1. Runtime of MergeSort
CSC 172
SPRING 2009
LECTURE 17 b

2. What is the RR for split? //O(1) //O(1) //O(1) //O(1) //O(1) //O(1)
public static Node split(Node head){
Node secondNode;
if (head == null) return null;
else if (head.getNext() == null) return null;
else {
secondNode = head.getNext();
head.setNext(secondNode.getNext());
secondNode.setNext(split(secondNode.getNext());
return secondNode; }
//O(1)
//O(1)
//O(1)
//O(1)
//O(1)
//O(1)

3. What is the RR for split? //O(1) //O(1) //O(1) //O(1) //O(1) //O(1)
public static Node split(Node head){
Node secondNode;
if (head == null) return null;
else if (head.getNext() == null) return null;
else {
secondNode = head.getNext();
head.setNext(secondNode.getNext());
secondNode.setNext(split(secondNode.getNext());
return secondNode; }
//O(1)
//O(1)
//O(1)
//O(1)
//O(1)
//O(1)
//O(1) + T(n-1)

4. What is the RR for Merge?
public static Node merge(Node list1, Node list2){
if (list1 == null) return list2;
else if (list2 == null) return list1;
else if (list1.getData.compareTo(list2.getData()) < 1) {
list1.setNext(merge(list1.getNext(),list2);
return list1;
} else {
list2.setNext(merge(list1,list2.getNext());
return list2; }

5. What is the RR for MergeSort?
public static Node mergeSort(Node list){
Node secondList;
if (list == null) return null;
else if (list.getNext() == null) return list;
else {
secondList = split(list);
return merge(mergeSort(list),mergeSort(secondList)); }
//O(1)
//O(1)
//O(1)
//O(n)

6. What is the RR for MergeSort?
public static Node mergeSort(Node list){
Node secondList;
if (list == null) return null;
else if (list.getNext() == null) return list;
else {
secondList = split(list);
return merge(mergeSort(list),mergeSort(secondList)); }
//O(1)
//O(1)
//O(1)
//O(n)
What is the length of list? secondList?
length(list) == length(secondList) = n/2

7. What is the RR for MergeSort?
public static Node mergeSort(Node list){
Node secondList;
if (list == null) return null;
else if (list.getNext() == null) return list;
else {
secondList = split(list);
return merge(mergeSort(list),mergeSort(secondList)); }
//O(1)
//O(1)
//O(1)
//O(n)
//O(n)
What is the length of list? secondList?
length(list) == length(secondList) = n/2
So, what is the cost of the call to merge?

8. What is the RR for MergeSort?
public static Node mergeSort(Node list){
Node secondList;
if (list == null) return null;
else if (list.getNext() == null) return list;
else {
secondList = split(list);
return merge(mergeSort(list),mergeSort(secondList)); }
//O(1)
//O(1)
//O(1)
//O(n)
//O(n)
If the cost of mergeSort is TmergeSort(n):
What is the cost of mergeSort is TmergeSort(list)?
TmergeSort(secondlist)?

9. What is the RR for MergeSort?
public static Node mergeSort(Node list){
Node secondList;
if (list == null) return null;
else if (list.getNext() == null) return list;
else {
secondList = split(list);
return merge(mergeSort(list),mergeSort(secondList) ); }
//O(1)
//O(1)
//O(1)
//O(n)
//O(n)
Tmergesort(n) = O(1)+O(n)+2Tmergesort(n/2)

10. What is the RR for MergeSort?

11. Recurrence Relation for MergeSort
Basis
n == 0
TmergeSort(0) = O(1)
Induction
TmergeSort(n) = O(n) + 2TmergeSort(n/2)

12. Replace O(1)s by concrete constants
TmergeSort(1) = a
TmergeSort(n) = bn + 2TmergeSort(n/2)
TmergeSort(n)/n = b + TmergeSort(n/2)/(n/2)
TmergeSort(n/2)/(n/2) = b + TmergeSort(n/4)/(n/4)
TmergeSort(n/4)/(n/4) = b + TmergeSort(n/8)/(n/8)
TmergeSort(n/8)/(n/8) = b + TmergeSort(n/16)/(n/16) …
TmergeSort(2)/(2) = b + TmergeSort(1)/(1)
How many calls?

13. How many calls? log2n calls
TmergeSort(n)/n = blog2n + TmergeSort(1)/(1)
TmergeSort(n) = nblog2n + na
TmergeSort(n) = O(nlog2n)