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

2. Announcements  HW #6 Due Today  Questions about Non-homogeneous RR?  Feedback from TAs:  Be sure to justify your answers / show your work.  Project #2  Quick questions?  Definition of pseudo-code  Early Day: Friday  Due: Monday

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  Prove the Master theorem

4. Divide and Conquer

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

6. Analysis of Binary Search

10. Analysis of Binary Search (cont.)

12. 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

13. Analysis of merge() procedure merge (U[1..p+1], V[1..q+1], A[1..n]) i,j  1 U[p+1],V[q+1]  sentinel for k  1 to p+q do if U[i] < V[ j] then A[k]  U[i]; i  i + 1 else A[k]  V[j]; j  j + 1 U A V How long to merge?

14. Analysis of merge() procedure merge (U[1..p+1], V[1..q+1], A[1..n]) i,j  1 U[p+1],V[q+1]  sentinel for k  1 to p+q do if U[i] < V[ j] then A[k]  U[i]; i  i + 1 else A[k]  V[j]; j  j + 1 U A V How long to merge?

15. Merge sort procedure mergesort (A[1..n]) if n is small enough then insertsort (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?

16. Merge sort procedure mergesort (A[1..n]) if n is small enough then insertsort (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?

17. Merge sort

20. 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

21. Efficiency of Mergesort

22. Proof of the Master Theorem How to proceed?

23. Proof of the Master Theorem

25. Proof: case 1

27. Proof: case 2

29. Proof: Conclusion

31. 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)