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Lecture No. 6

Fibonacci Sequences (Natural Models)

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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In this lecture we will cover the following:

• Fibonacci Problem and its Sequence

• Construction of Mathematical Model

• Explicit Formula Computing Fibonacci Numbers

• Recursive Algorithms

• Generalizations of Rabbits Problem and Constructing its Mathematical Models

• Applications of Fibonacci Sequences
**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

Today Covered

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• By studying Fibonacci numbers and constructing Fibonacci sequence we can imagine how mathematics is connected to apparently unrelated things in this universe.

• Even though these numbers were introduced in
1202 in Fibonacci’s book *Liber abaci*, but these
numbers and sequence are still fascinating and
mysterious to people of today.

• Fibonacci, who was born Leonardo da Pisa gave a problem in his book whose solution was the Fibonacci sequence as we will discuss it today.

**Fibonacci Sequence
**

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Statement: • Start with a pair of rabbits, one male and one female,

born on January 1. • Assume that all months are of equal length and that

rabbits begin to produce two months after their own birth. • After reaching age of two months, each pair produces

another mixed pair, one male and one female, and then another mixed pair each month, and no rabbit dies. How many pairs of rabbits will there be after one year?

Answer: *The Fibonacci Sequence!
*0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .

**Fibonacci’s Problem
**

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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**Construction of Mathematical Model
**

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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• Total pairs at level k = Total pairs at level k-1 + Total pairs born at level k (1)

• Since Total pairs born at level k = Total pairs at level k-2 (2)

• Hence by equation (1) and (2) Total pairs at level k = Total pairs at level k-1 + Total pairs at level k-2

• Now let us denote Fk = Total pairs at level k

• Now our recursive mathematical model will become Fk = Fk-1 + Fk-2

Construction of Mathematical Model

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Since Fk = Fk-1 + Fk-2 F0 = 0, F1= 1 • F2 = F1 + F0= 1 + 0 = 1 • F3 = F2 + F1= 1 + 1 = 2 • F4 = F3 + F2= 2 + 1 = 3 • F5 = F4 + F3= 3 + 2 = 5 • F6 = F5 + F4= 5 + 3 = 8 • F7 = F6 + F5= 8 + 5 = 13 • F8 = F7 + F6= 13 + 8 = 21 • F9 = F8 + F7= 21 + 13 = 34 • F10 = F9 + F8= 34 + 21 = 55 • F11 = F10 + F9= 55 + 34 = 89 • F12 = F11 + F10= 89 + 55 = 144 . . .

Computing Values using Mathematical Model

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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21 *kkk FFF
*

1 condition initialwith 2

10

21

*FF
kFFF kkk
*

Theorem: The fibonacci sequence F0,F1, F2,…. Satisfies the recurrence relation

Find the explicit formula for this sequence. Solution:

Let tk is solution to this, then characteristic equation

The given fibonacci sequence

**Explicit Formula Computing Fibonacci Numbers
**

012 *tt
***Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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2 51 ,

2 51

2 411

21

*tt
*

*t
*

**Fibonacci Sequence
**

For some real C and D fibonacci sequence satisfies the relation

0 1 0 F

2

51 2

51

0

0 2

51 2

51

0

0

00

0

*FDC
DC
*

*DCF
*

*n
*

*nDCF
nn
*

*n
*

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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1 2 1 2

51 2

51

2

51 2

51F

1 Now

1

1

*FDC
*

*DC
*

*n
*

*nn
*

*D
*

52 51

5 1

2 51

5 1F

Hence 5

1, 5

1C

get usly wesimultaneo 2 and 1 Solving

n

**Fibonacci Sequence
**

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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After simplifying we get

which is called the explicit formula for the Fibonacci sequence recurrence relation.

*nn
*

2 51

5 1

2 51

5 1Fn

**Fibonacci Sequence
**

5

1 5

1F

then 2

51 and 2

51Let

nn n

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Example: Compute F3

then 2

51 and 2

51 where 5

1 5

1F Since nnn

33

3 2 51

5 1

2 51

5 1F

**Verification of the Explicit Formula
**

8 555.1.35.1.31

5 1

8 555.1.35.1.31

5 1F Now,

22

3

555.1.35.1.31 5.8

1555.1.35.1.31 5.8

1F3

2555.1.35.1.31555.1.35.1.31 5.8

1F3

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Fibo-R(*n*)
**if ***n *= 0

**then **0
**if ***n *= 1

**then **1
**else **Fibo-R(*n*-1) + Fibo-R(*n*-2)

Recursive Algorithm Computing Fibonacci Numbers

*Terminating conditions
*

*Recursive calls
*

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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• Least Cost: To find an asymptotic bound of computational cost of this algorithm, we can use a simple trick to solve this recurrence containing big oh expressions

• Simply drop the big O from the recurrence, solve the recurrence, and put the O back. Our recurrence

will be refined to

2 n )2()1(

2 if )1( )(

*nTnT
*

*nO
nT
*

Running Time of Recursive Fibonacci Algorithm

2 n )2()1(

2 if 1 )(

*nTnT
*

*n
nT
*

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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**Construction of Mathematical Model
**

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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• Guess that Fn+1 is the least cost to solve this recurrence. Why this guess? n 0, T(n) Fn+1

then Fn+1 will be minimum cost for this recurrence • We prove it by mathematical induction Base Case There are two base cases

For n = 0, T(0) = 1 and F1 = 1, hence T(0) F1 For n = 1, T(1) = 1 and F2 = 1, hence T(1) F2

Running Time of Recursive Fibonacci Algorithm

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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• Inductive Hypothesis Let us suppose that statement is true some k 1

T(k) Fk+1 , for k =0, 1, 2,. . . and k 1 • Now we show that statement is true for k + 1 • Now, T(k + 1) = T(k) + T(k -1) By definition on T(n)

T(k + 1) = T(k) + T(k -1) Fk+1 + Fk = Fk+2 Assumption T(k + 1) Fk+2

• Hence the statement is true for k + 1. • We can now say with certainty that running time of

this recursive Fibonacci algorithm is at least (Fn+1).

Running Time of Recursive Fibonacci Algorithm

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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• Now we have proved that T(n) Fn+1 , n 0 (1)

• We already proved in solution to recursive relation that

It can be easily verified that Fn n/5 (3/2)n

From the equations (1) and (2), T(n) Fn+1 Fn (3/2)n

Hence we can conclude that running time of our recursive Fibonacci Algorithm is:

T(n) = (3/2)n

Running Time of Recursive Fibonacci Algorithm

(2) 2

51 and 2

51 w 5

1 5

1F nnn

*here
*

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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• W say that two quantities, x and y, (x < y), are in the golden ratio if the ratio between the sum, x + y, of these quantities and the larger one, y, is the same as the ratio between the larger one, y, and the smaller one x.

• Mathematicians have studied the golden ratio because of its unique and interesting properties.

Golden Ratio

62.1
*x
y
*

*y
yx
*

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Golden Ratio

01 1

1 1

i.e. y

yx andy x course of

1 ,62.01

62.1 2

51

2

*x
y
*

*y
x
*

1

1

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Recursion Tree

Drawback in Recursive Algorithms

F(n)

F(n-1) F(n-2)

F(0) F(1)

F(n-2) F(n-3) F(n-3) F(n-4)

F(1) F(0)

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Statement: • Start with a pair of rabbits, one male and one female,

born on January 1. • Assume that all months are of equal length and that

rabbits begin to produce two months after their own birth. • After reaching age of two months, each pair produces

two other mixed pairs, two male and two female, and then two other mixed pair each month, and no rabbit dies. How many pairs of rabbits will there be after one year?

Answer: *Generalization of Fibonacci Sequence!
*0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, . . .

Generalization of Rabbits Problem

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Construction of Mathematical Model

F0 = 0

F1 = 1

**F2 = 1
**

F3 = 3

F4 = 5

F5 = 11

F6 = 21

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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• Total pairs at level k = Total pairs at level k-1 + Total pairs born at level k (1)

• Since Total pairs born at level k =

2 x Total pairs at level k-2 (2) • By (1) and (2), Total pairs at level k =

Total pairs at level k-1 + 2 x Total pairs at level k-2 • Now let us denote

Fk = Total pairs at level k • Our recursive mathematical model:

Fk = Fk-1 + 2.Fk-2 • General Model (m pairs production): Fk = Fk-1 + m.Fk-2

Construction of Mathematical Model

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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• Recursive mathematical model (one pair production)

Fk = Fk-1 + Fk-2 • Recursive mathematical model

(two pairs production) Fk = Fk-1 + 2.Fk-2

• Recursive mathematical model (m pairs production)

Fk = Fk-1 + m.Fk-2

Generalization

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Since Fk = Fk-1 + 2.Fk-2 F0 = 0, F1 = 1 • F2 = F1 + 2.F0= 1 + 0 = 1 • F3 = F2 + 2.F1= 1 + 2 = 3 • F4 = F3 + 2.F2= 3 + 2 = 5 • F5 = F4 + 2.F3= 5 + 6 = 11 • F6 = F5 + F4= 11 + 10 = 21 • F7 = F6 + F5= 21 + 22 = 43 • F8 = F7 + F6= 43 + 42 = 85 • F9 = F8 + F7= 85 + 86 = 171 • F10 = F9 + F8= 171 + 170 = 341 • F11 = F10 + F9= 341 + 342 = 683 • F12 = F11 + F10= 683 + 682 = 1365 . . .

Computing Values using Mathematical Model

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Statement: • Start with a different kind of pair of rabbits, one male and

one female, born on January 1. • Assume all months are of equal length and that rabbits

begin to produce three months after their own birth. • After reaching age of three months, each pair produces

another mixed pairs, one male and other female, and then another mixed pair each month, and no rabbit dies. How many pairs of rabbits will there be after one year?

Answer: *Generalization of Fibonacci Sequence!
*0, 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, . . .

Another Generalization of Rabbits Problem

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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**Construction of Mathematical Model
**

F3 = 1

F4 = 2

F6 = 4

F8 = 9

F1 = 1

F0 = 0

F5 = 3

F2 = 1

F7 = 6

F9 = 13

F10 = 19
**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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• Total pairs at level k = Total pairs at level k-1 + Total pairs born at level k (1)

• Since Total pairs born at level k = Total pairs at level k-3 (2)

• By (1) and (2) Total pairs at level k = Total pairs at level k-1 + Total pairs at level k-3

• Now let us denote Fk = Total pairs at level k

• This time mathematical model: Fk = Fk-1 + Fk-3

Construction of Mathematical Model

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Since Fk = Fk-1 + Fk-3 F0 = 0, F1= F2= 1 • F3 = F2 + F0= 1 + 0 = 1 • F4 = F3 + F1= 1 + 1 = 2 • F5 = F4 + F2= 2 + 1 = 3 • F6 = F5 + F3= 3 + 1 = 4 • F7 = F6 + F4= 4 + 2 = 6 • F8 = F7 + F5= 6 + 3 = 9 • F9 = F8 + F6= 9 + 4 = 13 • F10 = F9 + F7= 13 + 6 = 19 • F11 = F10 + F8= 19 + 9 = 28 • F12 = F11 + F9= 28 + 13 = 41 . . .

Computing Values using Mathematical Model

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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• Recursive mathematical model (one pair, production after three months)

Fk = Fk-1 + Fk-3 • Recursive mathematical model

(two pairs, production after three months) Fk = Fk-1 + 2.Fk-3

• Recursive mathematical model (m pairs, production after three months)

Fk = Fk-1 + m.Fk-3 • Recursive mathematical model

(m pairs, production after n months) Fk = Fk-1 + m.Fk-n

More Generalization

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Fibonacci sequences • Are used in trend analysis • By some pseudorandom number generators • The number of petals is a Fibonacci number. • Many plants show the Fibonacci numbers in the

arrangements of the leaves around the stems. • Seen in arrangement of seeds on flower heads • Consecutive Fibonacci numbers give worst case

behavior when used as inputs in Euclid’s algorithm. • As n approaches infinity, the ratio F(n+1)/F(n)

approaches the golden ratio: =1.6180339887498948482...

Applications of Fibonacci Sequences

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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Fibonacci sequences • The Greeks felt that rectangles whose sides are in

the golden ratio are most pleasing • The Fibonacci number F(n+1) gives the number of

ways for 2 x 1 dominoes to cover a 2 x n checkerboard.

• Sum of the first n Fibonacci numbers is F(n+2)-1. • The shallow diagonals of Pascal’s triangle sum to

Fibonacci numbers. • Except n = 4, if F(n) is prime, then n is prime. • Equivalently, if n not prime, then F(n) is not prime. • gcd( F(n), F(m) ) = F( gcd(n, m) )

Applications of Fibonacci Sequences

**Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
**

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