Substitution Method
We make a guess for the solution and then we use mathematical induction to prove the guess is correct or incorrect.
For example consider the recurrence T(n) = 2T(n/2) + n
We guess the solution as T(n) = O(nLogn). Now we use induction to prove our guess.
We need to prove that T(n) <= cnLogn. We can assume that it is true for values smaller than n.
T(n) = 2T(n/2) + n
<= 2cn/2Log(n/2) + n
= cnLogn – cnLog2 + n
= cnLogn – cn + n
<= cnLogn
Recurrence Relations Notes for GATE Exam [2024]Advanced master theorem for divide and conquer recurrences:
Recurrence relations are the mathematical backbone of algorithmic analysis, providing a systematic way to express the time complexity of recursive algorithms. As GATE Exam 2024 approaches, a profound understanding of recurrence relations becomes imperative for tackling questions that demand a deep comprehension of algorithmic efficiency. These notes aim to present a concise and illuminating guide to recurrence relations, covering key concepts and techniques that are likely to be assessed in the GATE examination.
Table of Content
- What is Recurrence Relations?
- What is Linear Recurrence Relation?
- How to Solve Recurrence Relations?
- 1. Substitution Method:
- 2. Recurrence Tree Method:
- 3. Master Method:
- Different types of recurrence relations and their solutions:
- Previously Asked GATE Questions on Recurrence Relations: