Defining a unique-pairs procedure

Question

From the section called Nested Mappings

Exercise 2.40

Define a procedure unique-pairs that, given an integer n, generates the sequence of pairs (i,j) with 1 < j < i < n. Use unique-pairs to simplify the definition of prime-sum-pairs given above.

I wrote the following:

(define (prime-sum-pairs n)
 (filter (lambda (seq)
 (prime? (+ (car seq) (cadr seq))))
 (unique-pairs n)))
(define (enumerate-integers start end)
 (if (>= start end)
 (list end)
 (cons start (enumerate-integers (+ 1 start) end))))
(define (unique-pairs n)
 (flat-map (lambda (i) 
 (map (lambda (j) (list i j)) 
 (enumerate-integers 1 (- i 1))))
 (enumerate-integers 2 n)))
(define (filter test-fn seq)
 (if (null? seq) null
 (if (test-fn (car seq)) 
 (cons (car seq) 
 (filter test-fn (cdr seq)))
 (filter test-fn (cdr seq)))))
(define (accumulate op initial seq)
 (if (null? seq)
 initial
 (op (car seq)
 (accumulate op initial (cdr seq)))))
(define (flat-map f seq)
 (accumulate append
 null
 (map (lambda (x) (f x)) seq)))
(define (prime? n) (= (smallest-divisor n) n))
(define (divisible? n i) (= 0 (remainder n i)))
(define (square x) (* x x))
(define (smallest-divisor n)
 (define (rec i)
 (cond ((> n (square i)) n)
 ((divisible? n i) i)
 (else (rec (+ 1 i)))))
 (rec 2))

Can this be improved in any way?

Answer 1

Your code

(define (enumerate-integers start end)
 (if (>= start end)
 (list end)
 (cons start (enumerate-integers (+ 1 start) end))))
(define (unique-pairs n)
 (flat-map (lambda (i) 
 (map (lambda (j) (list i j)) 
 (enumerate-integers 1 (- i 1))))
 (enumerate-integers 2 n)))

looks fine to me. If you want to massage some details, you could rewrite enumerate-integers e.g. to:

(define (enumerate-integers start end)
 (if (> start end) '()
 (cons start (enumerate-integers (+ 1 start) end))))

which is cleaner because you don't have (list end) as a special case, and you can correctly produce an empty list of integers if start> end.

If you want to be even cleaner, you can do:

(define (enumerate-integers start end)
 (define (iter n)
 (if (> n end) '()
 (cons n (iter (+ n 1)))))
 (iter start))

This is a good pattern in case of more complex procedures.

Your flat-map is more complex than needed, your code:

(define (flat-map f seq)
 (accumulate append
 null
 (map (lambda (x) (f x)) seq)))

can be replaced with:

(define (flat-map f seq)
 (accumulate append
 null
 (map f seq)))

because (lambda (x) (f x)) is equal to f.