recursion
play

Recursion 1 Recursion n A process by which a function calls itself - PowerPoint PPT Presentation

Dec 08, 2023 •9 likes •519 views

Recursion 1 Recursion n A process by which a function calls itself repeatedly Either directly. n X calls X Or cyclically in a chain. n X calls Y, and Y calls X n Used for repetitive computations in which each action is stated in terms of a

  1. Recursion 1

  2. Recursion n A process by which a function calls itself repeatedly ¨ Either directly. n X calls X ¨ Or cyclically in a chain. n X calls Y, and Y calls X n Used for repetitive computations in which each action is stated in terms of a previous result fact(n) = n * fact (n-1) 2

  3. Contd. n For a problem to be written in recursive form, two conditions are to be satisfied: ¨ It should be possible to express the problem in recursive form n Solution of the problem in terms of solution of the same problem on smaller sized data ¨ The problem statement must include a stopping condition Stopping condition fact(n) = 1, if n = 0 = n * fact(n-1), if n > 0 Recursive definition 3

  4. n Examples: ¨ Factorial: fact(0) = 1 fact(n) = n * fact(n-1), if n > 0 ¨ GCD: gcd (m, m) = m gcd (m, n) = gcd (m%n, n), if m > n gcd (m, n) = gcd (n, n%m), if m < n ¨ Fibonacci series (1,1,2,3,5,8,13,21,….) fib (0) = 1 fib (1) = 1 fib (n) = fib (n-1) + fib (n-2), if n > 1 4

  5. Factorial long int fact (int n) { if (n == 1) return (1); else return (n * fact(n-1)); } 5

  6. Factorial Execution long int fact (int n) { if (n = = 1) return (1); else return (n * fact(n-1)); } 6

  7. Factorial Execution fact(4) long int fact (int n) { if (n = = 1) return (1); else return (n * fact(n-1)); } 7

  8. Factorial Execution fact(4) if (4 = = 1) return (1); else return (4 * fact(3)); long int fact (int n) { if (n = = 1) return (1); else return (n * fact(n-1)); } 8

  9. Factorial Execution fact(4) if (4 = = 1) return (1); else return (4 * fact(3)); if (3 = = 1) return (1); else return (3 * fact(2)); long int fact (int n) { if (n = = 1) return (1); else return (n * fact(n-1)); } 9

  10. Factorial Execution fact(4) if (4 = = 1) return (1); else return (4 * fact(3)); if (3 = = 1) return (1); else return (3 * fact(2)); if (2 = = 1) return (1); else return (2 * fact(1)); long int fact (int n) { if (n = = 1) return (1); else return (n * fact(n-1)); } 10

  11. Factorial Execution fact(4) if (4 = = 1) return (1); else return (4 * fact(3)); if (3 = = 1) return (1); else return (3 * fact(2)); if (2 = = 1) return (1); else return (2 * fact(1)); long int fact (int n) { if (1 = = 1) return (1); if (n = = 1) return (1); else return (n * fact(n-1)); } 11

  12. Factorial Execution fact(4) if (4 = = 1) return (1); else return (4 * fact(3)); if (3 = = 1) return (1); else return (3 * fact(2)); if (2 = = 1) return (1); else return (2 * fact(1)); 1 long int fact (int n) { if (1 = = 1) return (1); if (n = = 1) return (1); else return (n * fact(n-1)); } 12

  13. Factorial Execution fact(4) if (4 = = 1) return (1); else return (4 * fact(3)); if (3 = = 1) return (1); else return (3 * fact(2)); 2 if (2 = = 1) return (1); else return (2 * fact(1)); 1 long int fact (int n) { if (1 = = 1) return (1); if (n = = 1) return (1); else return (n * fact(n-1)); } 13

  14. Factorial Execution fact(4) if (4 = = 1) return (1); else return (4 * fact(3)); if (3 = = 1) return (1); else return (3 * fact(2)); 2 if (2 = = 1) return (1); else return (2 * fact(1)); 1 long int fact (int n) { if (1 = = 1) return (1); if (n = = 1) return (1); else return (n * fact(n-1)); } 14

  15. Factorial Execution fact(4) if (4 = = 1) return (1); else return (4 * fact(3)); 6 if (3 = = 1) return (1); else return (3 * fact(2)); 2 if (2 = = 1) return (1); else return (2 * fact(1)); 1 long int fact (int n) { if (1 = = 1) return (1); if (n = = 1) return (1); else return (n * fact(n-1)); } 15

  16. Factorial Execution fact(4) 24 if (4 = = 1) return (1); else return (4 * fact(3)); 6 if (3 = = 1) return (1); else return (3 * fact(2)); 2 if (2 = = 1) return (1); else return (2 * fact(1)); 1 long int fact (int n) { if (1 = = 1) return (1); if (n = = 1) return (1); else return (n * fact(n-1)); } 16

  17. Look at the variable addresses (a slightly different program) ! void main() { Output int x,y; 4 scanf("%d",&x); F: data = 4, &data = 3221224528 y = fact(x); &val = 3221224516 printf ("M: x= %d, y = %d\n", x,y); F: data = 3, &data = 3221224480 } &val = 3221224468 int fact(int data) F: data = 2, &data = 3221224432 { int val = 1; &val = 3221224420 printf("F: data = %d, &data = %u \n F: data = 1, &data = 3221224384 &val = %u\n", data, &data, &val); &val = 3221224372 if (data>1) val = data*fact(data-1); return val; M: x= 4, y = 24 } 17

  18. Fibonacci Numbers Fibonacci recurrence: fib(n) = 1 if n = 0 or 1; = fib(n – 2) + fib(n – 1) otherwise; int fib (int n){ if (n == 0 or n == 1) return 1; [BASE] return fib(n-2) + fib(n-1) ; [Recursive] } 18

  19. int fib (int n) { Fibonacci recurrence: if (n == 0 || n == 1) return 1; fib(n) = 1 if n = 0 or 1; return fib(n-2) + fib(n-1) ; = fib(n – 2) + fib(n – 1) } otherwise; fib (5) fib (3) fib (4) fib (1) fib (2) fib (2) fib (3) fib (0) fib (1) fib (1) fib (2) fib (0) fib (1) fib (1) fib (0) 19

  20. int fib (int n) { Fibonacci recurrence: if (n == 0 || n == 1) return 1; fib(n) = 1 if n = 0 or 1; return fib(n-2) + fib(n-1) ; = fib(n – 2) + fib(n – 1) } otherwise; fib (5) fib (3) fib (4) fib (1) fib (2) fib (2) fib (3) 1 fib (0) fib (1) fib (1) fib (2) fib (0) fib (1) 1 1 1 1 1 fib (1) fib (0) 1 1 20 fib.c

  21. int fib (int n) { Fibonacci recurrence: if (n==0 || n==1) return 1; fib(n) = 1 if n = 0 or 1; return fib(n-2) + fib(n-1) ; = fib(n – 2) + fib(n – 1) } otherwise; 8 fib (5) 3 5 fib (3) fib (4) 2 3 1 2 fib (1) fib (2) fib (2) fib (3) 1 2 1 1 1 1 1 fib (0) fib (1) fib (1) fib (2) fib (0) fib (1) 1 1 1 1 1 1 1 fib (1) fib (0) 1 1 21

  22. Sum of Squares int sumSquares (int m, int n) { int middle ; if (m == n) return m*m; else { middle = (m+n)/2; return sumSquares(m,middle) + sumSquares(middle+1,n); } } 22

  23. Annotated Call Tree 355 sumSquares(5,10) sumSquares(5,10) 245 110 sumSquares(5,10) sumSquares(8,10) sumSquares(5,7) 100 49 145 61 sumSquares(10,10) sumSquares(8,9) sumSquares(7,7) sumSquares(5,6) 36 81 25 64 sumSquares(9,9) sumSquares(6,6) sumSquares(8,8) sumSquares(5,5) 49 100 36 81 25 64 23

  24. Towers of Hanoi Problem 1 2 3 4 5 LEFT CENTER RIGHT 24

  25. n Initially all the disks are stacked on the LEFT pole n Required to transfer all the disks to the RIGHT pole ¨ Only one disk can be moved at a time. ¨ A larger disk cannot be placed on a smaller disk n CENTER pole is used for temporary storage of disks 25

  26. n Recursive statement of the general problem of n disks ¨ Step 1: n Move the top (n-1) disks from LEFT to CENTER ¨ Step 2: n Move the largest disk from LEFT to RIGHT ¨ Step 3: n Move the (n-1) disks from CENTER to RIGHT 26

  27. Tower of Hanoi A B C 27

  28. Tower of Hanoi A B C 28

  29. Tower of Hanoi A B C 29

  30. Tower of Hanoi A B C 30

  31. Towers of Hanoi function void towers (int n, char from, char to, char aux) { /* Base Condition */ if (n==1) { printf (“Disk 1 : %c à &c \n”, from, to) ; return ; } /* Recursive Condition */ towers (n-1, from, aux, to) ; ……………………. ……………………. } 31

  32. Towers of Hanoi function void towers (int n, char from, char to, char aux) { /* Base Condition */ if (n==1) { printf (“Disk 1 : %c à &c \n”, from, to) ; return ; } /* Recursive Condition */ towers (n-1, from, aux, to) ; printf (“Disk %d : %c à %c\n”, n, from, to) ; ……………………. } 32

  33. Towers of Hanoi function void towers (int n, char from, char to, char aux) { /* Base Condition */ if (n==1) { printf (“Disk 1 : %c à %c \n”, from, to) ; return ; } /* Recursive Condition */ towers (n-1, from, aux, to) ; printf (“Disk %d : %c à %c\n”, n, from, to) ; towers (n-1, aux, to, from) ; } 33

  34. TOH runs void towers(int n, char from, char to, char aux) { if (n==1) Output { printf ("Disk 1 : %c -> %c \n", from, to) ; 3 return ; Disk 1 : A -> C } Disk 2 : A -> B towers (n-1, from, aux, to) ; Disk 1 : C -> B printf ("Disk %d : %c -> %c\n", n, from, to) ; Disk 3 : A -> C towers (n-1, aux, to, from) ; Disk 1 : B -> A } Disk 2 : B -> C void main() Disk 1 : A -> C { int n; scanf("%d", &n); towers(n,'A',‘C',‘B'); } 34

  35. More TOH runs Output 4 Disk 1 : A -> B void towers(int n, char from, char to, char aux) Disk 2 : A -> C { if (n==1) Disk 1 : B -> C { printf ("Disk 1 : %c -> %c \n", from, to) ; Disk 3 : A -> B return ; Disk 1 : C -> A } towers (n-1, from, aux, to) ; Disk 2 : C -> B printf ("Disk %d : %c -> %c\n", n, from, to) ; Disk 1 : A -> B towers (n-1, aux, to, from) ; Disk 4 : A -> C } Disk 1 : B -> C void main() Disk 2 : B -> A { int n; Disk 1 : C -> A scanf("%d", &n); Disk 3 : B -> C towers(n,'A',‘C',‘B'); Disk 1 : A -> B } Disk 2 : A -> C Disk 1 : B -> C 35

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Recursion in C++ Stephen P. Carl - CS 242 1 Recursion Defined Recursion is a technique for

Recursion in C++ Stephen P. Carl - CS 242 1 Recursion Defined Recursion is a technique for

Recursion in C++ Stephen P. Carl - CS 242 1 Recursion Defined Recursion is a technique for defining data structures or algorithms in terms of themselves . A recursive algorithm is a form of decomposition where rather than choosing an arbitrary

846 views • 22 slides
Topics Simple Recursion Chapter 13 Recursion with a Return Value Binary Search

Topics Simple Recursion Chapter 13 Recursion with a Return Value Binary Search

Topics Simple Recursion Chapter 13 Recursion with a Return Value Binary Search Revisited Recursion Recursion Versus Iteration Simple Recursion Recursive Methods When solving a problem using recursion, the idea is to

626 views • 13 slides
Recursion Ch 14 Recursion There are two important parts of recursion: -A stopping case that

Recursion Ch 14 Recursion There are two important parts of recursion: -A stopping case that

Recursion Ch 14 Recursion There are two important parts of recursion: -A stopping case that ends the recursion -A reduction case that reduces the problem What are the base and stopping cases for the Fibonacci numbers? (sum of the previous

279 views • 8 slides
Recursion Ch 14 Recursion There are two important parts of recursion: -A stopping case that

Recursion Ch 14 Recursion There are two important parts of recursion: -A stopping case that

Recursion Ch 14 Recursion There are two important parts of recursion: -A stopping case that ends the recursion -A reduction case that reduces the problem What are the base and stopping cases for the Fibonacci numbers? (sum of the previous

296 views • 11 slides
Recursion Ch 14 Announcements Midterm graded on gradescope Highlights - recursion Recursion

Recursion Ch 14 Announcements Midterm graded on gradescope Highlights - recursion Recursion

Recursion Ch 14 Announcements Midterm graded on gradescope Highlights - recursion Recursion No fancy blue words or classes this chapter Recursion is simply calling a method from inside itself This copy will re-run the method on any new

363 views • 24 slides
Chapter 17 Recursion Chapter Scope The concept of recursion Recursive methods

Chapter 17 Recursion Chapter Scope The concept of recursion Recursive methods

Chapter 17 Recursion Chapter Scope The concept of recursion Recursive methods Infinite recursion When to use (and not use) recursion Using recursion to solve problems Solving a maze Towers of Hanoi Java Foundations,

602 views • 41 slides
Definition Recursion See &quot;Recursion&quot;. Recursion If you still don't get it, see:

Definition Recursion See &quot;Recursion&quot;. Recursion If you still don't get it, see:

Advanced Programming Recursion Recursion Summary n Definition of recursion and divide et impera n Simple recursive algorithms n Merge Sort n Quicksort n More complex recursive algorithms 2 1 Advanced Programming Recursion

893 views • 60 slides
Recursion Recursion Recursion means to define something in terms of itself. In order to

Recursion Recursion Recursion means to define something in terms of itself. In order to

CS206 CS206 Recursion Recursion Recursion means to define something in terms of itself. In order to understand recursion, one must A directory is a collection of files and directories. first understand recursion. Anonymous

60 views • 4 slides
CSSE 220 Recursion Checkout Recursion project from SVN Recursion A solution technique where

CSSE 220 Recursion Checkout Recursion project from SVN Recursion A solution technique where

CSSE 220 Recursion Checkout Recursion project from SVN Recursion A solution technique where the same computation occurs repeatedly as the problem is solved recurs Examples: Sierpinski Triangle:

699 views • 7 slides
Patterns of recursion Readings: none. Topics: Simple recursion Using accumulative recursion

Patterns of recursion Readings: none. Topics: Simple recursion Using accumulative recursion

Patterns of recursion Readings: none. Topics: Simple recursion Using accumulative recursion Recognizing generative recursion 1/22 09: Patterns CS 135 Simple vs. general recursion All of the recursion we have done to date has followed a

386 views • 22 slides
Recursion READING: LC textbook chapter 7 Recursion Recursive algorithms A

Recursion READING: LC textbook chapter 7 Recursion Recursive algorithms A

Summary Topics recursion overview simple examples Sierpinski gasket counting blobs in a grid csci 210: Data Structures Hanoi towers Recursion READING: LC textbook chapter 7 Recursion Recursive

308 views • 7 slides
Tail Recursion and Accumulators Recursion Should now be

Tail Recursion and Accumulators Recursion Should now be

Tail Recursion and Accumulators Recursion Should now be comfortable with recursion: No harder than using a loop (Maybe?) OAen much easier than

346 views • 24 slides
Recursion Ch 14 Highlights - recursion Recursion No fancy blue words or classes this chapter

Recursion Ch 14 Highlights - recursion Recursion No fancy blue words or classes this chapter

Recursion Ch 14 Highlights - recursion Recursion No fancy blue words or classes this chapter Recursion is simply calling a method from inside itself This copy will re-run the method on any new arguments or information (See:

158 views • 14 slides
Objectives become familiar with the idea of recursion learn to use recursion as a

Objectives become familiar with the idea of recursion learn to use recursion as a

Objectives become familiar with the idea of recursion learn to use recursion as a programming tool Recursion become familiar with the binary search algorithm as an example of recursion Chapter 11 become familiar with the merge

616 views • 18 slides
Recursion, tail recursion Recursion is the lifeblood of repetition in lisp Even loop

Recursion, tail recursion Recursion is the lifeblood of repetition in lisp Even loop

Recursion, tail recursion Recursion is the lifeblood of repetition in lisp Even loop functions are implemented recursively Typical function setup: Error-check parameters Check for base cases Handle general/recursive cases

429 views • 11 slides
CS200: Recursion and induction Prichard Ch. 6.1 &amp; 6.3 CS200 - Recursion 1 CS200 -

CS200: Recursion and induction Prichard Ch. 6.1 &amp; 6.3 CS200 - Recursion 1 CS200 -

CS200: Recursion and induction Prichard Ch. 6.1 &amp; 6.3 CS200 - Recursion 1 CS200 - Recursion 2 Backtracking n Problem solving technique that involves moves: guesses at a solution. n Depth First Search: in case of failure retrace

272 views • 23 slides
Advanced Counting Techniques CS1200, CSE IIT Madras Meghana Nasre April 9, 2020 CS1200, CSE IIT

Advanced Counting Techniques CS1200, CSE IIT Madras Meghana Nasre April 9, 2020 CS1200, CSE IIT

Advanced Counting Techniques CS1200, CSE IIT Madras Meghana Nasre April 9, 2020 CS1200, CSE IIT Madras Meghana Nasre Advanced Counting Techniques Advanced Counting Techniques Principle of Inclusion-Exclusion Recurrences and its

797 views • 52 slides
STACK FRAMES Stack Frames void Prog1( int q ) { int s; ... y Prog2( s, q ); x ret_addr2

STACK FRAMES Stack Frames void Prog1( int q ) { int s; ... y Prog2( s, q ); x ret_addr2

STACK FRAMES Stack Frames void Prog1( int q ) { int s; ... y Prog2( s, q ); x ret_addr2 ... ret_addr2 Prog2 } b a void Prog2( int a, int b ) s { int x, y; ret_addr1 Prog1 q .... } S. Prasitjutrakul 1994 Recursive Programs

220 views • 10 slides
INF421, Lecture 6 Recursion Leo Liberti LIX, Ecole Polytechnique, France INF421, Lecture 3

INF421, Lecture 6 Recursion Leo Liberti LIX, Ecole Polytechnique, France INF421, Lecture 3

INF421, Lecture 6 Recursion Leo Liberti LIX, Ecole Polytechnique, France INF421, Lecture 3 p. 1/37 Course Objective : teach notions AND develop intelligence Evaluation : TP not en salle info, Contrle la fin. Note: max( CC, 3 4 CC

790 views • 52 slides
For Wednesday Read chapter 10, section 3 Homework Chapter 10, exercise 10 Program 4

For Wednesday Read chapter 10, section 3 Homework Chapter 10, exercise 10 Program 4

For Wednesday Read chapter 10, section 3 Homework Chapter 10, exercise 10 Program 4 Any questions? Research Paper Any questions? Bin Packing Off-Line Strategies First Fit Decreasing Best Fit Decreasing Divide and

477 views • 10 slides
Tirgul 2 Asymptotic Analysis In other words, g ( n ) bounds f ( n ) from above (for large n

Tirgul 2 Asymptotic Analysis In other words, g ( n ) bounds f ( n ) from above (for large n

Big - O Tirgul 2 Asymptotic Analysis In other words, g ( n ) bounds f ( n ) from above (for large n s) up to a constant. Examples: 1 ) 1000000 = O ( 1 ) = 2 ) 0 . 5 n O ( n ) = 3 ) 10000 n O ( n ) = 4 )

279 views • 5 slides
Recursion n A problem solving technique where an algorithm is defined in terms of itself n A

Recursion n A problem solving technique where an algorithm is defined in terms of itself n A

1 Recursion n A problem solving technique where an algorithm is defined in terms of itself n A recursive method is a method that calls itself n A recursive algorithm breaks down the input or the search space and applies the same logic to a

649 views • 15 slides
Recursion: How It Works 7 January 2019 OSU CSE 1 Question Considered Before How should you

Recursion: How It Works 7 January 2019 OSU CSE 1 Question Considered Before How should you

Recursion: How It Works 7 January 2019 OSU CSE 1 Question Considered Before How should you think about recursion so you can use it to develop elegant recursive methods to solve certain problems? Answer : Pretend there is a FreeLunch

843 views • 50 slides
Refinement of Trace Abstraction for Real-Time Programs September 9, 2017 Franck Cassez 2 , Peter

Refinement of Trace Abstraction for Real-Time Programs September 9, 2017 Franck Cassez 2 , Peter

Refinement of Trace Abstraction for Real-Time Programs September 9, 2017 Franck Cassez 2 , Peter G. Jensen 1 , 2 and Kim G. Larsen 1 pgj@cs.aau.dk Department of Computer Science, Aalborg University Department of Computing, Macquarie University

293 views • 28 slides

More recommend