Recursive Methods Noter ch.2 Recursive Methods Recursive problem - - PowerPoint PPT Presentation

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Recursive Methods Noter ch.2 Recursive Methods Recursive problem - - PowerPoint PPT Presentation

Aug 08, 2023 182 likes •798 views

Recursive Methods Noter ch.2 Recursive Methods Recursive problem solution Problems that are naturally solved by recursion Examples: Recursive function: Fibonacci numbers Recursive graphics: Fractals Mutual recursion:

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SLIDE 1

Recursive Methods

Noter ch.2

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SLIDE 2

Recursive Methods

  • Recursive problem solution

– Problems that are naturally solved by recursion

  • Examples:

– Recursive function: Fibonacci numbers – Recursive graphics: Fractals – Mutual recursion: Expression evaluation – Randomization and recursion: Random plants/trees

  • General aspects

– Termination – Recursion versus iteration: simplicity vs efficiency

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SLIDE 3

Recursion in trees and flowers

Tree

=

3 smaller trees/branches trunk

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SLIDE 4

Natural Recursion: Derivative of Rational Function

  • →
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SLIDE 5

QUIZ

What is the proper recursive definition of the factorial function? (Assume n ≥ 1) 1. n! = n * (n-1)! 2. n! = 3. n! = 4. n! = 5. n! = 6. None of the above 7. I don’t know

Factorial function: n! = 1 * 2 * 3 * ... * (n-1) * n (n-1)! For n > 1 1 For n = 1 (n-1)! For n > 1 n-1 For n = 1 n*(n-1)! For n > 1 1 For n = 1 n*(n-1) For n > 1 (n-1)! For n = 1 Recursive definition

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SLIDE 6

Factorial function

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

n! = Java method for computing factorial function: n*(n-1)! For n > 1 1 For n ≤ 1

public static long factorial(int n) { if (n<=1) return 1; else return n*factorial(n-1); }

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SLIDE 7

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3

1.call

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SLIDE 8

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3 result =

1.call

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SLIDE 9

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3 result = recursiveCall =

1.call

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SLIDE 10

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3 result = recursiveCall = 2nd call: n = 2

1.call 2.call

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SLIDE 11

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3 result = recursiveCall = 2nd call: n = 2 result =

1.call 2.call

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SLIDE 12

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3 result = recursiveCall = 2nd call: n = 2 result = recursiveCall =

1.call 2.call

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SLIDE 13

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3 result = recursiveCall = 2nd call: n = 2 result = recursiveCall = 3rd call: n = 1

1.call 2.call 3.call

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SLIDE 14

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3 result = recursiveCall = 2nd call: n = 2 result = recursiveCall = 3rd call: n = 1 result =

1.call 2.call 3.call

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SLIDE 15

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3 result = recursiveCall = 2nd call: n = 2 result = recursiveCall = 3rd call: n = 1 result = 1

1.call 2.call 3.call

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SLIDE 16

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3 result = recursiveCall = 2nd call: n = 2 result = recursiveCall = 1 3rd call: (finished) n = 1 result = 1

1.call 2.call 3.call

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SLIDE 17

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3 result = recursiveCall = 2nd call: n = 2 result = 2 recursiveCall = 1 3rd call: (finished) n = 1 result = 1

1.call 2.call 3.call

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SLIDE 18

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3 result = recursiveCall = 2 2nd call: (finished) n = 2 result = 2 recursiveCall = 1 3rd call: (finished) n = 1 result = 1

1.call 2.call 3.call

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SLIDE 19

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: n = 3 result = 6 recursiveCall = 2 2nd call: (finished) n = 2 result = 2 recursiveCall = 1 3rd call: (finished) n = 1 result = 1

1.call 2.call 3.call

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SLIDE 20

public static long factorial(int n) { long result; if (n<=1) { result = 1; } else { long recursiveCall = factorial(n-1); result = n*recursiveCall; } return result; }

1st call: (finished) n = 3 result = 6 recursiveCall = 2 2nd call: (finished) n = 2 result = 2 recursiveCall = 1 3rd call: (finished) n = 1 result = 1

1.call 2.call 3.call

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SLIDE 21

QUIZ

public static void foo(int x) { System.out.println(x); if (x>1) foo(x-1); } What is the result of the method call foo(4) ? 1. Prints 4 2. Prints 1 3. Prints 1 2 3 4 4. Prints 4 3 2 1 5. Prints something else 6. Compiler error: method foo is not allowed to call itself 7. Runtime error: method foo is not allowed to call itself 8. I don’t know Recursive method call

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SLIDE 22

Recursive Function: Fibonacci Numbers

  • The sequence of Fibonacci numbers is
  • First 10 terms are

– 1, 1, 2, 3, 5, 8, 13, 21, 34, 55

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SLIDE 23

Recursive Function: Fibonacci Numbers

  • Recursive method corresponding to recursive

definition of Fibonacci numbers

public long fib(int n) { if (n <= 1) return 1; else return fib(n - 1) + fib(n - 2); }

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SLIDE 24

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4

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SLIDE 25

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 2nd call: n = 3

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SLIDE 26

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 2nd call: n = 3

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SLIDE 27

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 2nd call: n = 3 4th call: n = 1

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SLIDE 28

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 2nd call: n = 3 4th call: n = 1 returns 1

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SLIDE 29

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 2nd call: n = 3 5th call: n = 0 4th call: n = 1 returns 1

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SLIDE 30

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 2nd call: n = 3 5th call: n = 0 returns 1 4th call: n = 1 returns 1

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SLIDE 31

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 returns 2 2nd call: n = 3 5th call: n = 0 returns 1 4th call: n = 1 returns 1

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SLIDE 32

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 returns 2 2nd call: n = 3 6th call: n = 1 5th call: n = 0 returns 1 4th call: n = 1 returns 1

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SLIDE 33

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 returns 2 2nd call: n = 3 6th call: n = 1 returns 1 5th call: n = 0 returns 1 4th call: n = 1 returns 1

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SLIDE 34

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 returns 2 2nd call: n = 3 returns 3 6th call: n = 1 returns 1 5th call: n = 0 returns 1 4th call: n = 1 returns 1

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SLIDE 35

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 returns 2 7th call: n = 2 2nd call: n = 3 returns 3 6th call: n = 1 returns 1 5th call: n = 0 returns 1 4th call: n = 1 returns 1

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SLIDE 36

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 returns 2 7th call: n = 2 2nd call: n = 3 returns 3 6th call: n = 1 returns 1 8th call: n = 1 5th call: n = 0 returns 1 4th call: n = 1 returns 1

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SLIDE 37

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 returns 2 7th call: n = 2 2nd call: n = 3 returns 3 6th call: n = 1 returns 1 8th call: n = 1 returns 1 5th call: n = 0 returns 1 4th call: n = 1 returns 1

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SLIDE 38

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 returns 2 7th call: n = 2 2nd call: n = 3 returns 3 6th call: n = 1 returns 1 9th call: n = 0 8th call: n = 1 returns 1 5th call: n = 0 returns 1 4th call: n = 1 returns 1

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SLIDE 39

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 returns 2 7th call: n = 2 2nd call: n = 3 returns 3 6th call: n = 1 returns 1 9th call: n = 0 returns 1 8th call: n = 1 returns 1 5th call: n = 0 returns 1 4th call: n = 1 returns 1

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SLIDE 40

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 3rd call: n = 2 returns 2 7th call: n = 2 returns 2 2nd call: n = 3 returns 3 6th call: n = 1 returns 1 9th call: n = 0 returns 1 8th call: n = 1 returns 1 5th call: n = 0 returns 1 4th call: n = 1 returns 1

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SLIDE 41

Structure of method calls

public long fib(int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); }

1st call: n = 4 returns 5 3rd call: n = 2 returns 2 7th call: n = 2 returns 2 2nd call: n = 3 returns 3 6th call: n = 1 returns 1 9th call: n = 0 returns 1 8th call: n = 1 returns 1 5th call: n = 0 returns 1 4th call: n = 1 returns 1

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SLIDE 42

Tree Structure of Method Call fib(4)

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Thinking recursively vs efficiency

  • Recursive solution may be natural and simple to program

public long fib(int n) { if (n <= 1) return 1; else return fib(n - 1) + fib(n - 2); }

  • But iterative solution may be more efficient

public long iterativeFib(int n) { if (n <= 1) return 1; long fold = 1; long fold2 = 1; long fnew = 1; for (int i = 2; i <= n; i++) { fnew = fold + fold2; fold2 = fold; fold = fnew; } return fnew; }

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SLIDE 44

QUIZ

/* @precondition n>=1 */ public int area(int n) { } a) return n*(n+1)/2; b) if (n==1) return 1; else return area(n-1); c) return n+area(n-1); d) if (n==1) return 1; else return n+area(n-1); n Implement recursion Which lines should replace to

  • btain a correct and recursive

computation of triangle area ? (for n=4, the area is 10) 1. a 2. b 3. c 4. d 5. None of a,b,c,d 6. More than one of a,b,c,d could be used 7. I don’t know

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SLIDE 45

Recursive graphics

  • The snowflake is built

from 3 Koch lines of

  • rder 4.
  • The Koch line is

defined recursively

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SLIDE 46

Fractals: Koch lines of order 1-4

  • A line of order 0 is a

straight line

  • A line of order n

consists of 4 lines of

  • rder n-1 each of 1/3

length

n>=1

Order n−1 Order n−1 Order n−1 Order n−1 Order

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Crayon class

  • State of Crayon object

– Colour – Width (pixel) – Position (coordinates) – Direction (0-360 deg)

  • Public methods

– Relative changes – move(d): draw line of lenght d (from position in direction to new position) – jump(d): change position without drawing – turn(a): add a degrees to direction – Absolute changes – moveto(x,y): – jumpto(x,y): – turnto(a):

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KochLineTest

public class KochLineTest { public static void main(String[] args) { c = new Crayon(Color.red,1); c.jumpto(50,150); kochLine(4,300); c.turn(120); kochLine(4,300); c.turn(120); kochLine(4,300); c.turn(120); } private static Crayon c; private static void kochLine(int order, double len) {...} }

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SLIDE 49

recursive method kochLine

void kochLine(int order, double len) { if (order == 0) c.move(len); else if (order > 0) { kochLine(order-1,len/3); c.turn(-60); kochLine(order-1,len/3); c.turn(120); kochLine(order-1,len/3); c.turn(-60); kochLine(order-1,len/3); } } n>=1

Order n−1 Order n−1 Order n−1 Order n−1 Order

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SLIDE 50

QUIZ

a b Sierpinski Triangle Which of these recursions can be used to draw a Sierpiski Triangle? 1. Neither 2. a 3. b 4. c 5. a+b 6. b+c 7. a+c 8. a+b+c 9. I don’t know c Order 0 Order 1

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SLIDE 51

Recursion in general: Termination

  • Recursive call need not terminate

public void loopingRecursiveMethod() { loopingRecursiveMethod(); }

  • Any reasonable recursive method should have recursion

condition

public long fib(int n) { if (n <= 1) return 1; else return fib(n - 1) + fib(n - 2); } private void kochLine(int order, double len) { if (order == 0) no recursive call else if (order > 0) recursive call } no recursive call recursive call

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Mutual Recursion and termination

  • up and down are mutually recursive:

public void down(int n) { while(n%2==0) n = n/2; up(n); } public void up(int n) { if (n>1) { n = 3*n+1; down(n); } }

  • It is unknown whether the call down(m) terminates for all m! (Collatz

conjecture, 1937)

  • A recursion condition does not necessarily ensure termination
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SLIDE 53

QUIZ

Precondition for both methods: n >= 0 public boolean even(int n) { if (n==0) return true; return odd(n-1); } public boolean odd(int n) { if (n==0) return false; return even(n-1); } Which assertions are correct? 1. Both odd and even always stops 2.

  • dd always stops, but even may loop indefinitely

3. even always stops, but odd may loop indefinitely 4. Both odd and even may loop indefinitely 5. I don’t know Mutual recursion and termination

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Random trees and flowers

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Random trees and flowers

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Recursive trees

1 leaf/branch/trunk Order 2 6 5 4 3

  • leaf, branch, trunk differ by colors and thickness

Tree

=

3 smaller trees/branches trunk

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Variation by randomization

  • Each tree consists of a trunk and up to

BRANCH_MAX=6 smaller trees

  • Each of the subtrees are drawn with probability

BRANCH_PROB=0.4

  • Subtrees are tilted with angle to neighbor being

BRANCH_ANGLE=13 deg.

Order n−1 Order n−1 Order trunk Order n−1 n>=1 BRANCH_ANGLE leaf

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SLIDE 58

class CCrayon

  • CCrayon is a variant of Crayon with

2 extra methods:

– setColor – setWidth

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class TreeTest

public class TreeTest { public static void main(String[] args) { c = new CCrayon(); c.jumpto(200,400); c.turn(-90); tree(6,40); } private static CCrayon c; private static void tree(int order, int len) {...} private static void leaf(int len) { ... } private final static int BRANCH_MAX = 6; private final static int BRANCH_ANGLE = 13; private final static double BRANCH_PROB = 0.4; private final static double LEAF_PROB = 0.3; }

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SLIDE 60

Recursive method tree

public void tree(int order, int len) { if (order==0) leaf(len/2); else { int bias = (int) (2*Math.random()); if (order+bias >=6) c.setColor(Color.black); else if (order+bias >=4) c.setColor(Color.gray); else c.setColor(Color.green); c.setWidth(2*order); c.move(len); c.turn((BRANCH_ANGLE*(BRANCH_MAX-1))/2.0); for (int i = 1 ; i<=BRANCH_MAX ; i = i+1 ) { if (Math.random()<BRANCH_PROB) tree(order-1, len-2); c.turn(-BRANCH_ANGLE); } c.turn((BRANCH_ANGLE*(BRANCH_MAX+1))/2.0); c.turn(180); c.jump(len); c.turn(-180); //return pen to base of tree } }

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SLIDE 61

Method leaf

void leaf(int len) { if (Math.random()<LEAF_PROB) { if (Math.random()<0.5) c.setColor(Color.red); else c.setColor(Color.yellow); c.setWidth(2); c.turn(-BRANCH_ANGLE/2.0); c.move(len); c.turn(BRANCH_ANGLE); c.move(len); c.turn(180-BRANCH_ANGLE); c.move(len); c.turn(BRANCH_ANGLE); c.move(len); c.turn(180-BRANCH_ANGLE/2.0); } }