Understanding the Master Theorem for Recursion Equations

The Master Theorem is a powerful tool for analyzing recursion equations commonly found in divide and conquer algorithms. It provides a framework for solving recurrence relations of the form T(n) = aT(n/b) + f(n). By examining different cases and comparing functions with powers of n, we can determine the time complexity of algorithms efficiently.

• Recursion
• Master Theorem
• Divide and Conquer
• Time Complexity

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1. The Master Theorem

2. A family of Recursion Equations Divide and conquer frequently lead to recursions of the form ?(?) = ??(?/?) + ?(?)

3. A family of Recursion Equations Solve Recurrence:

4. A family of Recursion Equations

5. A family of Recursion Equations

6. A family of Recursion Equations Total is log?? 1 ???(?/??)) + ??log?? ( ?=0 Need to compare f with power of n in order to see what dominates

7. A family of Recursion Equations ?(?) = ?(?log?? ?) ?(?) = (?log??) ?(?) = (?log??) ?(?) = (?log??log?) ?(?) = (?log??+?)and??(?/?) ??(?)eventually ?(?) = (?(?))

8. A family of Recursion Equations There are gaps between the three cases, where the master theorem does not apply

9. Examples ?(?) = 2?(?/2) + ? ?log22= ? = ?(?)Case 2 ?(?) = (?log?)

10. Examples ?(?) = 3?(?/2) + ? log23 = 1.58496 ? = ?(?log23 0.1) ?(?) = (?log23)

11. Examples ?(?) = ?(?/2) + ? ? = 1,? = 2 log21 = 0 ? = (?0+1/2) ?(?) = (?)

12. Examples ?(?) = 3?(?/3) + ?log? ? = 3? = 3so compare with? ?log? (?1+?) ?log? (?) Falls into the gap between Case 2 and Case 3