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2.3 Recursion

2.3 Recursion. M. C. Escher, 1948. Overview. What is recursion? When one function calls itself directly or indirectly. Why learn recursion? Occurs in nature New way of thinking about programming Powerful programming paradigm. Many computations are naturally self-referential.

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2.3 Recursion

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  1. 2.3 Recursion M. C. Escher, 1948

  2. Overview What is recursion? When one function calls itself directly or indirectly. Why learn recursion? • Occurs in nature • New way of thinking about programming • Powerful programming paradigm. Many computations are naturally self-referential. • Mergesort, FFT, gcd. • A folder contains files and other folders. Closely related to mathematical induction. M. C. Escher, 1948

  3. Greatest Common Divisor Gcd. Find largest integer that evenly divides into p and q. Ex. gcd(4032, 1272) = 24. Applications. • Simplify fractions: 1272/4032 = 53/168. • RSA cryptosystem. 4032 = 26 32 711272 = 23 31 531 gcd = 23 31 = 24

  4. Greatest Common Divisor Gcd. Find largest integer that evenly divides into p and q. Euclid's algorithm. [Euclid 300 BCE] base case reduction step,converges to base case gcd(4032, 1272) = gcd(1272, 216) = gcd(216, 192) = gcd(192, 24) = gcd(24, 0) = 24. 4032= 3  1272+216

  5. Greatest Common Divisor Gcd. Find largest integer d that evenly divides into p and q. Java implementation. base case reduction step,converges to base case public staticint gcd(int p,int q) { if(q ==0)return p; elsereturn gcd(q, p % q); } base case reduction step

  6. How-To’s on Writing Recursive Functions Base Case: • You must check if we’ve reached the base case before doing another level of recursion! Make Progress Towards Base Case: • Your recursive calls must be on a smaller or simpler input. • Eventually this must reach the base case (and not miss it). Multiple recursive calls: • Sometimes more than one recursive call. Sometimes not. • Are their return values chosen, combined? Learn and follow these rules carefully!

  7. Recursive Graphics

  8. H-tree Used in integrated circuits to distribute the clock signal. H-tree of order n. • Draw an H. • Recursively draw 4 H-trees of order n-1, one connected to each tip. tip and half the size size order 3 order 1 order 2

  9. Htree in JavaDr. Java Demo public class Htree { public static void draw(int n, double sz, double x, double y) { if (n == 0) return; double x0 = x - sz/2, x1 = x + sz/2; double y0 = y - sz/2, y1 = y + sz/2; StdDraw.line(x0, y, x1, y); StdDraw.line(x0, y0, x0, y1); StdDraw.line(x1, y0, x1, y1); draw(n-1, sz/2, x0, y0); draw(n-1, sz/2, x0, y1); draw(n-1, sz/2, x1, y0); draw(n-1, sz/2, x1, y1); } public static void main(String[] args) { int n = Integer.parseInt(args[0]); draw(n, .5, .5, .5); } } draw the H, centered on (x, y) recursively draw 4 half-size Hs

  10. Computational Cost of the H-Tree Program • H-tree of Order 1 • First invocation of draw() calls itself 4 times • H-tree of Order 2?

  11. Towers of Hanoi http://en.wikipedia.org/wiki/Image:Hanoiklein.jpg

  12. Towers of Hanoi Move all the discs from the leftmost peg to the rightmost one. • Only one disc may be moved at a time. • A disc can be placed either on empty peg or on top of a larger disc. start finish Towers of Hanoi demo Edouard Lucas (1883)

  13. Move largest disc left. cyclic wrap-around Towers of Hanoi: Recursive Solution Move n-1 smallest discs right. Move n-1 smallest discs right.

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