CS 312: Algorithm Analysis

1. This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License. CS 312: Algorithm Analysis Lecture #9: Solving Divide and Conquer Recurrence Relations with Change of Variable Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, Sean Warnick

2. Announcements • HW #6 Due Today • Questions about Non-homogeneous RR? • Project #2 • Questions? • Early Day: Wednesday • Due: Friday

3. Objectives • Understand how to analyze recurrence relations that are typical of divide and conquer algorithms • Learn to use the “change of variable” technique to solve such recurrences • Review Merge sort (quickly!) • Theoretical analysis using RRs • Time permitting, prove the Master theorem

4. Divide and Conquer

5. Analysis Approach: Reduction Divide and Conquer Recurrence Relation Change of Variable Non-Homogeneous Recurrence Relation with Geometric Forcing Function LTI Homogeneous Recurrence Relation

6. Analysis of Binary Search

8. Analysis Approach: Reduction Divide and Conquer Recurrence Relation Change of Variable Non-Homogeneous Recurrence Relation with Geometric Forcing Function “Unwindthe Stack” LTI Homogeneous Recurrence Relation

10. Analysis of Binary Search (cont.)

11. 3 1 4 1 5 9 2 6 5 3 5 8 9 3 1 4 1 5 9 2 6 5 3 5 8 9 2 3 5 5 6 8 9 1 1 3 4 5 9 1 1 2 3 3 4 5 5 5 6 8 9 9 Review: Merge sort • Split the array in 1/2 • Sort each 1/2 • Merge the sorted halves • Adhoc(): Use insertion sort when sub-arrays are small

12. 3 1 4 1 5 9 2 6 5 3 5 8 9 3 1 4 1 5 9 2 6 5 3 5 8 9 2 3 5 5 6 8 9 1 1 3 4 5 9 1 1 2 3 3 4 5 5 5 6 8 9 9 Analysis of merge() proceduremerge (U[1..p+1], V[1..q+1], A[1..n]) i,j 1 U[p+1],V[q+1]  sentinel for k  1 top+qdo if U[i] < V[ j] then A[k]  U[i]; i  i + 1 else A[k]  V[j]; j  j + 1 How long to merge? U V A

13. Merge sort proceduremergesort (A[1..n]) if n is small enough theninsertsort (A) else array U[1..1+floor(n/2)], V[1..1+ceil(n/2)] U[1..floor(n/2)]  A[1..floor(n/2)] V[1..ceil(n/2)]  A[1+floor(n/2)..n] mergesort (U) mergesort (V) merge (U,V,A) What is the efficiency of Mergesort?

14. Merge sort

17. Using the Master Theorem a = number of sub-instances that must be solved n = original instance size (variable) n/b = size of subinstances d = polynomial order of g(n), where g(n) = cost of dividing and recombining

18. Efficiency of Mergesort

19. Another Useful Logarithm Identity Proof: Statement Reason • and are inverses • commutativity of • and are inverses

20. (Sketch of the)Proof of the Master Theorem How to proceed?

21. Proof of the Master Theorem

22. Proof of the Master Theorem

23. Proof: case 1

24. Proof: case 2

25. Proof: case 2

26. Proof: Conclusion

27. Proof: Conclusion

28. Assignment • Read: Section 2.3 in the textbook • HW #7: • Part III Exercises (Section 3.2) • Analyze 3-part mergesort using recurrence relations techniques • Problem 2.4 in the textbook (using the master theorem where possible)