Exercise 3.70 merge-weighted and weighted-pairs.rkt

#lang racket
	
; Exercise 3.70.  It would be nice to be able to generate streams in which the pairs appear
; in some useful order, rather than in the order that results from an ad hoc interleaving
; process. We can use a technique similar to the merge procedure of exercise 3.56, if we
; define a way to say that one pair of integers is "less than" another. One way to do
; this is to define a "weighting function" W(i,j) and stipulate that (i1,j1) is less than
; (i2,j2) if W(i1,j1) < W(i2,j2). Write a procedure merge-weighted that is like merge,
; except that merge-weighted takes an additional argument weight, which is a procedure
; that computes the weight of a pair, and is used to determine the order in which elements
; should appear in the resulting merged stream. Using this, generalize pairs to a
; procedure weighted-pairs that takes two streams, together with a procedure that computes
; a weighting function, and generates the stream of pairs, ordered according to weight.
; Use your procedure to generate

; a. The stream of all pairs of positive integers (i,j) with i < j ordered according to
; the sum i + j

; b. the stream of all pairs of positive integers (i,j) with i < j, where neither i nor j
; is divisible by 2, 3, or 5, and the pairs are ordered according to the sum 2i + 3j + 5ij.

; S O L U T I O N

(define (weighted-pairs s t weight)
	(stream-cons
		(list (stream-first s) (stream-first t))
		(merge-weighted
			(stream-map (lambda (x) (list (stream-first s) x)) (stream-rest t))
			(weighted-pairs (stream-rest s) (stream-rest t) weight)
			weight
		)
	)
)

(define (merge-weighted s1 s2 weight)
	(cond
		((stream-empty? s1) s2)
		((stream-empty? s2) s1)
		(else
			(let ((s1car (stream-first s1)) (s2car (stream-first s2)))
				(cond
					((< (weight s1car) (weight s2car))
						(stream-cons s1car (merge-weighted (stream-rest s1) s2 weight))
					)
					((> (weight s1car) (weight s2car))
						(stream-cons s2car (merge-weighted s1 (stream-rest s2) weight))
					)
					(else
						(stream-cons
							s1car
							(stream-cons
								s2car
								(merge-weighted (stream-rest s1) (stream-rest s2) weight)
							)
						)
					)
				)
			)
		)
	)
)

; The following procedure expects two ascending infinite streams,
; one of which needs to be a subset of the other. Let S be the stream with the superset
; and let T be the stream with the subset. Then this procedure will produce a new stream
; that contains all those elements that are present in S but not in T
(define (s-minus-t s t)
	; Assumes that S and T are ordered ascending and T is a subset of S
	(cond
		((stream-empty? t) s)
		((stream-empty? s) empty-stream)
		(else
			(let ((scar (stream-first s)) (tcar (stream-first t)))
				(cond
					((< scar tcar)
						(stream-cons
							scar
							(s-minus-t (stream-rest s) t)
						)
					)
					((> scar tcar)
						(s-minus-t s (stream-rest t))
					)
					(else
						(s-minus-t (stream-rest s) (stream-rest t))
					)
				)
			)
		)
	)
)

; The following proc copied from SICP Exercise 3.56
(define (merge s1 s2)
	(cond
		((stream-empty? s1) s2)
		((stream-empty? s2) s1)
		(else
			(let ((s1car (stream-first s1)) (s2car (stream-first s2)))
				(cond
					((< s1car s2car)
						(stream-cons s1car (merge (stream-rest s1) s2))
					)
					((> s1car s2car)
						(stream-cons s2car (merge s1 (stream-rest s2)))
					)
					(else
						(stream-cons s1car
							(merge (stream-rest s1) (stream-rest s2))
						)
					)
				)
			)
		)
	)
)

; Implementation of procedure triples
; The infinite streams S, T and U are represented as follows:

; S0		T0		U0
; S1		T1		U1
; S2		T2		U2
; S3		T3		U3
; S4		T4		U4
; .		.		.
; .		.		.
; .		.		.

; All triples (Si, Tj, Uk) such that i < j < k can be produced by combining the following:

; 1.	(S0, T1, U2)
; 2.	S0 combined with all the pairs produced using the streams (T1, T2, T3, ...) and
; 		(U1, U2, U3, ...) such that for every pair (Tj, Uk), j < k. We need to exclude the 
; 		first element from this combined stream because the first element will be
; 		(S0, T1, U2) which is already accounted for.
; 3. 	triples called recursively on the streams (S1, S2, ...) (T1, T2, ...) and (U1, U2, ...)

(define (triples s t u)
	(stream-cons
		(list
			(stream-first s)
			(stream-first (stream-rest t))
			(stream-first (stream-rest (stream-rest u)))
		)
		(interleave
			(stream-map
				(lambda (x) (cons (stream-first s) x))
				(stream-rest (less-than-pairs (stream-rest t) (stream-rest u)))
			)
			(triples (stream-rest s) (stream-rest t) (stream-rest u))
		)
	)
)

(define (is-pythagorean-triple? triple)
	; This procedure assumes that the elements in the triple are in increasing order 
	(= (+ (square (car triple)) (square (car (cdr triple)))) (square (car (cdr (cdr triple)))))
)

; This procedures produces all pairs (Si, Tj) where i < j
(define (less-than-pairs s t)
	(stream-cons
		(list (stream-first s) (stream-first (stream-rest t)))
		(interleave
			(stream-map (lambda (x) (list (stream-first s) x)) (stream-rest (stream-rest t)))
			(less-than-pairs (stream-rest s) (stream-rest t))
		)
	)
)

(define (all-pairs s t)
	(stream-cons
		(list (stream-first s) (stream-first t))
		(interleave
			(stream-map (lambda (x) (list (stream-first s) x)) (stream-rest t))
			(interleave
				(stream-map (lambda (x) (list x (stream-first t))) (stream-rest s))
				(all-pairs (stream-rest s) (stream-rest t))
			)
		)
	)
)

(define (pairs s t)
	(stream-cons
		(list (stream-first s) (stream-first t))
		(interleave
			(stream-map (lambda (x) (list (stream-first s) x)) (stream-rest t))
			(pairs (stream-rest s) (stream-rest t))
		)
	)
)

(define (interleave s1 s2)
	(if (stream-empty? s1)
		s2
		(stream-cons
			(stream-first s1)
			(interleave s2 (stream-rest s1))
		)
	)
)

(define (partial-sums S)
	(stream-cons (stream-first S) (add-streams (partial-sums S) (stream-rest S)))
)

(define (ln2-summands n)
	(stream-cons
		(/ 1.0 n)
		(stream-map - (ln2-summands (+ n 1)))
	)
)

(define ln2-stream
	(partial-sums (ln2-summands 1))
)

(define (sqrt x tolerance)
	(stream-limit (sqrt-stream x) tolerance)
)

(define (stream-limit s t)
	(let ((s0 (stream-ref s 0)) (s1 (stream-ref s 1)))
			(if (< (abs (- s1 s0)) t)
			s1
			(stream-limit (stream-rest s) t)
		)
	)
)

(define (sqrt-stream x)
	(define guesses
		(stream-cons
			1.0
			(stream-map
				(lambda (guess) (sqrt-improve guess x))
				guesses
			)
		)
	)
	guesses
)

(define (sqrt-improve guess x)
	(average guess (/ x guess))
)

(define (average x y) (/ (+ x y) 2))
(define (square x) (* x x))

(define (euler-transform s)
	(let ((s0 (stream-ref s 0))          ; Sn-1
		 (s1 (stream-ref s 1))           ; Sn
		 (s2 (stream-ref s 2)))          ; Sn+1
		(stream-cons
			(-
				s2
				(/ (square (- s2 s1)) (+ s0 (* -2 s1) s2))
			)
			(euler-transform (stream-rest s))
		)
	)
)

(define (make-tableau transform s)
	(stream-cons
		s
		(make-tableau transform (transform s))
	)
)

(define (accelerated-sequence transform s)
	(stream-map stream-first (make-tableau transform s))
)

(define (scale-stream stream factor)
	(stream-map (lambda (x) (* x factor)) stream)
)

(define (stream-map proc . argstreams)
	; (displayln "Entered stream-map")
	(if (stream-empty? (car argstreams))
		empty-stream
		(stream-cons
			(apply proc (map stream-first argstreams))
			(apply stream-map (cons proc (map stream-rest argstreams)))
		)
	)
)

(define (stream-filter pred stream)
	(cond
		((stream-empty? stream) empty-stream)
		((pred (stream-first stream))
			(stream-cons
				(stream-first stream)
				(stream-filter pred (stream-rest stream))
			)
		)
		(else
			(stream-filter pred (stream-rest stream))
		)
	)
)

(define (stream-ref s n)
	; (display "Entered stream-ref with n = ")
	; (display n)
	; (newline)
	(if (= n 0)
		(stream-first s)
		(stream-ref (stream-rest s) (- n 1))
	)
)

; This procedure produces a stream with a maximum number of count elements from
; the supplied stream
(define (truncate-stream s count)
	(if (<= count 0)
		empty-stream
		(stream-cons
			(stream-first s)
			(truncate-stream (stream-rest s) (- count 1))
		)
	)
)

; This procedure displays a finite number of elements from the supplied stream
; as specified by 'count'
(define (display-stream-elements count s)
	(if (= 0 count)
		(begin
			(newline)
			'done
		)
		(begin
			(newline)
			(display (stream-first s))
			(display-stream-elements (- count 1) (stream-rest s))
		)
	)
)

(define (display-stream s)
	(stream-for-each display-line s)
)

(define (display-line x)
	(newline)
	(display x)
)

(define (indent-and-display pair)
	(display-spaces (car pair))
	(display pair)
	(newline)
)

(define (display-spaces n)
	(if (<= n 0)
		(void)
		(begin
			(display " ")
			(display-spaces (- n 1))
		)
	)
)

(define (div-series dividend-series divisor-series)
	(cond
		((= 0 (stream-first divisor-series))
			(error "Denominator should not have a zero constant: " (stream-first divisor-series))
		)
		(else
			(mul-series
				dividend-series
				(invert-unit-series (scale-stream divisor-series (/ 1 (stream-first divisor-series))))
			)
		)
	)
)

(define (invert-unit-series s)
	(stream-cons
		1
		(mul-series
			(scale-stream (stream-rest s) -1)
			(invert-unit-series s)
		)
	)
)

(define (mul-series s1 s2)
	(stream-cons
		; (a0 * b0) (The following is the constant term of the series resulting from
		; the multiplication)
		(* (stream-first s1) (stream-first s2))
		; The following is the rest of the series starting with the x^1 term
		(add-streams
			; {a0 * (b1x + b2x^2 + b3x^3 + ...)} +
			; {b0 * (a1x + a2x^2 + a3x^3 + ...)} +
			(add-streams
				(scale-stream (stream-rest s2) (stream-first s1))
				(scale-stream (stream-rest s1) (stream-first s2))
			)
			; {(a1x + a2x^2 + a3x^3 + ...) * (b1x + b2x^2 + b3x^3 + ...)}
			(stream-cons
				; 0 needs to be prepended to this stream so that the first term of the stream
				; is the x^1 term. Only then the outer add-streams will add like terms in the
				; two series supplied to it
				0
				(mul-series (stream-rest s1) (stream-rest s2))
			)
		)
	)
)

(define exp-series
	(stream-cons 1 (integrate-series exp-series))
)

; the integral of negative sine is cosine
(define cosine-series
	(stream-cons 1 (integrate-series (scale-stream sine-series -1)))
)

; the integral of cosine is sine
(define sine-series
	(stream-cons 0 (integrate-series cosine-series))
)

(define tan-series 
	(div-series sine-series cosine-series)
)

(define (integrate-series s)
	(div-streams s integers)
)

(define (add-streams s1 s2)
	(stream-map + s1 s2)
)

(define (div-streams s1 s2)
	(stream-map / s1 s2)
)

(define ones (stream-cons 1 ones))

(define integers (stream-cons 1 (add-streams ones integers)))

; Test Driver

(define (run-test return-type proc . args)

	(define (print-item-list items first-time?)
		(cond
			((not (pair? items)) (void))
			(else
				(if (not first-time?)
					(display ", ")
					(void)
				)
				(print (car items))
				(print-item-list (cdr items) false)
			)
		)
	)

	(display "Applying ")
	(display proc)
	(if (not (null? args))
		(begin
			(display " on: ")
			(print-item-list args true)
		)
		(void)
	)
	(newline)
	(let ((result (apply proc args)))
		(if (not (eq? return-type 'none))
			(display "Result: ")
			(void)
		)
		(cond
			((procedure? result) ((result 'print)))
			; ((eq? return-type 'deque) (print-deque result))
			((eq? return-type 'none) (void))
			(else
				(print result)
				(newline)
			)
		)
	)
	(newline)
)

(define (execution-time proc . args)
	(define start-time (current-milliseconds))
	; (display start-time)
	; (display " ")
	(apply proc args)
	(define end-time (current-milliseconds))
	; (display end-time) 
	(display "Execution time of ")
	(display proc)
	(display ": ")
	(- end-time start-time)
)

; Tests

; Test Results

Welcome to DrRacket, version 6.11 [3m].
Language: racket, with debugging; memory limit: 4096 MB.
> ; Part a

(define (pair-weight-as-sum pair)
	(+ (car pair) (cadr pair))
)

> (display-stream-elements 85 (weighted-pairs integers integers pair-weight-as-sum))

(1 1)
(1 2)
(1 3)
(2 2)
(1 4)
(2 3)
(1 5)
(2 4)
(3 3)
(1 6)
(2 5)
(3 4)
(1 7)
(2 6)
(3 5)
(4 4)
(1 8)
(2 7)
(3 6)
(4 5)
(1 9)
(2 8)
(3 7)
(4 6)
(5 5)
(1 10)
(2 9)
(3 8)
(4 7)
(5 6)
(1 11)
(2 10)
(3 9)
(4 8)
(5 7)
(6 6)
(1 12)
(2 11)
(3 10)
(4 9)
(5 8)
(6 7)
(1 13)
(2 12)
(3 11)
(4 10)
(5 9)
(6 8)
(7 7)
(1 14)
(2 13)
(3 12)
(4 11)
(5 10)
(6 9)
(7 8)
(1 15)
(2 14)
(3 13)
(4 12)
(5 11)
(6 10)
(7 9)
(8 8)
(1 16)
(2 15)
(3 14)
(4 13)
(5 12)
(6 11)
(7 10)
(8 9)
(1 17)
(2 16)
(3 15)
(4 14)
(5 13)
(6 12)
(7 11)
(8 10)
(9 9)
(1 18)
(2 17)
(3 16)
(4 15)
'done
> ; Part b
(define (pair-weight-custom pair)
	(+ (* 2 (car pair)) (* 3 (cadr pair)) (* 5 (car pair) (cadr pair)))
)
> (define integer-stream-not-multiple-of-2-3-5
	(s-minus-t
		(s-minus-t
			(s-minus-t integers (scale-stream integers 2))
			(scale-stream integers 3)
		)
		(scale-stream integers 5)
	)
)
> (display-stream-elements 20 integer-stream-not-multiple-of-2-3-5)

1
7
11
13
17
19
23
29
31
37
41
43
47
49
53
59
61
67
71
73
'done
> (display-stream-elements 50
	(weighted-pairs
		integer-stream-not-multiple-of-2-3-5
		integer-stream-not-multiple-of-2-3-5
		pair-weight-custom
	)
)

(1 1)
(1 7)
(1 11)
(1 13)
(1 17)
(1 19)
(1 23)
(1 29)
(1 31)
(7 7)
(1 37)
(1 41)
(1 43)
(1 47)
(1 49)
(1 53)
(7 11)
(1 59)
(1 61)
(7 13)
(1 67)
(1 71)
(1 73)
(1 77)
(1 79)
(7 17)
(11 11)
(1 83)
(1 89)
(1 91)
(7 19)
(11 13)
(1 97)
(1 101)
(1 103)
(1 107)
(1 109)
(7 23)
(1 113)
(13 13)
(1 119)
(1 121)
(11 17)
(1 127)
(1 131)
(1 133)
(1 137)
(1 139)
(7 29)
(11 19)
'done