Math Problem Statement
Problem 1: Recurrence Relation with Division by 3 and Linear Term Solve the recurrence relation: T(N)=T(N/3)+N with the base case T(1)=1. Solution to Problem 1: T(N)=O(N) Hint: The summation of the geometric series will be ..which is eventually Final Answer: T(N)=O(N) Problem 2: Recurrence Relation with Multiplication by 2 and Linear Term Problem: Solve the recurrence relation: T(N)=2T(N/2)+N with the base case T(1)=1. Solution to Problem 2: Final Answer: T(N)=O(NlogN)
Solution
Solution to Problem 1: Recurrence Relation with Division by 3 and Linear Term
The recurrence relation given is:
with the base case .
Approach
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Recursive Expansion: Expand the recurrence repeatedly: Substitute for : Repeat times:
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Geometric Series: The summation converges as , and the sum becomes: Thus:
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Asymptotic Analysis: The dominant term is , so:
Solution to Problem 2: Recurrence Relation with Multiplication by 2 and Linear Term
The recurrence relation is:
with the base case .
Approach
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Recursive Expansion: Expand the recurrence repeatedly: Substitute for : Simplify: Repeat times:
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Stopping Condition: When , we have . At this point:
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Final Expression: Substitute : Simplify :
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Asymptotic Analysis: The dominant term is , so:
Summary
- Problem 1 Final Answer:
- Problem 2 Final Answer:
Let me know if you need any clarifications or details about these solutions!
Questions for Deeper Understanding:
- What happens if the base case is not but some other value?
- How does changing the constant in affect the solution?
- Can the recurrence be solved using a Master Theorem?
- What are the real-world scenarios where these recurrences arise?
- How would the analysis change if is not a power of 3 or 2?
Tip:
When solving recurrences, always check if the Master Theorem applies first—it can save a lot of time in analysis!
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Math Problem Analysis
Mathematical Concepts
Recurrence Relations
Geometric Series
Asymptotic Analysis
Divide and Conquer
Formulas
Geometric series sum: S = a / (1 - r) for |r| < 1
Master Theorem for divide-and-conquer recurrences: T(N) = aT(N/b) + O(N^d)
Theorems
Master Theorem
Geometric Series Formula
Suitable Grade Level
Undergraduate Level (Computer Science, Discrete Mathematics)