Logic, Algorithms and Data Structures Recursion and Stacks. By: Jonas Öberg

Transcription

1 Logic, Algorithms and Data Structures Recursion and Stacks M1 By: Jonas Öberg

2 What is recursion? Quick answer: A recursive function is a function which uses itself Example We define by the use of f( n ) = 2 f (n -1) if n >= 1 otherwise 1

3 What is recursion? What does it do?? f(3) { 3 >= 1 } f(3) = 2 * f(3-1) = 2 * f(2) f(3) = 2 * f(2) f(2) { 2 >= 1 } f(2) = 2 * f(2-1) = 2 * f(1) f(3) = 2 * 2 * f(1) f(1) { 1 >= 1} f(1) = 2 * f(1-1) = 2 * f(0) f( n ) = 2 f (n -1) if n >= 1 1 otherwise f(3) = 2 * 2 * 2 * f(0) f(0) { 0 >= 1 } f(0) = 1 f(3) = 2 * 2 * 2 * 1 f(3) = 8

4 How do we write recursively? Problem: Write a function that computes the smallest element in a set. Sample data: A = { 7, 3, 5, 9, 2, 12 } Stated goal: f({7, 3, 5, 9, 2, 12}) = 2

5 How to think Assume that the function already does its job for a subset of A, ie. A-{x} f (A-{x}) Our parts: f (A-{x}) and x To this, we only need to add some glue and/or gaff tape

6 How to think Given: A = {7, 3, 5, 9, 2, 12} x = 7 We assume that: f(a-{x}) = 2 We want a function such that: f(a) = f(a-{x})... x = 2

7 How to think So, we want to find a function which solves: f(a) = f(a-{x})... x = 2 f(a) = = 2 This is what we wanted: write a function that computes the smallest element in a set. f(a) = min(2, 7) = 2

8 How to think min(x, y) = if x <= y then x else y So, min(x, y) will be our glue! f(a) = min(f(a-{x}), x)

9 How to think f( A ) = min(f (A-{x}), x) But there's something missing here!

10 The Base Case Computing f({3, 5, 7}) f({3, 5, 7}) = min(3, f({5, 7})) = min(3, min(5, f({7}))) = min(3, min(5, min(7, f({})))) = f ( A ) = min( x, f(a {x})) min(3, min(5, min(7, min({}, f({}))))) = min(3, min(5, min(7, min({}, min({}, f({})))))))... We need to define f({}) This is called the Base Case

11 Implementing the base case What's the smallest value of an empty set? f( A ) =, if A = {} min(f (A-{x}), x), otherwise

12 Implementation f( A ) =, if A = {} min(f (A-{x}), x), otherwise int setmin(set a) { if (a.isempty()) { return Integer.MAX_VALUE; } } Iterator<Integer> i = a.iterator(); int x = i.next(); i.remove(); Return Math.min(x, setmin(a));

13 Computing faculty f(5) = 5 * 4 * 3 * 2 * 1 = 120 Assume that f(4) = 24 already works Then, f(5) = 5 * f(4) f(x) = x * f(x-1)

14 Faculty base case f( x ) = 1, if x = 1 x * f (x-1), otherwise int faculty(int x) { if (x == 1) { return 1; } } return x * faculty(x - 1); f(5) = 5 * f(4) = 5 * 4 * f(3) = 5 * 4 * 3 * f(2) = 5 * 4 * 3 * 2 * f(1) = 5 * 4 * 3 * 2 * 1 = 120

15 Stacks

16 When is the stack used? main() function(1, Kalle, 34); function(a, b, c) List l = new ArrayList(); l.add(b); 1 Kalle 34 b Stack

17 Stacks when recursing f( x ) = 1, if x = 1 x * f (x-1), otherwise faculty(5) 5 * faculty(4) 5 * 4 * faculty(3) 5 * 4 * 3 * faculty(2) 5 * 4 * 3 * 2 * faculty(1) 5 * 4 * 3 * 2 * Stack

18 public class RecHelloWorld { public static void write_string (String s) { System.out.printf("\n%s\n", s); write_string(s); } public static void main(string[] args) { write_string("hello World"); } } Hello World Hello World Hello World Hello World Hello World Exception in thread "main" java.lang.stackoverflowerror

19 Tower of Hanoi

20 Algorithm Total of rings = n Step 1: Move rings n-1 to Help tower Step 2: Move ring n to Goal tower Step 3: Move rings n-1 to Goal tower

21 Algorithm Base case: moving ring 1 from Start to Goal Assume that we have a function to move rings n-1. Solve the problem for moving ring n.

22 Algorithm public static void move(int nr, String from, String to) { System.out.printf("%2d: Moving ring nr %d from %s to %s\n", i, nr, from, to); }

23 Algorithm public static void tower(int nr, String start, String goal, String help) { if (nr == 1) //Base case move(1, start, goal); else { tower(nr 1, start, help, goal); move(nr, start, goal); tower(nr 1, help, goal, start); } } Step 1: Move rings n-1 to Help tower Step 2: Move ring n to Goal tower Step 3: Move rings n-1 to Goal tower

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25 public class Hanoi { public static int i = 0; // global variable to keep track of method calls public static void move(int nr, String from, String to) { i++; System.out.printf("%2d: Moving ring nr %d from %s to %s\n", i, nr, from, to); } public static void tower(int nr, String start, String goal, String help) { if (nr == 1) //Base case move(1, start, goal); else { tower(nr 1,start, help, goal); move(nr, start, goal); tower(nr 1, help, goal, start); } } } public static void main(string[] args) { int nr = Integer.parseInt(args[0]); tower(nr,"start Tower", "Goal Tower", "Help Tower"); }

26 java Hanoi 3 1: Moving ring nr 1 from Start Tower to Goal Tower 2: Moving ring nr 2 from Start Tower to Help Tower 3: Moving ring nr 1 from Goal Tower to Help Tower 4: Moving ring nr 3 from Start Tower to Goal Tower 5: Moving ring nr 1 from Help Tower to Start Tower 6: Moving ring nr 2 from Help Tower to Goal Tower 7: Moving ring nr 1 from Start Tower to Goal Tower

27 Call tree

28 Labyrinth

29 Code snippet

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31 RPN calculator * 72 / Stack

32 RPN calculator * * Stack

33 Koch snowflakes 1. Start with an equilateral triangle 2. Divide each line in turn in three parts 3. Draw another equilateral triangle extending from the middle 4. Repeat 2

34 Fractal grammar F = Move forward + = turn left 60 degrees - = turn right 60 degrees

35 Turtle graphics Imagine that you're a turtle! F + + F + + F

36 Fractal grammar F = Move forward + = turn left 60 degrees - = turn right 60 degrees Koch snowflake Axiom: F + + F + + F Rules: F := F F + + F - F

37 Alternatives to Koch Cesaro fractal

38 Serpinski triangle A, B = Move forward + = turn left 60 degrees - = turn right 60 degrees Serpinski triangle Axiom: A Rules: A := B A B and B := A + B + A

39 Fractal plant F = Move forward -/+ = turn right/left 25 degrees X = noop [ = save position and angle to stack ] = restore position and angle from stack Fractal plant Axiom: X Rules: X := F-[[X]+X]+F[+FX]-X and F := FF

40 Fractal plant result

41 Of course, there's work to do..