1. Dr Nazir A. Zafar Advanced Algorithms Analysis and Design Advanced Algorithms Analysis and Design By Dr. Nazir Ahmad Zafar

2. Lecture No. 6 Fibonacci Sequences (Natural Models) Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

3. 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

4. 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

5. 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

6. Construction of Mathematical Model Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

7. 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 F k = Total pairs at level k Now our recursive mathematical model will become F k = F k-1 + F k-2 Construction of Mathematical Model Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

8. Since F k = F k-1 + F k-2 F 0 = 0, F 1 = 1 F 2 = F 1 + F 0 = 1 + 0 = 1 F 3 = F 2 + F 1 = 1 + 1 = 2 F 4 = F 3 + F 2 = 2 + 1 = 3 F 5 = F 4 + F 3 = 3 + 2 = 5 F 6 = F 5 + F 4 = 5 + 3 = 8 F 7 = F 6 + F 5 = 8 + 5 = 13 F 8 = F 7 + F 6 = 13 + 8 = 21 F 9 = F 8 + F 7 = 21 + 13 = 34 F 10 = F 9 + F 8 = 34 + 21 = 55 F 11 = F 10 + F 9 = 55 + 34 = 89 F 12 = F 11 + F 10 = 89 + 55 = 144... Computing Values using Mathematical Model Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

9. Theorem: The fibonacci sequence F 0,F 1, F 2,…. Satisfies the recurrence relation Find the explicit formula for this sequence. Solution: Let t k is solution to this, then characteristic equation The given fibonacci sequence Explicit Formula Computing Fibonacci Numbers Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

10. Fibonacci Sequence For some real C and D fibonacci sequence satisfies the relation Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

11. Fibonacci Sequence Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

12. After simplifying we get which is called the explicit formula for the Fibonacci sequence recurrence relation. Fibonacci Sequence Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

13. Example: Compute F 3 Verification of the Explicit Formula Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

14. 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

15. 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 Running Time of Recursive Fibonacci Algorithm Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

16. Construction of Mathematical Model Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

17. Guess that F n+1 is the least cost to solve this recurrence. Why this guess?  n  0, T(n)  F n+1 then F n+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 F 1 = 1, hence T(0)  F 1 For n = 1, T(1) = 1 and F 2 = 1, hence T(1)  F 2 Running Time of Recursive Fibonacci Algorithm Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

18. Inductive Hypothesis Let us suppose that statement is true some k  1 T(k)  F k+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)  F k+1 + F k = F k+2 Assumption T(k + 1)  F k+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  (F n+1 ). Running Time of Recursive Fibonacci Algorithm Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

19. Now we have proved that T(n)  F n+1, n  0(1) We already proved in solution to recursive relation that It can be easily verified that F n   n /5  (3/2) n From the equations (1) and (2), T(n)  F n+1  F n  (3/2) n Hence we can conclude that r unning time of our recursive Fibonacci Algorithm is: T(n) =  (3/2) n Running Time of Recursive Fibonacci Algorithm Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

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

21. Golden Ratio Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

22. 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-4) F(1)F(0) Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

23. 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

24. Construction of Mathematical Model Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

25. 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 F k = Total pairs at level k Our recursive mathematical model: F k = F k-1 + 2.F k-2 General Model (m pairs production): F k = F k-1 + m.F k-2 Construction of Mathematical Model Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

26. Recursive mathematical model (one pair production) F k = F k-1 + F k-2 Recursive mathematical model (two pairs production) F k = F k-1 + 2.F k-2 Recursive mathematical model (m pairs production) F k = F k-1 + m.F k-2 Generalization Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

27. Since F k = F k-1 + 2.F k-2 F 0 = 0, F 1 = 1 F 2 = F 1 + 2.F 0 = 1 + 0 = 1 F 3 = F 2 + 2.F 1 = 1 + 2 = 3 F 4 = F 3 + 2.F 2 = 3 + 2 = 5 F 5 = F 4 + 2.F 3 = 5 + 6 = 11 F 6 = F 5 + F 4 = 11 + 10 = 21 F 7 = F 6 + F 5 = 21 + 22 = 43 F 8 = F 7 + F 6 = 43 + 42 = 85 F 9 = F 8 + F 7 = 85 + 86 = 171 F 10 = F 9 + F 8 = 171 + 170 = 341 F 11 = F 10 + F 9 = 341 + 342 = 683 F 12 = F 11 + F 10 = 683 + 682 = 1365... Computing Values using Mathematical Model Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

28. 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

29. Construction of Mathematical Model Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

30. 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 F k = Total pairs at level k This time mathematical model:F k = F k-1 + F k-3 Construction of Mathematical Model Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

31. Since F k = F k-1 + F k-3 F 0 = 0, F 1 = F 2 = 1 F 3 = F 2 + F 0 = 1 + 0 = 1 F 4 = F 3 + F 1 = 1 + 1 = 2 F 5 = F 4 + F 2 = 2 + 1 = 3 F 6 = F 5 + F 3 = 3 + 1 = 4 F 7 = F 6 + F 4 = 4 + 2 = 6 F 8 = F 7 + F 5 = 6 + 3 = 9 F 9 = F 8 + F 6 = 9 + 4 = 13 F 10 = F 9 + F 7 = 13 + 6 = 19 F 11 = F 10 + F 8 = 19 + 9 = 28 F 12 = F 11 + F 9 = 28 + 13 = 41... Computing Values using Mathematical Model Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

32. Recursive mathematical model (one pair, production after three months) F k = F k-1 + F k-3 Recursive mathematical model (two pairs, production after three months) F k = F k-1 + 2.F k-3 Recursive mathematical model (m pairs, production after three months) F k = F k-1 + m.F k-3 Recursive mathematical model (m pairs, production after n months) F k = F k-1 + m.F k-n More Generalization Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

33. 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

34. 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