where, a ≥ 1, b > 1, and f(n) > 0

3. Suppose, there’s a time function of input size n
Let’s say we divide that into b sub-problems, then:
and constant time will be taken for each merge
operation :

5. Master method
Where:
• T(n) is the running time of the algorithm on input size n
• a is the number of subproblems generated by the algorithm
• n/b is the size of each subproblem
• f(n) is the time spent on the algorithm's work outside of the recursive calls.
• f(n) = θ(nklogpn), k >= 0 and p is a real number
The Master theorem provides a formula to calculate the time complexity of the
algorithm based on the values of a, b, and f(n).
)
(
)
( n
f
b
n
aT
n
T 








6. Master method
Case 1: If a > bk, then T(n) = θ (nlog
b
a)
Case 2: If a = bk and
• If p < -1, then T(n) = θ (nlog
b
a)
• If p = -1, then T(n) = θ (nlog
b
a.log2n)
• If p > -1, then T(n) = θ (nlog
b
a.logp+1n)
Case 3: If a < bk and
• If p < 0, then T(n) = O (nk)
• If p >= 0, then T(n) = θ (nklogpn)
To solve recurrence relations using Master’s theorem, we compare a with bk. Then, we follow the
following cases-

7. Problems using the Master theorem:
Problem 1: T(n) = 2T(n/2) + n
a = 2, b = 2, k = 1, p = 0, log22 = 1
Compare nlog
2
2 with f(n) = n
 f(n) = θ(n)  Case 2
 T(n) = θ(n log n)