Effective Algorithm for n Fibonacci Number By: Professor Lili Saghafi
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The document discusses the recursive implementation of the Fibonacci sequence, its definition, and the associated issues such as inefficiency and potential for infinite loops. It presents a naive algorithm for calculating Fibonacci numbers and explains the computational inefficiency due to excessive recursive calls. Additionally, the document suggests an efficient approach by storing computed values to reduce redundancy.
Effective Algorithm for n Fibonacci Number By: Professor Lili Saghafi
- 1. A recursive implementation and its problems for calculating the Fibonacci sequence of length N. Data Structures and Algorithms Presented By: Professor Lili Saghafi proflilisaghafi@gmail.com @Lili_PLSSimon Fraser University SFU May 2019
- 2. Agenda • Understand the definition of the Fibonacci numbers. • Understand the definition of the Recursive / Recursive Functions • Show that the naive algorithm for computing them is slow. • Efficiently create algorithms to compute large Fibonacci numbers.
- 3. Definition: It is the integer sequence in which every number after the first two is the sum of the two preceding numbers. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, . . .
- 4. The Fibonacci Sequence • It is the integer sequence in which every number after the first two is the sum of the two preceding numbers. • 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
- 6. puzzle that Fibonacci posed • Leonardo of Pisa, known as Fibonacci. • Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: – a single newly born pair of rabbits (one male, one female) are put in a field; – rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; – rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. • The puzzle that Fibonacci posed was: how many pairs will there be in one year?
- 7. we have the following recursive definition for the nth Fibonacci number, F0=0 F1=1 Fn:=Fn−1+Fn−2
- 9. Examples
- 11. Recursion • Have you ever seen a set of Russian dolls? At first, you see just one figurine, usually painted wood, that looks something like this:
- 12. • You can remove the top half of the first doll, and what do you see inside? Another, slightly smaller, Russian doll!
- 13. • You can remove that doll and separate its top and bottom halves. And you see yet another, even smaller, doll:
- 14. • And once more:
- 15. • And you can keep going. Eventually you find the teeniest Russian doll. It is just one piece, and so it does not open: We started with one big Russian doll, and we saw smaller and smaller Russian dolls, until we saw one that was so small that it could not contain another.
- 16. Recursion What do Russian dolls have to do with algorithms? • Just as one Russian doll has within it a smaller Russian doll, which has an even smaller Russian doll within it, all the way down to a tiny Russian doll that is too small to contain another, we'll see how to design an algorithm to solve a problem by solving a smaller instance of the same problem, unless the problem is so small that we can just solve it directly. • We call this technique recursion. • Recursion has many, many applications
- 17. What is Recursive Function • A recursive function is a function that calls itself during its execution. • This enables the function to repeat itself several times, outputting the result at the end of each iteration.
- 18. Example of a recursive function function Count (integer N) if (N <= 0) return "Must be a Positive Integer"; if (N > 9) return "Counting Completed"; else return Count (N+1); end function
- 19. • The function Count() uses recursion to count from any number between 1 and 9, to the number 10. • For example, Count(1) would return 2,3,4,5,6,7,8,9,10. • Count(7) would return 8,9,10. • The result could be used as a roundabout way to subtract the number from 10.
- 20. Downside • Recursive functions are common in computer science because they allow programmers to write efficient programs using a minimal amount of code. • The downside is that they can cause infinite loops and other unexpected results if not written properly. • For example, in previous example, the function is terminated if the number is 0 or less or greater than 9. • If proper cases are not included in the function to stop the execution, the recursion will repeat forever, causing the program to crash, or worse yet, hang the entire computer system.
- 21. Question? • How you can compute any term of the series of Fibonacci numbers in any programming languages using recursive functions?
- 22. Naive Algorithm • Recursive functions are those functions which, basically, call themselves. • The most basic method to do This is because the computing power required to calculate larger terms of the series is immense. • The number of times the function is called causes a stack overflow in most languages.
- 24. Basic Algorithm • A recursive function F (F for Fibonacci): to compute the value of the next term. • 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … • Our function will take n as an input, • which will refer to the nth term of the sequence that we want to be computed. • F(4) should return the fourth term of the sequence.
- 25. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … • The code should, regardless the language, look something like this: function F(n) if n = 0 return 0 if n = 1 return 1 else return F(n-1) + F(n-2) • The term 0 of the sequence will be considered to be 0, so the first term will be 1; the second term, 1; the third term, 2; and so on • Term is the position of the number
- 26. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … • If it gets 0 as an input, it returns 0. • If it gets 1, it returns 1. • If it gets 2… in that case it falls into the else statement, which will call the function again for terms 2–1 (1) and 2–2 (0). • That will return 1 and 0, and the two results will be added, returning 1. function F(n) if n = 0 return 0 if n = 1 return 1 else return F(n-1) + F(n-2)
- 27. Problem with recursive functions • If you wanted the 100th term of the sequence. • The function would call itself for the 99th and the 98th, which would themselves call the function again for the 98th and 97th, and 97th and 96th terms…and so on. • It would be really slow.
- 28. Downside • Recursion allows for the design of powerful, elegant algorithms, but it uses up time and space for the call stack. – Efficiency can sometimes be improved by eliminating recursion. – A tail-recursive algorithm can easily be converted into a loop. – If the algorithm is only returning a value (as opposed to modifying an existing data structure), redundant computation can be avoided by storing the results of previous invocation in a table.
- 29. Recursive Algorithm for Fibonacci numbers It is the integer sequence in which every number after the first two is the sum of the two preceding numbers
- 30. What are the number of lines of code executed • Let T(n) denote the number of lines of code executed by FibRecurs(n).
- 31. number of lines of code executed •Let T(n) denote the number of lines of code executed by FibRecurs(n).
- 32. Running Time T(n) denote the number of lines of code executed by FibRecurs(n). If n>=2
- 33. Question ! • How many digits do you think that T(100) has? A- Less than 10 B- 10 to 20 C- 20 to 30 T is for the number of line of code to be executed.
- 34. Answer • How many digits do you think that T(100) has? C- 20 to 30 T is for the number of line of code to be executed.
- 35. Running Time T is for the number of line of code to be executed. T of n is at least to the length of Fibonacci numbers
- 36. Why slow? Look at all recursive calls this algorithm needs We need recursive call to n-1 and for doing that we need n- 2 and n-3 We need recursive call to n-2 and for doing that we need n-3 and n-4 And it just keep going………and create a big tree of recursive calls
- 37. Because it seems that we compute things over and over Every time we ask to do it since it is a new recursive call , we compute the whole thing over and over
- 38. Lets blow up the tree , F n-4 computes 5 times!
- 39. It is not right! • just because it looks right, it doesn’t mean it is. • not every problem that looks like a recursion-solvable should be solved with recursion
- 41. Efficient Algorithm Just look at the last two element and add them
- 42. New Algorithm When I filled out the entire list , I am going to return the last element Need to create an array to store all the numbers First element of array set to 0 Second element of array set to 1 I runs from 2 to n We need to set i element to SUM of two i-1 and i-2
- 43. How fast is this New Algorithm? How many lines of code? 3 lines of code at the beginning and one return statement means 4 line code We have 4 statement that went through n-1 time and each time we have to execute 2 lines of code so adding everything we reach that T(n)=2n+2 and for n=100 it takes 202 line of code.
- 44. Summary
- 45. Conclusion • The right algorithm makes all the difference.
- 46. Python def F(n): if n == 0: return 0 if n == 1: return 1 else: return F(n-1) + F(n-2) function F(n) if n = 0 return 0 if n = 1 return 1 else return F(n-1) + F(n-2)
- 47. Java public static int F(int n) { if(n == 0) { return 0; } if(n == 1) { return 1; } else { return F(n-1) + F(n-2); }} function F(n) if n = 0 return 0 if n = 1 return 1 else return F(n-1) + F(n-2)
- 48. C++ int F(int n) { if(n == 0) { return 0; } if(n == 1) { return 1; } else { return F(n-1) + F(n-2); }} function F(n) if n = 0 return 0 if n = 1 return 1 else return F(n-1) + F(n-2)
- 49. JavaScript function F(n) { if(n == 0) { return 0; } if(n == 1) { return 1; } else { return F(n-1) + F(n-2); }} function F(n) if n = 0 return 0 if n = 1 return 1 else return F(n-1) + F(n-2)
- 50. A recursive implementation and its problems for calculating the Fibonacci sequence of length N. Data Structures and Algorithms Presented By: Professor Lili Saghafi proflilisaghafi@gmail.com @Lili_PLS