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Recursion II Fundamentals of Computer Science Outline Recursion A method calling itself A new way of thinking about a problem A powerful programming paradigm Examples: Last time: Factorial, binary search, H-tree,

  1. Recursion II Fundamentals of Computer Science

  2. Outline  Recursion  A method calling itself  A new way of thinking about a problem  A powerful programming paradigm  Examples:  Last time:  Factorial, binary search, H-tree, Fibonacci  Today:  Greatest Common Divisor (GCD)  Brownian Motion  Sorting

  3. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) return; //Pop mystery(n - 1); //Push mystery(n - 2); //Push } //Pop public static void main(String[] args) { mystery(3); //Push } //Pop }

  4. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) return; //Pop mystery(n - 1); //Push mystery(n - 2); //Push 3 } //Pop mystery(3) public static void main(String[] args) { mystery(3); //Push } //Pop }

  5. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) return; //Pop mystery(n - 1); //Push 2 mystery(n - 2); //Push mystery(3-1) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push } //Pop }

  6. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) 1 return; //Pop mystery(n - 1); //Push mystery(2-1) 2 mystery(n - 2); //Push mystery(3-1) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop }

  7. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); 0 if (n <= 0) mystery(1-1) 1 return; //Pop mystery(n - 1); //Push mystery(2-1) 2 mystery(n - 2); //Push mystery(3-1) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 }

  8. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); 0 if (n <= 0) mystery(1-1) 1 return; //Pop mystery(n - 1); //Push mystery(2-1) 2 mystery(n - 2); //Push mystery(3-1) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0

  9. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) 1 return; //Pop mystery(n - 1); //Push mystery(2-1) 2 mystery(n - 2); //Push mystery(3-1) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0

  10. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); -1 if (n <= 0) mystery(1-2) 1 return; //Pop mystery(n - 1); //Push mystery(2-1) 2 mystery(n - 2); //Push mystery(3-1) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0

  11. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); -1 if (n <= 0) mystery(1-2) 1 return; //Pop mystery(n - 1); //Push mystery(2-1) 2 mystery(n - 2); //Push mystery(3-1) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0

  12. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) 1 return; //Pop mystery(n - 1); //Push mystery(2-1) 2 mystery(n - 2); //Push mystery(3-1) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0 -1

  13. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) return; //Pop mystery(n - 1); //Push 2 mystery(n - 2); //Push mystery(3-1) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0 -1

  14. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) 0 return; //Pop mystery(n - 1); //Push mystery(2-2) 2 mystery(n - 2); //Push mystery(3-1) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0 -1

  15. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) return; //Pop mystery(n - 1); //Push 2 mystery(n - 2); //Push mystery(3-1) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0 -1 0

  16. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) return; //Pop mystery(n - 1); //Push mystery(n - 2); //Push 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0 -1 0

  17. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) return; //Pop mystery(n - 1); //Push 1 mystery(n - 2); //Push mystery(3-2) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0 -1 0

  18. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) 0 return; //Pop mystery(n - 1); //Push mystery(1-1) 1 mystery(n - 2); //Push mystery(3-2) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0 -1 0 1

  19. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) return; //Pop mystery(n - 1); //Push 1 mystery(n - 2); //Push mystery(3-2) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0 -1 0 1 0

  20. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) -1 return; //Pop mystery(n - 1); //Push mystery(1-2) 1 mystery(n - 2); //Push mystery(3-2) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0 -1 0 1 0

  21. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) return; //Pop mystery(n - 1); //Push 1 mystery(n - 2); //Push mystery(3-2) 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0 -1 0 1 0

  22. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) return; //Pop mystery(n - 1); //Push mystery(n - 2); //Push 3 } //Pop mystery(3) public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0 -1 0 1 0

  23. Recursion Walkthrough public class RecursiveMystery Call Stack n { public static void mystery(int n) { System. out.println(n); if (n <= 0) return; //Pop mystery(n - 1); //Push mystery(n - 2); //Push } //Pop public static void main(String[] args) { 3 mystery(3); //Push 2 } //Pop 1 } 0 -1 0 1 0

  24. Brownian Motion  Models many natural and artificial phenomenon  Motion of pollen grains in water  Price of stocks  Rugged shapes of mountains and clouds

  25. Simulating Brownian Motion  Midpoint displacement method:  Track interval (x 0 , y 0 ) to (x 1 , y 1 )  Choose d displacement randomly from Gaussian  Divide in half, x m = (x 0 +x 1 )/2 and y m = (y 0 +y 1 )/2 + d  Recur on the left and right intervals

  26. Recursive Midpoint Displacement Algorithm void curve( double x0, double y0, double x1, double y1, double var) { if (x1 - x0 < .005) { StdDraw. line(x0, y0, x1, y1); base case return ; } double xm = (x0 + x1) / 2.0; double ym = (y0 + y1) / 2.0; ym = ym + StdRandom. gaussian(0, Math.sqrt(var)); curve(x0, y0, xm, ym, var / 2.0); reduction step curve(xm, ym, x1, y1, var / 2.0); }

  27. Plasma Cloud  Same idea, but in 2D  Each corner of square has some color value  Divide into four sub-squares  New corners: avg of original corners, or all 4 + random  Recur on four sub-squares

  28. 28

  29. Brownian Landscape 29

  30. Divide and Conquer  Divide and conquer paradigm  Break big problem into small sub-problems  Solve sub-problems recursively  Combine results “ Divide et impera. Vendi, vidi, vici. ” -Julius Caesar  Used to solve many important problems  Sorting things, mergesort: O(N log N)  Parsing programming languages  Discrete FFT, signal processing  Multiplying large numbers  Traversing multiply linked structures (stay tuned)

  31. Divide and Conquer: Sorting  Goal: Sort by number, ignore suit, aces high Approach 1) Split in half (or as close as possible) 2) Give each half to somebody to sort 3) Take two halves and merge together Unsorted pile #1 Unsorted pile #2

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