1. Compsci 201 Recursion, Recurrences, & Trees
Owen Astrachan
Jeff Forbes
October 25, 2017
10/25/17
Compsci 201, Fall 2017, Recurrences

2. CompSci 201, Fall 2017, Recurrences
P is for …
Parallel
Better and more complicated than serial?
Planning
Come up with a set of actions that will achieve a goal
Probabilistic Inference
Markovian text recognition?
PriorityQueue
Efficiently finding minimum
10/25/17
CompSci 201, Fall 2017, Recurrences

3. CompSci 201, Fall 2017, Recurrences
Plan for the Week
How do we calculate the Big-Oh of recursive algorithms?
What’s a tree and how does it help solve problems efficiently?
10/25/17
CompSci 201, Fall 2017, Recurrences

4. Big-Oh for recursive functions
public int size(ListNode list) {
if (list == null) return 0;
return 1 + size(list.next); }
What’s the Big Oh for a list with n nodes?
Express cost of algorithm for input of size n as recurrence T(n)
T(0) Base case!
T(n) = if n > 0
Repeated substitution to solve for a closed form answer

5. Solving recurrence relations
plug, simplify, reduce, guess, verify?
T(n) = T(n-1) + 1
T(0) = 1
T(n) = T(n-k) + k Find the pattern!
Now, let k=n, then T(n) = T(0)+n = 1+n
get to base case, solve the recurrence: O(n)
T(n-1) = T(n-1-1) + 1
T(n) = [T(n-2) + 1] + 1 = T(n-2)+2
T(n-2) = T(n-2-1) + 1
T(n) = [(T(n-3) + 1) + 1] + 1 = T(n-3)+3

6. CompSci 201, Fall 2017, Recurrences
ListNRev revisited
public ListNode nlist(int n) {
if (n <= 0) return null;
ListNode first = new ListNode(n);
ListNode last = first;
for(int k=0; k < n-1; k++) {
last.next = new ListNode(n);
last = last.next; }
return first;
10/25/17
CompSci 201, Fall 2017, Recurrences

7. ListNRev revisited – circular
public ListNode nlistCirc(int n) {
if (n <= 0) return null;
ListNode first = new ListNode(n);
ListNode last = first;
for(int k=0; k < n-1; k++) {
last.next = new ListNode(n);
last = last.next; }
last.next = first;
return last;
10/25/17
CompSci 201, Fall 2017, Recurrences

8. CompSci 201, Fall 2017, Recurrences
ListNRev Complexity?
What’s changed?
What are the new recurrences for each version?
public ListNode make(int n) {
if (n <= 0) return null;
ListNode list = nlist(n);
ListNode last = list;
while (last.next != null)
last = last.next;
last.next = make(n-1);
return list; }
public ListNode makeCirc(int n) {
if (n <= 0) return null;
ListNode list = nlistCirc(n);
ListNode head = list.next;
list.next = makeCirc(n-1);
return head; }
10/25/17
CompSci 201, Fall 2017, Recurrences

9. CompSci 201, Fall 2017, Recurrences
Towers of Hanoi
// Move numDisks disks from -> to using aux
void move(int from, int to, int aux, int numDisks) {
if (numDisks == 1)
   System.out.println(from + " to " + to);
else {
    move(from, aux, to, numDisks-1);
    move(from, to, aux, 1);
    move(aux, to, from, numDisks-1); }
How many recursive calls?
How fast are we progressing to the solution?
10/25/17
CompSci 201, Fall 2017, Recurrences

10. CompSci 201, Fall 2017, Recurrences
Towers of Hanoi
// Move numDisks disks from -> to using aux
void move(int from, int to, int aux, int numDisks) {
if (numDisks == 1)
   System.out.println(from + " to " + to);
else {
    move(from, aux, to, numDisks-1);
    move(from, to, aux, 1);
    move(aux, to, from, numDisks-1); }
T(n-1)
T(1)
T(n-1)
10/25/17
CompSci 201, Fall 2017, Recurrences

11. CompSci 201, Fall 2017, Recurrences
Hanoi Recurrence
Plug and chug
Let i = n - 1
10/25/17
CompSci 201, Fall 2017, Recurrences

12. Recognizing Recurrences
Solve once, re-use in new contexts
T must be explicitly identified
n must be some measure of size of input/parameter
T(n) is the time for algortihmto run on an n-element vector
T(n) = T(n/2) + O(1) binary search O( )
T(n) = T(n-1) + O(1) sequential search O( )
T(n) = 2T(n/2) + O(1) tree traversal O( )
T(n) = 2T(n/2) + O(n) quicksort O( )
T(n) = T(n-1) + O(n) selection sort O( )
T(n) = 2T(n-1) + O(1) Towers of Hanoi O( )
Remember the algorithm, re-derive complexity
log n n n
n log n n2 2n

13. Another recurrence http://bit.ly/201-f17-boom int boom(int m, int x) {
if (m == 0)
 return h(x);
return boom(m-1, q(x)) +
boom(m-1, r(x)); }
Assume h(x), q(x), and r(x) are all O(1)
Express cost of algorithm
How do m and x matter?

14. SpreadingNews APT: greedy, recursive

15. How to write SpreadingNews?
Assume three subordinates: 5 min, 4 min, 7 min
Who do we tell first? Second?
How long does it take us to finish?
Suppose subordinates take 3 min, 3 min, 3 min?
How do we computer how long it takes for sub?
Do we have a way of doing this?
Is there some way out of always asking again?
Putting ideas into code: greedy and recursion!
Where is base case?

16. CompSci 201, Fall 2017, Recurrences
Towards Trees
Trees are powerful recurring structure
Quick-union and SpreadingNews have tree structure
Binary search trees are used in TreeSet and TreeMap
10/25/17
CompSci 201, Fall 2017, Recurrences

17. Binary Tree Node myRoot = null; public class Node {
public int myValue;
public Node myLeft;
public Node myRight;
public Node(int val) {
myValue = val; }

18. Binary Tree

19. Binary Tree 1 2