lecture 16 more recursion
play

Lecture 16: More Recursion! CS 1110 Introduction to Computing - PowerPoint PPT Presentation

Oct 28, 2022 •141 likes •588 views

http://www.cs.cornell.edu/courses/cs1110/2019sp Lecture 16: More Recursion! CS 1110 Introduction to Computing Using Python [E. Andersen, A. Bracy, D. Gries, L. Lee, S. Marschner, C. Van Loan, W. White] Recursion Recursive Function : A

  1. http://www.cs.cornell.edu/courses/cs1110/2019sp Lecture 16: More Recursion! CS 1110 Introduction to Computing Using Python [E. Andersen, A. Bracy, D. Gries, L. Lee, S. Marschner, C. Van Loan, W. White]

  2. Recursion Recursive Function : A function that calls itself (directly or indirectly) Recursive Definition : A definition that is defined in terms of itself 2

  3. A Mathematical Example: Factorial Non-recursive definition: n! = n � n-1 � … � 2 � 1 = n (n-1 � … � 2 � 1) Recursive definition: n! = n (n-1)! for n > 0 Recursive case 0! = 1 Base case What happens if there is no base case? 3

  4. Recursive Call Frames factorial def factorial(n): 1 """Returns: factorial of n. n 3 Precondition: n ≥ 0 an int""" if n == 0: 1 return 1 2 return n*factorial(n-1) 3 factorial(3) 4

  5. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Precondition: n ≥ 0 an int""" if n == 0: 1 return 1 2 return n*factorial(n-1) 3 Call: factorial(3) 5

  6. Recursion factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Precondition: n ≥ 0 an int""" if n == 0: 1 return 1 2 Now what? return n*factorial(n-1) 3 Each call is a new frame. factorial(3) 6

  7. What happens next? (Q) A: B: factorial factorial 1, 3 1, 3, 1 û n n 3 3 2 factorial 1 D: n 2 factorial 1, 3, 1 û n 3 2 C: ERASE FRAME factorial 1, 3 factorial factorial 1 1 n 3 n n 3 2 7

  8. What happens next? (A) A: B: CORRECT factorial factorial 1, 3 1, 3, 1 û n n 3 3 2 factorial 1 D: n 2 factorial 1, 3, 1 û n 3 2 C: ERASE FRAME factorial 1, 3 factorial factorial 1 1 n 3 n n 3 2 8

  9. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Pre: n ≥ 0 an int""" factorial 1 if n == 0: 1 n 2 return 1 2 return n*factorial(n-1) 3 Call: factorial(3) 9

  10. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 return 1 2 return n*factorial(n-1) 3 Call: factorial(3) 10

  11. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 return 1 2 factorial 1 return n*factorial(n-1) n 1 3 Call: factorial(3) 11

  12. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 return 1 2 factorial 1, 3 return n*factorial(n-1) n 1 3 Call: factorial(3) 12

  13. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 return 1 2 factorial 1, 3 return n*factorial(n-1) n 1 3 factorial 1 Call: factorial(3) n 0 13

  14. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 return 1 2 factorial 1, 3 return n*factorial(n-1) n 1 3 factorial 1, 2 Call: factorial(3) n 0 14

  15. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 return 1 2 factorial 1, 3 return n*factorial(n-1) n 1 3 factorial 1, 2 Call: factorial(3) n 0 RETURN 1 15

  16. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 return 1 2 factorial 1, 3 return n*factorial(n-1) n 1 3 factorial 1, 2 Call: factorial(3) n 0 RETURN 1 16

  17. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 return 1 2 factorial 1, 3 return n*factorial(n-1) n 1 RETURN 1 3 factorial 1, 2 Call: factorial(3) n 0 RETURN 1 17

  18. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 return 1 2 factorial 1, 3 return n*factorial(n-1) n 1 RETURN 1 3 factorial 1, 2 Call: factorial(3) n 0 RETURN 1 18

  19. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 2 RETURN return 1 2 factorial 1, 3 return n*factorial(n-1) n 1 RETURN 1 3 factorial 1, 2 Call: factorial(3) n 0 RETURN 1 19

  20. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 2 RETURN return 1 2 factorial 1, 3 return n*factorial(n-1) n 1 RETURN 1 3 factorial 1, 2 Call: factorial(3) n 0 RETURN 1 20

  21. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 6 RETURN Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 2 RETURN return 1 2 factorial 1, 3 return n*factorial(n-1) n 1 RETURN 1 3 factorial 1, 2 Call: factorial(3) n 0 RETURN 1 21

  22. Recursive Call Frames factorial def factorial(n): 1, 3 """Returns: factorial of n. n 3 6 RETURN Pre: n ≥ 0 an int""" factorial 1, 3 if n == 0: 1 n 2 2 RETURN return 1 2 factorial 1, 3 return n*factorial(n-1) n 1 RETURN 1 3 factorial 1, 2 Call: factorial(3) n 0 RETURN 1 22

  23. Divide and Conquer Goal : Solve problem P on a piece of data data Idea : Split data into two parts and solve problem data 1 data 2 Solve Problem P Solve Problem P Combine Answer! 23

  24. Example: Reversing a String def reverse(s): H e l l o ! """Returns: reverse of s Precondition: s a string""" # 1. Handle base case ! o l l e H # 2. Break into two parts # 3. Combine the result 24

  25. Example: Reversing a String def reverse(s): H e l l o ! """Returns: reverse of s Precondition: s a string""" # 1. Handle base case left H e l l o ! # 2. Break into two parts right ! o l l e left = reverse(s[0]) right = reverse(s[1:]) If this is how we break it up…. # 3. Combine the result How do we combine it? 25

  26. How to Combine? (Q) def reverse(s): H e l l o ! """Returns: reverse of s Precondition: s a string""" # 1. Handle base case left H e l l o ! # 2. Break into two parts right ! o l l e left = reverse(s[0]) right = reverse(s[1:]) # 3. Combine the result return A: left + right B: right + left C: left D: right 26

  27. How to Combine? (A) def reverse(s): H e l l o ! """Returns: reverse of s Precondition: s a string""" # 1. Handle base case left H e l l o ! # 2. Break into two parts right ! o l l e left = reverse(s[0]) right = reverse(s[1:]) CORRECT # 3. Combine the result return A: left + right B: right + left C: left D: right 27

  28. Example: Reversing a String def reverse(s): H e l l o ! """Returns: reverse of s Precondition: s a string""" # 1. Handle base case left H e l l o ! # 2. Break into two parts right ! o l l e left = reverse(s[0]) right = reverse(s[1:]) # 3. Combine the result return right+left 28

  29. What is the Base Case? (Q) def reverse(s): H e l l o ! """Returns: reverse of s Precondition: s a string""" # 1. Handle base case A: if s == "": B: if len(s) <= 2: C: if len(s) <= 1: return s return s return s # 2. Break into two parts D: Either A or C left = reverse(s[0]) would work right = reverse(s[1:]) # 3. Combine the result E: A, B, and C return right+left would all work 29

  30. What is the Base Case? (A) def reverse(s): H e l l o ! """Returns: reverse of s Precondition: s a string""" # 1. Handle base case CORRECT A: if s == "": B: if len(s) <= 2: C: if len(s) <= 1: return s return s return s # 2. Break into two parts D: Either A or C left = reverse(s[0]) would work right = reverse(s[1:]) # 3. Combine the result E: A, B, and C return right+left would all work 30

  31. Example: Reversing a String def reverse(s): """Returns: reverse of s Precondition: s a string""" # 1. Handle base case if len(s) <= 1: Base Case return s # 2. Break into two parts left = reverse(s[0]) s[0] right = reverse(s[1:]) Recursive Case # 3. Combine the result return right+left 31

  32. Alternate Implementation (Q) def reverse(s): """Returns: reverse of s Does this work? Precondition: s a string""" # 1. Handle base case A: YES if len(s) <= 1: return s B: NO # 2. Break into two parts half = len(s)//2 left = reverse(s[:half]) right = reverse(s[half:]) # 3. Combine the result return right+left 32

Recommend

Trees Last time: recursion In the last lecture, we learned about recursion &amp;

Trees Last time: recursion In the last lecture, we learned about recursion &amp;

Trees Last time: recursion In the last lecture, we learned about recursion &amp; divide-and-conquer Split the problem into smaller parts Solve each of the smaller parts separately: easier to code &amp; understand! Apply these

782 views • 38 slides
Recursion: Introduction and Correctness Lecture B Jones MWF 9-9:50am PCNYN 109 Lecture D

Recursion: Introduction and Correctness Lecture B Jones MWF 9-9:50am PCNYN 109 Lecture D

Recursion: Introduction and Correctness Lecture B Jones MWF 9-9:50am PCNYN 109 Lecture D Russell MWF 4-4:50am Center 101 http://cseweb.ucsd.edu/classes/sp16/cse21-bd/ April 11, 2016 Today's plan 1. What's recursion? 2. Correctness of

716 views • 40 slides
Recursion Contents Recursion in general Simple examples The need to move closer to a

Recursion Contents Recursion in general Simple examples The need to move closer to a

Lecture 4 Recursion Contents Recursion in general Simple examples The need to move closer to a solution List processing predicates Recursion in general Recursion is the idea of defining something in terms of itself. It

474 views • 15 slides
Foundations of Computer Science Lecture 7 Recursion Powerful but Dangerous Recursion and

Foundations of Computer Science Lecture 7 Recursion Powerful but Dangerous Recursion and

Foundations of Computer Science Lecture 7 Recursion Powerful but Dangerous Recursion and Induction Recursive Sets and Structures Last Time 1 With induction, it may be easier to prove a stronger claim. 2 Leaping induction. n 3 &lt; 2 n for

183 views • 16 slides
Recursion and Networking CS 118 Computer Network Fundamentals Peter Reiher Lecture 12 CS 118

Recursion and Networking CS 118 Computer Network Fundamentals Peter Reiher Lecture 12 CS 118

Recursion and Networking CS 118 Computer Network Fundamentals Peter Reiher Lecture 12 CS 118 Page 1 Winter 2016 Outline Preview and motivation What is recursion? The basic block concept Stacks, hourglasses, and DAGs Lecture

1.32k views • 69 slides
1 Recursive Definitions Infinite Recursion The recursive part of the LIST definition is used

1 Recursive Definitions Infinite Recursion The recursive part of the LIST definition is used

CSC 2014 Java Bootcamp Lecture 9 Recursion 2 Recursion Recursion is a fundamental programming technique that can provide an elegant solution certain kinds of problems RECURSIVE THINKING 3 4 Recursive Thinking Recursive Definitions

148 views • 4 slides
15-112 Fundamentals of Programming Week 5 - Lecture 2: Recursion June 14, 2016 What is

15-112 Fundamentals of Programming Week 5 - Lecture 2: Recursion June 14, 2016 What is

15-112 Fundamentals of Programming Week 5 - Lecture 2: Recursion June 14, 2016 What is recursion? Recursion: To understand recursion, you have to first understand recursion. What is recursion? Recursion: To understand recursion, you have

1.51k views • 118 slides
Lecture 15: Recursion (Sections 5.8-5.10) CS 1110 Introduction to Computing Using Python [E.

Lecture 15: Recursion (Sections 5.8-5.10) CS 1110 Introduction to Computing Using Python [E.

http://www.cs.cornell.edu/courses/cs1110/2019sp Lecture 15: Recursion (Sections 5.8-5.10) CS 1110 Introduction to Computing Using Python [E. Andersen, A. Bracy, D. Gries, L. Lee, S. Marschner, C. Van Loan, W. White] Recursion Recursive

488 views • 28 slides
15-112 Fundamentals of Programming Week 5 - Lecture 3: More Advanced Recursion June 15, 2016

15-112 Fundamentals of Programming Week 5 - Lecture 3: More Advanced Recursion June 15, 2016

15-112 Fundamentals of Programming Week 5 - Lecture 3: More Advanced Recursion June 15, 2016 Recursion vs Iteration 15-112 View Recursion Iteration - Elegance + - Performance + Debugability - + Memoization def fib(n): if (n &lt;

762 views • 62 slides
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

296 views • 11 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
Lecture Slides for MAT-60556 Part IX: Recursion theory Henri Hansen November 7, 2013 1

Lecture Slides for MAT-60556 Part IX: Recursion theory Henri Hansen November 7, 2013 1

Lecture Slides for MAT-60556 Part IX: Recursion theory Henri Hansen November 7, 2013 1 Recursion theory Recursion theory mostly characterizes computabil- ity . It is a study of functions that can be computed us- ing simple logical

654 views • 44 slides
CSE 105Theory of Computability Fall, 2006 Lecture 20November 30 Recursion Theorem

CSE 105Theory of Computability Fall, 2006 Lecture 20November 30 Recursion Theorem

CSE 105Theory of Computability Fall, 2006 Lecture 20November 30 Recursion Theorem Instructor: Neil Rhodes Recursion Theorem Informal idea: a Turing Machine can obtain its own description and compute with it Formally: Given a TM T

155 views • 4 slides
Principles of Programming CM10227 Lecture D.2.: Conditionals, Recursion, Interesting Functions

Principles of Programming CM10227 Lecture D.2.: Conditionals, Recursion, Interesting Functions

Conditionals More on Functions Recursion Principles of Programming CM10227 Lecture D.2.: Conditionals, Recursion, Interesting Functions Dr. Marina De Vos University of Bath Ext: 5053 Academic Year 2012-2013 Lecture D.2. (MDV) Programming

1.21k views • 76 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
Lecture #6: Recursion Last modified: Tue Feb 2 15:56:07 2016 CS61A: Lecture #6 1 Philosophy of

Lecture #6: Recursion Last modified: Tue Feb 2 15:56:07 2016 CS61A: Lecture #6 1 Philosophy of

Lecture #6: Recursion Last modified: Tue Feb 2 15:56:07 2016 CS61A: Lecture #6 1 Philosophy of Functions (I) Syntactic specification (signature) Precondition def sqrt(x): &quot;&quot;&quot;Assuming X &gt;= 0, returns approximation to square

792 views • 10 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
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
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
Objectives Basic Recursion Your goal for this lecture is to understand recursion at least, to

Objectives Basic Recursion Your goal for this lecture is to understand recursion at least, to

1 in aa + a aa 1 a in aa + a let aa = a * a let aa = a * a Objectives Function Calls Iterative Recursion Recursion and Lists Objectives Function Calls Iterative Recursion Recursion and Lists Objectives Basic Recursion Your goal for

286 views • 5 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
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

More recommend