Lecture 4: Recursive Definitions CS200: Computer Science University of Virginia Computer Science David Evans http://www.cs.virginia.edu/evans Menu Defining Recursive Procedures Problem Set 1 Problem Set 2 24 January 2003

CS 200 Spring 2003 2 Defining Recursive Procedures 1. Be optimistic. Assume you can solve it. If you could, how would you solve a bigger problem. 2. Think of the simplest version of the problem, something you can already solve. (This is the base case.) 3. Combine them to solve the problem. 24 January 2003

CS 200 Spring 2003 3 Example Define (find-closest goal numbers) that evaluates to the number in the list numbers list that is closest to goal: > (find-closest 200 (list 101 110 120 201 340 588)) 201 > (find-closest 12 (list 1 11 21)) 11 > (find-closest 12 (list 95)) 95 24 January 2003 CS 200 Spring 2003

4 Find Closest Number Be optimistic! Assume you can define: (find-closest-number goal numbers) that finds the closest number to goal from the list of numbers. What if there is one more number? Can you write a function that finds the closest number to match from newnumber and numbers? 24 January 2003 CS 200 Spring 2003 5

Find Best Match Strategy: If the new number is better, than the best match with the other number, use the new number. Otherwise, use the best match of the other numbers. 24 January 2003 CS 200 Spring 2003 6 Optimistic Function (define (find-closest goal numbers) (if (< (abs (- goal (first numbers)))

(abs (- goal (find-closest goal (rest numbers))))) (first numbers) (find-closest goal (rest numbers)))) 24 January 2003 CS 200 Spring 2003 7 Defining Recursive Procedures 2. Think of the simplest version of the problem, something you can already solve. If there is only one number, that is the

best match. 24 January 2003 CS 200 Spring 2003 8 The Base Case Same as before (define (find-closest goal numbers) (if (= 1 (length numbers)) (first numbers) (if (< (abs (- goal (first numbers))) (abs (- goal

(find-closest goal (rest numbers))))) (first numbers) (find-closest goal (rest numbers)))) 24 January 2003 CS 200 Spring 2003 9 Testing (define (find-closest goal numbers) (if (= 1 (length numbers)) (first numbers)

(if (< (abs (- goal (first numbers))) (abs (- goal (find-closest goal (rest numbers))))) (first numbers) (find-closest goal (rest numbers)))) > (find-closest-number 200 (list 101 110 120 201 340 588)) 201 > (find-closest-number 0 (list 1)) 1 > (find-closest-number 0 (list )) first: expects argument of type ; given () 24 January 2003

CS 200 Spring 2003 10 Seen Anything Like This? define (find-best-match sample tiles color-comparator) (if (null? tiles) ;;; If there are no tiles, (error "No tiles to match!") ;;; we must lose. (if (= (length tiles) 1) ;;; If there is just one tile, (first tiles) ;;; that tile is the best match. (pick-better-match ;;; Otherwise, the best match is

sample ;;; either the first tile in tiles, (first tiles) ;;; or the best match we would find (find-best-match ;;; from looking at the rest of the sample ;;; tiles. Use pick-better-match (rest tiles) ;;; to determine which one is better. color-comparator) color-comparator)))) 24 January 2003 CS 200 Spring 2003

11 Fibonaccis Problem Filius Bonacci, 1202 in Pisa: Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year? 24 January 2003 CS 200 Spring 2003 12

Rabbits From http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html 24 January 2003 CS 200 Spring 2003 13 Fibonacci Numbers GEB p. 136: These numbers are best defined recursively by the pair of formulas FIBO (n) = FIBO (n 1) + FIBO (n 2) for n > 2

FIBO (1) = FIBO (2) = 1 Can we turn this into a Scheme function? Note: SICP defines Fib with Fib(0)= 0 and Fib(1) = 1 for base case. Same function except for Fib(0) is undefined in GEB version. 24 January 2003 CS 200 Spring 2003 14 Defining Recursive Procedures Slide 3 Returns 1. Be optimistic. Assume you can solve it. If you could, how would you solve a bigger

problem. 2. Think of the simplest version of the problem, something you can already solve. (This is the base case.) 3. Combine them to solve the problem. 24 January 2003 CS 200 Spring 2003 15 Defining FIBO 1. Be optimistic - assume you can solve it, if you could, how would you solve a bigger problem.

2. Think of the simplest version of the problem, something you can already solve. 3. Combine them to solve the problem. 24 January 2003 These numbers are best defined recursively by the pair of formulas FIBO (n) = FIBO (n 1) + FIBO (n 2) for n > 2 FIBO (1) = FIBO (2) = 1

CS 200 Spring 2003 16 Defining fibo ;;; (fibo n) evaluates to the nth Fibonacci ;;; number (define (fibo n) FIBO (1) = FIBO (2) = 1 (if (or (= n 1) (= n 2)) 1 ;;; base case FIBO (n) = FIBO (n 1) (+ (fibo (- n 1)) + FIBO (n 2) (fibo (- n 2))))) for n > 2

24 January 2003 CS 200 Spring 2003 17 Fibo Results > (fibo 2) 1 Why cant our > (fibo 3) 100,000x Apollo 2 Guidance > (fibo 4) 3

Computer calculate > (fibo 10) (fibo 100)? 55 > (fibo 100) Still working after 4 hours To be continued Monday (answer is in SICP, 1.2) 24 January 2003 CS 200 Spring 2003 18 Problem Set 1 24 January 2003

CS 200 Spring 2003 19 Question 2 Without Evaluation Rules, Question 2 was guesswork Now you know the Evaluation Rules, you can answer Question 2 without any guessing! 24 January 2003 CS 200 Spring 2003 20

2d (100 + 100) Evaluation Rule 3. Application. a. Evaluate all the subexpressions 100 100 b. Apply the value of the first subexpression to the values of all the other subexpressions Error: 100 is not a procedure, we only have apply rules for procedures! 24 January 2003 CS 200 Spring 2003

21 2h (if (not "cookies") "eat" "starve") Evaluation Rule 4-if. Evaluate Expression0. If it evaluates to #f, the value of the if expression is the value of Expression2. Otherwise, the value of the if expression is the value of Expression1. Evaluate (not "cookies") 24 January 2003 CS 200 Spring 2003 22 Evaluate (not "cookies")

Evaluation Rule 3. Application. a. Evaluate all the subexpressions cookies The quotes really matter here! Without them what would cookies evaluate to? b. Apply the value of the first subexpression to the values of all the other subexpressions (not v) evaluates to #t if v is #f, otherwise it evaluates to #f. (SICP, p. 19) So, (not cookies) evaluates to #f 24 January 2003 CS 200 Spring 2003 23

Defining not (not v) evaluates to #t if v is #f, otherwise it evaluates to #f. (SICP, p. 19) (define (not v) (if v #f #t) 24 January 2003 CS 200 Spring 2003 24 2h (if (not "cookies") "eat" "starve") Evaluation Rule 4-if. Evaluate Expression0. If

it evaluates to #f, the value of the if expression is the value of Expression1. Otherwise, the value of the if expression is the value of Expression2. Evaluate (not "cookies") => #f So, value of if is value of Expression2 => starve 24 January 2003 CS 200 Spring 2003 25 Making Orange Ed Mitchells Answer (slightly adapted) (define (mix-one f a b)

(/ (+ (f a) (f b)) 2)) (define (mix-color color1 color2) (make-color (mix-one get-red color1 color2) (mix-one get-green color1 color2) (mix-one get-blue color1 color2))) 24 January 2003 CS 200 Spring 2003 26 Making Orange (define orange (mix-color red yellow)) (show-color orange) (define reddish-orange (mix-color red orange))

(show-color reddish-orange) 24 January 2003 CS 200 Spring 2003 27 brighter? Chalermpong Worawannotais answer (define (brightness color) (+ (get-red color) (get-green color) (get-blue color))) (define (brighter? color1 color2)

(> (brightness color1) (brightness color2))) 24 January 2003 CS 200 Spring 2003 28 closer-color? (define (closer-color? sample color1 color2) (< (+ (abs (- (get-red color1) (get-red sample))) (abs (- (get-blue color1) (get-blue sample))) (abs (- (get-green color1) (get-green sample)))) (+ (abs (- (get-red color2) (get-red sample))) (abs (- (get-blue color2) (get-blue sample))) (abs (- (get-green color2) (get-green sample))))

)) 24 January 2003 CS 200 Spring 2003 29 (+ (abs (- (get-red color1) (get-red sample))) (abs (- (get-blue color1) (get-blue sample))) (abs (- (get-green color1) (get-green sample)))) (define (closer-color? sample color1 color2) (< (+ (abs (- (get-red color2) (get-red sample))) (abs (- (get-blue color2) (get-blue sample))) (abs (- (get-green color2) (get-green sample)))) ))

24 January 2003 CS 200 Spring 2003 30 (lambda ( ) (+ (abs (- (get-red color1 ) (get-red sample ))) (abs (- (get-blue color1) (get-blue sample ))) (abs (- (get-green color1) (get-green sample)))) (define (closer-color? sample color1 color2) (< (+ (abs (- (get-red color2) (get-red sample))) (abs (- (get-blue color2) (get-blue sample)))

(abs (- (get-green color2) (get-green sample)))) )) 24 January 2003 CS 200 Spring 2003 31 (define color-difference (lambda (colora colorb) (+ (abs (- (get-red colora ) (get-red colorb ))) (abs (- (get-blue colora) (get-blue colorb ))) (abs (- (get-green colora) (get-green colorb )))) )) (define (closer-color? sample color1 color2) (< (color-difference color1 sample)

(+ (color-difference (abs (- (get-redcolor2 color2) (get-red sample))) sample) (abs (- (get-blue color2) (get-blue sample))) (abs (- (get-green color2) (get-green sample)))) )) 24 January 2003 CS 200 Spring 2003 32 (define color-difference (lambda (colora colorb) (+ (abs (- (get-red colora) (get-red colorb)))

(abs (- (get-green colora) (get-green colorb))) (abs (- (get-blue colora) (get-blue colorb)))))) (define (closer-color? sample color1 color2) (< (color-difference color1 sample) (color-difference color2 sample))) What if you want to use square instead of abs? 24 January 2003 CS 200 Spring 2003 33 (define color-difference (lambda (cf) (lambda (colora colorb) (+ (cf (- (get-red colora) (get-red colorb))) (cf (- (get-green colora) (get-green colorb)))

(cf (- (get-blue colora) (get-blue colorb))))))) (define (closer-color? sample color1 color2) (< (color-difference color1 sample) (color-difference color2 sample))) 24 January 2003 CS 200 Spring 2003 34 (define color-difference (lambda (cf) (lambda (colora colorb) (+ (cf (- (get-red colora) (get-red colorb)) (cf (- (get-green colora) (get-green colorb)) (cf (- (get-blue colora) (get-blue colorb)))))))) (define (closer-color? sample color1 color2)

(< ((color-difference square) color1 sample) ((color-difference square) color2 sample))) 24 January 2003 CS 200 Spring 2003 35 The Patented RGB RMS Method /* /* /* /* /* This is a variation of RGB RMS error. The final square-root has been eliminated to */ speed up the process. We can do this because we only care about relative error. */

HSV RMS error or other matching systems could be used here, as long as the goal of */ finding source images that are visually similar to the portion of the target image */ under consideration is met. */ for(i = 0; i > size; i++) { rt = (int) ((unsigned char)rmas[i] - (unsigned char)image->r[i]); gt = (int) ((unsigned char)gmas[i] - (unsigned char) image->g[i]; bt = (int) ((unsigned char)bmas[i] - (unsigned char)image->b[i]; result += (rt*rt+gt*gt+bt*bt); } 24 January 2003 Your code should never look like this!

Use new lines and indenting to make it easy to understand the structure of your code! (Note: unless you are writing a patent. Then the36 CS 200 Spring 2003 goal is to make it as hard to understand as possible.) The Patented RGB RMS Method rt = rmas[i] - image->r[i]; gt = gmas[i] - image->g[i]; bt = bmas[i] - image->b[i]; result += (rt*rt + gt*gt + bt*bt); Patent requirements: 1. new must not be previously available

(ancient Babylonians made mosaics) 2. useful 3. nonobvious about of the class came up with this method! (most of rest used abs instead, which works as well) 24 January 2003 CS 200 Spring 2003 37 PS1 Grading Scale Gold Star Excellent Work. You got everything I wanted on this PS. Green Star Good Work. You got most things on this PS, but some answers could be better.

Silver Star Some problems. Make sure you understand the solutions on todays slides. PS1 Average: 24 January 2003 CS 200 Spring 2003 38 No upper limit - Double Gold Star: exceptional work! Better than I expected anyone would do. - Triple Gold Star: Better than I thought possible (deserving of a patent!)

- Quadruple Gold Star: You have broken important new ground in CS which should be published in a major journal - Quintuple Gold Star: You deserve to win a Turing Award! (e.g., a fast, general way to make the best nonrepeating photomosaic) 24 January 2003 CS 200 Spring 2003 39 Problem Set 2 Unlike PS1, you should be able to understand all the provided code (except, dont worry

about the trigonometry) Main ideas: recursion, procedures We have covered everything you need after today As we progress, problem sets will expect you to write more code on your own. PS8 is Do something worthwhile using things you have learned from this class. (But will be a little bit more specific about what is expected.) 24 January 2003 CS 200 Spring 2003 40 Gosper Curves

Gosper Curve Level 0 Gosper Curve Level 1 Gosper Curve Level 50 24 January 2003 CS 200 Spring 2003 41 > (show-gosper smiley 0) > (show-gosper smiley 7) Curves are functions. We can transform them to make new curves. 24 January 2003 CS 200 Spring 2003

42 Charge Many, many important problems can be solved by recursive definitions Read GEB Chapter 5 before Mondays class: lots of great stuff in there! 24 January 2003 CS 200 Spring 2003 43