AoC 2018 Day 13: Mine Cart Madness

Source: Mine Cart Madness

Part 1: Load a minecart track that looks like this:

|   |  /----\
| /-+--+-\  |
| | |  | v  |
\-+-/  \-+--/

Assuming minecarts follow the tracks and alternate turning left, going straight, and turning right on each intersection (+), where does the first collision occur?

NOTE: Update carts top to bottom, left to right. Carts can collide mid update.

Okay. It’s an interesting data format problem mostly. And we have to deal a bit with not being imperative, in particular when we’re doing the updates, since having collisions happen halfway through an update is not so great. Let’s load it:

(struct point (x y) #:transparent)
(struct cart (location velocity next-turn) #:transparent)
(struct track (data carts top-left bottom-right) #:transparent)

The goal will be to load the tracks into a hash of point to character (so I know how the track turns / intersects) and separately store the carts. I’ll want to process the track after the initial load to replace carts with their underlying track and get the initial velocity as well. Finally, next-turn will always start as 'left, but we’ll deal with that later.

; Read a track from the current-input
(define (read-track [in (current-input-port)])
  ; Read the raw data
  (define raw-data
    (for/fold ([track (hash)])
              ([line (in-lines in)]
               [y (in-naturals)])
      (for/fold ([track track])
                ([c (in-string line)]
                 [x (in-naturals)]
                 #:unless (equal? c #\space))
        (hash-set track (point x y) c))))

  ; For each cart character, track to underlying track and current velocity
  (define cart-track-data
    (hash #\> (list #\-  1 0)
          #\< (list #\- -1 0)
          #\^ (list #\| 0 -1)
          #\v (list #\| 0  1)))

  ; Determine where the carts are
  (define-values (data carts)
    (for*/fold ([data raw-data]
                [carts (list)])
               ([(p c) (in-hash raw-data)]
                [ctd (in-value (hash-ref cart-track-data c #f))]
                #:when ctd)
       ; Overwrite the track point with the underlying track
       (hash-set data p (first ctd))
       ; Store the cart position and velocity
       (list* (cart p (point (second ctd) (third ctd)) 'left) carts))))

  ; Determine the bounds for the track
  (define-values (min-x max-x min-y max-y)
    (for/fold ([min-x #f] [max-x #f] [min-y #f] [max-y #f])
              ([(p c) (in-hash data)])
      (values (min (or min-x (point-x p)) (point-x p))
              (max (or max-x (point-x p)) (point-x p))
              (min (or min-y (point-y p)) (point-y p))
              (max (or max-y (point-y p)) (point-y p)))))

  ; Finally create the track structure
  (track data
         (point min-x min-y)
         (point max-x max-y)))

It’s a bit complicated and it turns out that the only reason that I need the bounds is to display the tracks for debug reasons, which amuses me somewhat. But we have it, so let’s use it. Speaking of display, here’s a function to take those bounds and write out what the current track looks like:

; Helper function to display a track
(define (display-track m)
  (for ([y (in-range (point-y (track-top-left m)) (add1 (point-y (track-bottom-right m))))])
    (for ([x (in-range (point-x (track-top-left m)) (add1 (point-x (track-bottom-right m))))])
        [(for/first ([c (in-list (track-carts m))]
                     #:when (equal? (cart-location c) (point x y)))
         => (λ (c)
               (match (cart-velocity c)
                 [(point 0  1) #\v]
                 [(point 0 -1) #\^]
                 [(point  1 0) #\>]
                 [(point -1 0) #\<])))]
         (display (hash-ref (track-data m) (point x y) #\.))]))

Helpful for debugging. match is lovely.

Okay, next we want a function to update a single cart. We’ll use this next to update each cart in turn. This got tricky. Initially, I was checking the carts against the track data (to check for collisions), but that doesn’t work, since we can have a cart collide against another one after both have already moved. The only way I’ve seen to track that is to specify the current cart locations to the update-cart function. There really should be a more elegant way to do this:

; Update a cart by one tick
; Return the new cart; but if it collided raise the cart as an exception insteadcurrent-frame
; c* Is a running updated list of carts, since collisions can happen with already moved carts
(define (update-cart m c c*)
  (match-define (cart (point x y) (point vx vy) rotation) c)

  (define new-location (point (+ x vx) (+ y vy)))

  (define-values (new-velocity new-rotation)
    (case (hash-ref (track-data m) new-location)
      [(#\/) (values (point (- vy) (- vx)) rotation)]
      [(#\\) (values (point vy vx) rotation)]
       (case rotation
         [(left)     (values (point vy (- vx)) 'straight)]
         [(straight) (values (point vx vy)     'right)]
         [(right)    (values (point (- vy) vx) 'left)])]
      [else  (values (point vx vy) rotation)]))

  (define updated-cart (cart new-location new-velocity new-rotation))

  ; Check for collisions
  (for ([c (in-list c*)]
        #:when (equal? new-location (cart-location c)))
    (raise updated-cart))


One thing that I actually enjoy here is that I’m using exception handling to deal with the collisions. The function itself only deals with returning the updated cart. If there’s a collision, we’ll raise the cart that caused the problem in order to catch it via with-handlers later.

I’m also pretty proud of the rotation code. There are a few other ways you can do this, for example with matrix multiplication or hard coding each case. But I worked it out on paper and it turns out that you flip x and y and negate one or the other. (It’s a symptom of the aforementioned matrix multiplication.)

In any case, we can now update the entire track:

; Update a track by one tick
; If two carts collide, an exception will be raised, catch it and return the cart
(define (update-track m)
  ; Update carts, has to be done oddly since they can collide half way through an update
  ; Otherwise, remove the third argument from update-cart and use:
  ;   (map (curry update-cart m) (sort (track-carts m) cart-location-<?))
  (define updated-carts
    (let loop ([done '()]
               [todo (sort (track-carts m) cart-location-<?)])
        [(null? todo) done]
         ; TODO: Make this more efficient than using append
         (loop (list* (update-cart m (first todo) (append done todo)) done)
               (rest todo))])))

  (track (track-data m)
         (track-top-left m)
         (track-bottom-right m)))

As the comment notes, I wish we could have done all the carts at a time. But that would lead to carts skipping past each other in cases like this: ->--<-. So it goes.

Now we can take it one step more and run the simulation until we get a collision. Because we’re using exceptions and we haven’t caught them yet:

; Run a track until collision, return the cart that collided
(define (update-track-until-collision m)
  (with-handlers ([cart? identity])
    (let loop ([m m])
      (loop (update-track m)))))

; Run the main program
(printf "[part1]\n")
(define input (read-track))
(update-track-until-collision input)

That’s fun.

$ cat input.txt | racket mine-cart-madness.rkt

(cart (point 118 66) (point 1 0) 'straight)


Part 2: Instead of ending the simulation, removing carts that crash into one another. What is the location of the final remaining cart?

This time around, the updated-carts function will be caught immediately in this function, triggering the removal of the cart that was removed (with the (rest todo) and all other carts at the same location (remove-by-location). Voila:

; Update a track by one tick
; If two carts collide, remove those two carts and contiue updating
(define (update-track/remove-collisions m)
  ; Helper to remove carts at a given location from a list
  (define (remove-by-location carts location)
    (for/list ([c (in-list carts)]
               #:unless (equal? (cart-location c) location))

  ; Calculate the new list of carts, some might collide
  (define updated-carts
    (let loop ([done '()]
               [todo (sort (track-carts m) cart-location-<?)])
        [(null? todo) done]
             ; Case where carts collided
             ; Remove that cart + all other carts with that coordinate from both lists
             ([cart? (λ (c)
                       (loop (remove-by-location done (cart-location c))
                             (remove-by-location (rest todo) (cart-location c))))])
           ; Otherwise
           (loop (list* (update-cart m (first todo) (append done todo)) done)
                 (rest todo)))])))

  (track (track-data m)
         (track-top-left m)
         (track-bottom-right m)))

The wrapper function to run until there is only one cart is only slightly longer:

; Run a track until there is only one cart left, removing carts that collide
(define (update-track-until-singleton m)
  (let loop ([m m])
      [(<= (length (track-carts m)) 1)
       (first (track-carts m))]
       (loop (update-track/remove-collisions m))])))

; Run the main program again
(printf "\n[part2]\n")
(current-frame 0)
(update-track-until-singleton input)


$ cat input.txt | racket mine-cart-madness.rkt

(cart (point 118 66) (point 1 0) 'straight)

(cart (point 70 129) (point 0 1) 'right)

As a bonus, I made a function that can render the simulation to a picture!

; Helper function to render a map as a frame to an image with ASCII graphics
(define (render-frame m prefix)
  (match-define (track data carts (point left top) (point right bottom)) m)
  (printf "rendering frame ~a:~a\n" prefix (current-frame))
     3 (add1 (- right left)) (add1 (- bottom top))
     (λ (x y)
         ; Carts are red
         [(for/first ([c (in-list carts)]
                      #:when (equal? (point x y) (cart-location c)))
          (vector 1 0 0)]
         ; Tracks are white
         [(hash-ref data (point x y) #f)
          (vector 1 1 1)]
         ; Background is black
          (vector 0 0 0)]))))
   (~a "frame-" prefix "-" (~a (current-frame) #:min-width 4 #:left-pad-string "0" #:align 'right) ".png")
  (current-frame (add1 (current-frame))))

Pretty pictures!

$ cat input.txt | racket mine-cart-madness.rkt --debug-frames

$ convert frame-collision-*.png -interpolate Nearest -filter point -resize 200% aoc-13-minecart.gif

Not super helpful at this scale, but pretty.

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