Recurrent Problem in Discrete Mathmatics

1. Recurrent Problem

2. Motivation
We will explore three sample problems that give a feel for what’s to come.
They have two traits in common:
 Investigated repeatedly
 Solutions all use the idea of recurrence
• Solution to each problem depends on the solutions to smaller instances of the same problem.

3. Problems with Recursive Solution
Easy to solve small problem.
Refers to previous steps and so on becomes hard to solve problems that are
relatively large.

4. Closed Form
An expression that can be expressed analytically in terms of a finite
number of certain "well-known" functions.
 Elementary Functions
o Constants
o One variable
o Finite number of exponentials or logarithms or roots of polynomial
o Four elementary operations (+ – × ÷).

5. Steps to Solve A Problem
Look at small cases.
Find and prove a mathematical expression for the quantity of interest.
Recurrence Relation.
Find and prove a closed form for our mathematical expression. Prove
the closed form using mathematical induction

6. Recurrent Problems
The Tower of Hanoi
The Lines in The Plane
The Josephus Problem

7. Tower of Hanoi Problem
Invented by the French mathematician Edouard Lucas in 1883.
Given a tower of n disks, initially stacked in decreasing size on one of three pegs.
The objective is
 Transfer the entire tower to one of the other pegs.
 Move only one disk at a time.
 Never move a larger one onto a smaller.

8. Tower of Hanoi : 3 Disks START
BEG AUX END

9. Tower of Hanoi : 3 Disks FINISH
BEG AUX END

10. Tower of Hanoi : Recursive Solution
Let’s call the three peg BEG(Source), AUX(Auxiliary) and END(Destination).
1) Move the top N – 1 disks from the BEG to AUX tower using END
2) Move the Nth disk from BEG to END tower
3) Move the N – 1 disks from AUX tower to END tower using BEG

11. Tower of Hanoi : 3 Disks
BEG AUX END
Move n-1 disks from BEG to AUX using END

12. Tower of Hanoi : 3 Disks
BEG AUX END

13. Tower of Hanoi : 3 Disks
BEG AUX END

14. Tower of Hanoi : 3 Disks
BEG AUX END
Move largest one from BEG to END

15. Tower of Hanoi : 3 Disks
BEG AUX END
Move n-1 disks from AUX to END using BEG

16. Tower of Hanoi : 3 Disks
BEG AUX END

17. Tower of Hanoi : 3 Disks
BEG AUX END

18. Tower of Hanoi : 3 Disks
BEG AUX END

19. Tower of Hanoi : No. of Moves, Tn
• for 1 disk it takes 1 move to transfer 1 disk from BEG to END;
• for 2 disks, it will take 3 moves: 2 Tn-1 + 1 = 2(1) + 1 = 3
• for 3 disks, it will take 7 moves: 2 Tn-1 + 1 = 2(3) + 1 = 7
• for 4 disks, it will take 15 moves: 2 Tn-1 + 1 = 2(7) + 1 = 15
• for 5 disks, it will take 31 moves: 2 Tn-1 + 1 = 2(15) + 1 = 31
• for 6 disks... ?

20. Tower of Hanoi : No. of Moves, Tn
• Explicit Pattern
Number of Disks Number of Moves
1 1
2 3
3 7
4 15
5 31
T0 = 0
Tn = 2 Tn-1 + 1,n>0

21. Tower of Hanoi : Recursive Solution Difficulties
T7 = ?
T7 = 2 T6 + 1
= 2(2 T5 + 1) + 1
= 4(2 T4 + 1) + 3
= 8(2 T3 + 1) + 7
= 16(2 T2 + 1) + 15
= 32(2 T1 + 1) + 31
= 64(2 T0 + 1) + 63
= 64 × 0 + 64+ 63
= 127
T50 = ?
T50 = 2 T49 + 1
= 2(2 T48 + 1) + 1
= 4(2 T47 + 1) + 3
= 8(2 T46 + 1) + 7
……………………….
=?

22. Tower of Hanoi : Finding Closed Form
• Explicit Pattern
Number of Disks Number of Moves
1 1
2 3
3 7
4 15
5 31
• Powers of two help reveal the pattern
Number of Disks Number of Moves
1 21 - 1 = 2 - 1 = 1
2 22 - 1 = 4 - 1 = 3
3 23 - 1 = 8 - 1 = 7
4 24 - 1 = 16 - 1 = 15
5 25 - 1 = 32 - 1 = 31
Tn = 2 n - 1
A Closed Form

23. Tower of Hanoi : Proving Closed Form
Basis : For n= 0, T0 = 2n – 1 = 20 – 1 = 0
Induction : Let for n = n-1 the expression is true, then Tn-1 = 2n-1 – 1
Hypothesis : For n = n ,
Tn = 2 Tn-1 + 1
= 2 (2n-1 – 1 ) + 1
= 2n-1+1 – 2 + 1
= 2n – 1

24. Lines in The Plane Problem
Popularly: How many slices of pizza can a person obtain by making n
straight cuts with a pizza knife?
Academically: What is the maximum number Ln of regions defined by n
lines in the plane?

25. Lines in The Plane : Recursive Solution
Draw lines so that they intersect each others
Calculate maximum number of regions created by lines

26. Lines in The Plane : N = 0
L0 = 1

27. Lines in The Plane : N = 1
L1 = 2

28. Lines in The Plane : N = 2
L2 = 4

29. Lines in The Plane : N = 3
L3 = 7

30. Lines in The Plane : N = 4
L4 = 11

31. Lines in The Plane : Maximum No. of Regions, Ln
• for 0 line it creates 1 region;
• for 1 lines, it creates 2 region; : Ln-1 + n = 1 + 1 = 2
• for 2 lines, it creates 4 region; : Ln-1 + n = 2 + 2 = 4
• for 3 lines, it will take 7 region: Ln-1 + n = 4 + 3 = 7
• for 4 lines, it will take 10 region: Ln-1 + n = 7 + 4 = 11
• for 5 lines... ?
L0 = 1
Ln = Ln-1 + n, n>0

32. Lines in The Plane : Recursive Solution Difficulties
L7 = ?
L7 = L6 + 7
= L5 + 6 + 7
= L4 + 5+ 13
= L3 + 4 + 18
= L2 + 3 + 22
= L1 + 2 + 25
= L0 + 1 + 27
= 1+ 28
= 29
L50 = ?
L50 = L49 + 50
= L48 + 49+ 50
= L47 + 48+ 99
= L46 + 47 +14 7
……………………….
=?

33. Lines in The Plane: Finding Closed Form
Ln = Ln-1 + n
= Ln-2 + (n-1) + n
= Ln-3 + (n-2) + (n-1) + n
= Ln-2 + (n-3) + (n-2) + (n-1) + n
……………………..
= L0 + 1 + 2 + …… + (n-3) + (n-2) + (n-1) + n
= 1 +
𝒏(𝒏+𝟏)
𝟐

34. Lines in The Plane : Proving Closed Form
Basis : For n= 0, L0 = 1 +
𝟎(𝟎+𝟏)
𝟐
= 1
Induction : Let for n = n-1 the expression is true, then Ln-1 = 1 +
𝒏(𝒏 − 𝟏)
𝟐
Hypothesis : For n = n ,
Ln = Ln-1 + n
= 1 +
𝒏(𝒏 − 𝟏)
𝟐
+ n
= 1 +
𝒏𝟐
− 𝒏 + 𝟐𝒏
𝟐
= 1 +
𝒏𝟐
+ 𝒏
𝟐
= 1 +
𝒏(𝒏 + 𝟏)
𝟐

35. Josephus Problem
Do it Yourself
Solve Exercises

36. The End