How many ways to make 50 cents? first-denomination Solution. CS61A Lecture 5. count-change. cc base cases. How many have you figured out?
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1 6/6/ CS6A Lecture -6-7 Colleen Lewis How many ways to make cents? first-denomination Solution (define (first-denomination kinds-of-coins) ((= kinds-of-coins ) ) ((= kinds-of-coins ) ) ((= kinds-of-coins ) ) ((= kinds-of-coins ) ) ((= kinds-of-coins ) ))) STk> (first-denomination ) STk> (first-denomination ) STk> (first-denomination ) count-change amount )) (first-denomination kinds-of-coins) cc base cases amount )) STk> ) STk> ) STk> ) How many have you figured out? A) B) C) D) count-change base cases ((= amount ) ) ((< amount ) ) ((= kinds-of-coins ) ) (else ((or (< amount )(= kinds-of-coins )) ) Designing the recursion Delegate (make a recursive call): Assuming you use the biggest coin (w/ highest value) Assuming you don t use the biggest coin Example: How many ways can you make cents?
2 6/6/ Use biggest coin Woah! The number of coins didn t change! - - Don t use biggest coin Woah! The amount didn t change! Continuing from previous slide count-change recursive cases Blank Blank (+ Reading Code (Like analyzing a poem) Start with simple parts e.g. (first-denomination) Read, re-read, re-read, re-read, test, re-read Re-read specific sections Highlight things you think are important Write down your hypotheses after each reading Tests your hypothesis when possible You re not going to understand it the first or third time you read it! Same is true for project specs!!!! Iterative versus Recursive Process (all of which use recursion!)
3 6/6/ STk>(evens ( 8 )) (evens ( 8 )) (se (evens ( 8 )) (se 8 Recursive process case I m not done (se (se 8 (se ()))) (evens ( )) (se (evens ()) () evens (define (evens sent) ((empty? sent) ()) ((even? (first sent)) Waiting (se (first sent) to do se (evens (bf sent)))) (else (evens (bf sent))))) STk>(evens ( 8 ) ) (evens-iter ( 8 ) ()) (evens-iter ( 8 ) ()) (evens-iter ( ) ) (evens-iter () ) Keep track of your as you go! case I m DONE! STk>(evens ( 8 ) ) (evens-iter ( 8 ) ()) (evens-iter ( 8 ) ()) (evens-iter ( ) ) (evens-iter () ) evens (define (evens sent) (define (evens-iter sent ) (if (empty? sent) (evens-iter (bf sent) (se (first sent))))) (evens sent '())) Do you like this version better? A) Yes B) No Factorial (if (= n ) (* n (factorial (- n )))))
4 6/6/ STk>(factorial ) (factorial ) (* (factorial ) (* Recursive process case I m not done (* (* (* ))) (factorial ) (* (factorial ) factorial version (define (fact-iter n ) (if (= n ) (fact-iter (- n )(* n )))) (fact-iter n )) Do you like this version better? A) Yes B) No Keep track of your as you go! case I m DONE! STk>(factorial ) (fact-iter ) (fact-iter ) (fact-iter ) (fact-iter ) RC = RC + RC R: R: R: R: R: R: R: 6 Pascal s Triangle Recursion (Cont.) RC = RC + RC (R,C) = (R-,C-) + (R-, C) (define (pascal row col) ((= col ) ) ((= col row) ) (else (+ (pascal (- row ) (- col )) (pascal (- row ) col)))) Summary count-change an example of: Practicing learning to read and trace code Tree recursion multiple recursive calls in a single case Recursive vs. Iterative Processes All of them are implemented with recursion! Recursive processes aren t done at the base case es keep their with them
5 6/6/ Base Cases How many ways can you count $. in change using types of coins? way -$. in change using types of coins? ways $. in change using types of coins? ways count-change recursive cases Blank Blank (+ (- amount(first-denomination kinds-of-coins)) kinds-of-coins) amount (- kinds-of-coins )))))) amount )) ((= amount ) ) ((or (< amount ) (= kinds-of-coins )) ) (else (+ (- amount (first-denomination kinds-of-coins)) kinds-of-coins) amount (- kinds-of-coins )))))) (define (first-denomination kinds-of-coins) ((= kinds-of-coins ) ) ((= kinds-of-coins ) ) ((= kinds-of-coins ) ) ((= kinds-of-coins ) ) ((= kinds-of-coins ) ))) Woah! This is overwhelming! evens Solution (define (evens sent) (define (evens-iter sent ) (if (empty? sent) (evens-iter (bf sent) (se (first sent))))) (evens sent '())) factorial version Solution (define (fact-iter n ) (if (= n ) (fact-iter (- n )(* n )))) (fact-iter n )) Recursion (Cont.) Solution RC = RC + RC (R,C) = (R-,C-) + (R-, C) (define (pascal row col) ((= col ) ) ((= col row) ) (else (+ (pascal (- row ) (- col )) (pascal (- row ) col))))
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