Railway circuit

Railway circuit is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Railway circuit

Given n sections of curve tracks, each one being an arc of 30° of radius R, the goal is to build and count all possible different railway circuits.

Constraints :

• n = 12 + k*4 (k = 0, 1 , ...)
• The circuit must be a closed, connected graph, and the last arc must joint the first one
• Duplicates, either by symmetry, translation, reflexion or rotation must be eliminated.
• Paths may overlap or cross each other.
• All tracks must be used.

Illustrations : http://www.echolalie.org/echolisp/duplo.html

Task:

Write a function which counts and displays all possible circuits Cn for n = 12, 16 , 20. Extra credit for n = 24, 28, ... 48 (no display, only counts). A circuit Cn will be displayed as a list, or sequence of n Right=1/Left=-1 turns.

Example:

C12 = (1,1,1,1,1,1,1,1,1,1,1,1) or C12 = (-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1)

Straight tracks (extra-extra credit)

Suppose we have m = k*2 sections of straight tracks, each of length L. Such a circuit is denoted Cn,m . A circuit is a sequence of +1,-1, or 0 = straight move. Count the number of circuits Cn,m with n same as above and m = 2 to 8 .

EchoLisp

<lang scheme>

R is turn counter in right direction

The nb of right turns in direction i

must be = to nb of right turns in direction i+6 (opposite)

(define (legal? R) (for ((i 6)) #:break (!= (vector-ref R i) (vector-ref R (+ i 6))) => #f #t))

equal circuits by rotation ?

(define (circuit-eq? Ca Cb) (for [(i (vector-length Cb))] #:break (eqv? Ca (vector-rotate! Cb 1)) => #t #f))

check a result vector RV of circuits

Remove equivalent circuits

(define (check-circuits RV) (define n (vector-length RV)) (for ((i (1- n))) #:continue (null? (vector-ref RV i)) (for ((j (in-range (1+ i) n ))) #:continue (null? (vector-ref RV j)) (when (circuit-eq? (vector-ref RV i) (vector-ref RV j)) (vector-set! RV j null)))))

global

*circuits* = result set = a vector

(define-values (*count* *calls* *circuits*) (values 0 0 null))

generation of circuit C[i] i = 0 .... maxn including straight (may be 0) tracks

(define (circuits C Rct R D n maxn straight ) (define _Rct Rct) ;; save area (define _Rn (vector-ref R Rct)) (++ *calls* )

(cond

   [(> *calls* 4_000_000) #f] ;; enough for maxn=24
   
   ;; hit !! legal solution
   [(and (= n maxn) ( zero? Rct ) (legal? R) (legal? D))

(++ *count*) (vector-push *circuits* (vector-dup C))];; save solution

    ;; stop
    [( = n maxn) #f]

    ;; important cutter - not enough right turns
    [(and (!zero? Rct) (< (+ Rct maxn ) (+ n straight 11))) #f] 

    [else

;; play right (vector+= R Rct 1) ; R[Rct] += 1 (set! Rct (modulo (1+ Rct) 12)) (vector-set! C n 1) (circuits C Rct R D (1+ n) maxn straight)

;; unplay it - restore values (set! Rct _Rct) (vector-set! R Rct _Rn) (vector-set! C n '-)

;; play left (set! Rct (modulo (1- Rct) 12)) (vector-set! C n -1) (circuits C Rct R D (1+ n) maxn straight)

;; unplay (set! Rct _Rct) (vector-set! R Rct _Rn) (vector-set! C n '-)

;; play straight line (when (!zero? straight) (vector-set! C n 0) (vector+= D Rct 1) (circuits C Rct R D (1+ n) maxn (1- straight))

;; unplay (vector+= D Rct -1) (vector-set! C n '-)) ]))

(generate max-tracks [ + max-straight])

(define (gen (maxn 20) (straight 0)) (define R (make-vector 12)) (define D (make-vector 12)) (define C (make-vector maxn '-)) (set!-values (*count* *calls* *circuits*) (values 0 0 (make-vector 0))) (vector-set! R 0 1) ;; play starter (always right) (vector-set! C 0 1) (circuits C 1 R D 1 (+ maxn straight) straight) (writeln 'gen-counters (cons *calls* *count*))

(check-circuits *circuits*) (set! *circuits* (for/vector ((c *circuits*)) #:continue (null? c) c)) (if (zero? straight) (printf "Number of circuits C%d : %d" maxn (vector-length *circuits*)) (printf "Number of circuits C%d,%d : %d" maxn straight (vector-length *circuits*))) (when (< (vector-length *circuits*) 20) (for-each writeln *circuits*))) </lang>

Output:

(gen 12)
gen-counters     (331 . 1)    
Number of circuits C12 : 1
#( 1 1 1 1 1 1 1 1 1 1 1 1)    

(gen 16)
gen-counters     (8175 . 6)    
Number of circuits C16 : 1
#( 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 -1 1)  
  
(gen 20)
gen-counters     (150311 . 39)    
Number of circuits C20 : 6
#( 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 1 1 -1 1 -1 1)    
#( 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 1 1)    
#( 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 1 1 -1 1)    
#( 1 1 1 1 1 -1 1 1 -1 1 1 1 1 1 1 -1 1 1 -1 1)    
#( 1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 -1 1 1 1 -1 1)    
#( 1 1 1 -1 1 1 1 1 -1 1 1 1 1 -1 1 1 1 1 -1 1)  
  
(gen 24)
gen-counters     (2574175 . 286)    
Number of circuits C24 : 35

(gen 12 4)  
Number of circuits C12,4 : 4
#( 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0)    
#( 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0)    
#( 1 1 1 1 0 1 1 0 1 1 1 1 0 1 1 0)    
#( 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0)