Discrete Mathematics Recursion and Sequences

1. Discrete MathematicsRecursion and Sequences

2. Recursion Recursion is a powerful computer programming technique in which a procedure (e.g. a subroutine) is defined so that it calls itself. Note that recursion is not available in all computer languages To introduce recursion, look at sequences of numbers.

3. Definitions • Sequence: is just an infinite list of numbers in a particular order • Like a set, but: • Elements can be duplicated • Elements are ordered • A sequence is a function from a subset of Z to a set S • an is a term in the sequence • {an} means the entire sequence • The same notation as sets!

4. Sequence examples • an = 3n • The terms in the sequence are a1, a2, a3, … • The sequence {an} is { 3, 6, 9, 12, … } • Arithmetic Progression • a, a+d, a+2d, …, a+nd, … • an = a + (n-1) d • bn = 2n • The terms in the sequence are b1, b2, b3, … • The sequence {bn} is { 2, 4, 8, 16, 32, … } • Geometric Progression • a, ar, ar2, ar3, …, arn-1, … • an = arn-1

5. Determining the sequence formula • Given values in a sequence, how do you determine the formula? • Steps to consider: • Is it an arithmetic progression (each term a constant amount from the last)? • Is it a geometric progression (each term a factor of the previous term)? • Does the sequence it repeat (or cycle)? • Does the sequence combine previous terms? • Are there runs of the same value?

6. Determining the sequence formula • 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, … • The sequence alternates 1’s and 0’s, increasing the number of 1’s and 0’s each time • 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, … • This sequence increases by one, but repeats all even numbers once • 1, 0, 2, 0, 4, 0, 8, 0, 16, 0, … • The non-0 numbers are a geometric sequence (2n) interspersed with zeros • 3, 6, 12, 24, 48, 96, 192, … • Each term is twice the previous: geometric progression • an = 3*2n-1

7. Non-Recursive Definition of a Sequence • 15, 8, 1, -6, -13, -20, -27, … • Each term is 7 less than the previous term • an = 22 - 7n • 3, 5, 8, 12, 17, 23, 30, 38, 47, … • The difference between successive terms increases by one each time • a1 = 3, an = an-1 + n • an = n(n+1)/2 + 2 • 2, 16, 54, 128, 250, 432, 686, … • Each term is twice the cube of n • an = 2*n3 • 2, 3, 7, 25, 121, 721, 5041, 40321 • Each successive term is about n times the previous • an = n! + 1 When a sequence is defined by a rule (may be arithmetic or geometric progression) in this way, we say the definition is non-recursive.

8. What is an algorithm? • An algorithm is “a finite set of precise instructions for performing a computation or for solving a problem” • A program is one type of algorithm • Directions to somebody’s house is an algorithm • A recipe for cooking a cake is an algorithm • The steps to compute the cosine of 90° is an algorithm

9. Some algorithms are harder than others • Some algorithms are easy • Finding the largest (or smallest) value in a list • Finding a specific value in a list • Some algorithms are a bit harder • Sorting a list • Some algorithms are very hard • Finding the shortest path between Edirneand Diyarbakır • Some algorithms are essentially impossible • Factoring large composite numbers

10. Algorithms to Generate Sequences A non-recursive definition can be used in algorithm to obtain the first m terms of a sequence. The algorithm that generates the first m terms of the sequence 15, 8, 1, -6, -13, -20, -27, … 1. Input m 2. For n = 1 tomdo 2.1 t:=22-7n 2.2 output t This algorithm uses the non-recursive definition an = 22 - 7n Note that the algortihm is written in pseudocode, a form of structured English which can be easily converted to code in the chosen programming language.

11. Algorithms to Generate Sequences A different algorithm to generate first m terms of 15, 8, 1, -6, -13, -20, -27, … 1. Inputm 2. t:=15 3. Output t 4. For n = 2 tomdo 4.1. t:=t – 7 4.2. Output t A trace of this algorithm for m = 5 shows that it does, indeed, output the first 5 terms of the sequence. Recursive definition of this sequence is: a1 = 15 (base) an = an-1 – 7 (n > 1) It is recursive, because the formula an contains the previous term an, so the formula calls itself.

12. More Examples Find the recursive and non-recursive definitions for the sequence 1, 3, 7, 15, 31, 63, 127, … The non-recursive formula is more difficult to find than the recursive formula, but it is easier to use once it has been found – this is fairly typical. Non-recursive formula: an = 2 n - 1 Recursive formula a1 = 1, an = 2an-1 + 1 (n > 1) This algorithm produces the above sequence. It is not a recursive algorithm. Instead it is an iterative algorithm. 1. Inputm 2. t:=1 3. Output t 4. For n = 1 tomdo 4.1 t:= 2t + 1 4.2 Output t The word “iterative” means to do something repeatedly, as in a For-do loop

13. Fibonacci Sequence This sequence was posed by the Italian mathematician Leonardo Pisano in 1202. A man put a pair of newborn rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits will be present at later times if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? The resulting sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …

14. Fibonacci Sequence This is known as the Fibonacci sequence which was the second name for Leonardo. The recursive definition of this sequence is a1 = 1, a2 = 1, an = an-1 + an-2 (n>2) It’s not straight forward to obtain a non-recursive definition of the Fibonacci sequence In fact, the formula is

15. The Collatz Sequence Let a1 = any chosen positive integer, and for n>1, define an by an = 3an-1 +1 if an-1 is odd an = 0.5an-1 if an-1 is even This is called the Collatz sequence (after Lothar Collatz (1910-90), a German mathematician) It is also known as the 3X + 1 sequence e.g. İf we choose a1 = 3, then the sequence is 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, … In fact for any starting number, if the sequence ever reaches 1, it will cycle through 1, 4 and 2. İf we choose a1 = 13, then the sequence is 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, … In all examples, the Collatz sequence eventually reached the value of 1.

16. Recursive FunctionsFactorial A simple example of a recursively defined function is the factorial function: n! = 1· 2· 3· 4 ···(n –2)·(n –1)·n i.e., the product of the first n positive numbers (by convention, the product of nothing is 1, so that 0! = 1). Q: Find a recursive definition for n!

17. Recursive FunctionsFactorial Initialization: 0!= 1 Recursion: n != n ·(n -1)! To compute the value of a recursive function, e.g. 5!, one plugs into the recursive definition obtaining expressions involving lower and lower values of the function, until arriving at the base case. EG: 5! =

18. Recursive FunctionsFactorial Initialization: 0!= 1 Recursion: n != n ·(n -1)! To compute the value of a recursive function, e.g. 5!, one plugs into the recursive definition obtaining expressions involving lower and lower values of the function, until arriving at the base case. EG: 5! = 5.4!

19. Recursive FunctionsFactorial Initialization: 0!= 1 Recursion: n != n ·(n -1)! To compute the value of a recursive function, e.g. 5!, one plugs into the recursive definition obtaining expressions involving lower and lower values of the function, until arriving at the base case. EG: 5! = 5.4! = 5.4.3!

20. Recursive FunctionsFactorial Initialization: 0!= 1 Recursion: n != n ·(n -1)! To compute the value of a recursive function, e.g. 5!, one plugs into the recursive definition obtaining expressions involving lower and lower values of the function, until arriving at the base case. EG: 5! = 5.4! = 5.4.3! = 5.4.3.2!

21. Recursive FunctionsFactorial Initialization: 0!= 1 Recursion: n != n ·(n -1)! To compute the value of a recursive function, e.g. 5!, one plugs into the recursive definition obtaining expressions involving lower and lower values of the function, until arriving at the base case. EG: 5! = 5.4! = 5.4.3! = 5.4.3.2! =5.4.3.2.1!

22. Recursive FunctionsFactorial Initialization: 0!= 1 Recursion: n != n ·(n -1)! To compute the value of a recursive function, e.g. 5!, one plugs into the recursive definition obtaining expressions involving lower and lower values of the function, until arriving at the base case. EG: 5! = 5.4! = 5.4.3! = 5.4.3.2! =5.4.3.2.1! = 5.4.3.2.1.0!

23. An Iterative Algorithm for n! 1. Input n 2. t:=1 3. For i = 1 to n do 3.1 t:=i x t 4. Output t This is an iterative algorithm because of the For-do loop, but it is not recursive (the algorithm does not call itself) Function factorial (n) 1. t:=1 2. For i = 1 to n do 2.1 t:= i x t 3. factorial:=t

24. A Recursive Algorithm for n! The function algorithm for n! İs an iterative algorithm, based on the recursive definition 0!= 1 n != n ·(n -1)! (n>0) Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) This is now a genuine recursive algorithm, because it contains a call to itself.

25. How does the Algorithm work? Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) Compute 5!

26. Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) How does the Algorithm work?

27. Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) How does the Algorithm work?

28. Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) How does the Algorithm work?

29. Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) How does the Algorithm work?

30. Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) How does the Algorithm work?

31. Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) How does the Algorithm work?

32. Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) How does the Algorithm work?

33. Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) How does the Algorithm work?

34. Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) How does the Algorithm work?

35. Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) How does the Algorithm work?

36. Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) How does the Algorithm work?

37. Function factorial (n) 1. If n = 0 then 1.1. factorial:=1 else 1.2. factorial:= n x factorial(n-1) How does the Algorithm work? Return 5! = 120

38. Another Recursive Algorithm Example: The sequence 2, 4, 9, 23, 64, 186, … can be defined recursively by: a1 = 2 an = 3an-1 – n (n>1) Write a recursive algorithm for this sequence. Solution: Function t(n) 1. If n = 1 then 1.1. t:=2 else 1.2. t:= 3t(n-1) - n

39. To Output the First m terms A way to output the 1st m terms of this sequence is to use an algorithm that calls the function: 1. Input m 2. For n = 1 to m do 2.1 x:= t(n) 2.2 Output x Although this works, it is not very efficient e.g. If m = 4, t(1) would be calculated four times, t(2) three times, and so on. This is because our algorithm don’t provide a means storing results that can be used in subsequent passes through the For-do loop.

40. An Efficient Approach Using the algorithm we already have, output the first m-1 terms, omit this step if m = 1 Evaluate t(m), the mth term of the sequence Output the value of t(m) output_sequence (m) 1. If m = 1 then 1.1 t:= 2 else 1.2 output_sequence(m-1) 1.3 t:= 3t – m 2. Output t

41. How does the Algorithm work? output_sequence (m) 1. If m = 1 then 1.1 t:= 2 else 1.2 output_sequence(m-1) 1.3 t:= 3t – m 2. Output t Output the first 5 terms

42. How does the Algorithm work? output_sequence (m) 1. If m = 1 then 1.1 t:= 2 else 1.2 output_sequence(m-1) 1.3 t:= 3t – m 2. Output t

43. How does the Algorithm work? output_sequence (m) 1. If m = 1 then 1.1 t:= 2 else 1.2 output_sequence(m-1) 1.3 t:= 3t – m 2. Output t

44. How does the Algorithm work? output_sequence (m) 1. If m = 1 then 1.1 t:= 2 else 1.2 output_sequence(m-1) 1.3 t:= 3t – m 2. Output t

45. How does the Algorithm work? output_sequence (m) 1. If m = 1 then 1.1 t:= 2 else 1.2 output_sequence(m-1) 1.3 t:= 3t – m 2. Output t

46. How does the Algorithm work? output_sequence (m) 1. If m = 1 then 1.1 t:= 2 else 1.2 output_sequence(m-1) 1.3 t:= 3t – m 2. Output t

47. How does the Algorithm work? output_sequence (m) 1. If m = 1 then 1.1 t:= 2 else 1.2 output_sequence(m-1) 1.3 t:= 3t – m 2. Output t 2

48. How does the Algorithm work? output_sequence (m) 1. If m = 1 then 1.1 t:= 2 else 1.2 output_sequence(m-1) 1.3 t:= 3t – m 2. Output t 2 4

49. How does the Algorithm work? output_sequence (m) 1. If m = 1 then 1.1 t:= 2 else 1.2 output_sequence(m-1) 1.3 t:= 3t – m 2. Output t 2 4 9

50. How does the Algorithm work? output_sequence (m) 1. If m = 1 then 1.1 t:= 2 else 1.2 output_sequence(m-1) 1.3 t:= 3t – m 2. Output t 2 4 9 23