#+title: Solution to p9, aka the shitshow

Load the data
#+begin_src emacs-lisp :results none
  (with-temp-buffer
    (insert-file-contents "input")
    (advent/replace-multiple-regex-buffer
     '(("," . " ")
       ("^" . "(")
       ("$" . ")")))
    (goto-char (point-min))
    (insert "(setq data '(")
    (goto-char (point-max))
    (insert "))")
    (eval-buffer))
#+end_src

General preparation; the area is symmetric so I consider only
half of the pairs of vertices.
#+begin_src emacs-lisp :results none
  (defun area (el)
    (let ((a (car el))
  	(b (cdr el)))
      (* (1+ (abs (- (car a) (car b))))
         (1+ (abs (- (cadr a) (cadr b)))))))

  (defun symmetric-pairs (list)
    (apply #'append (--map-indexed (-map (lambda (other) (cons it other))
  				       (-drop (1+ it-index) list))
  				 list)))
#+end_src

This is for part 1.  Easy.
#+begin_src emacs-lisp
  (-max (-map 'area (symmetric-pairs data)))
#+end_src

#+RESULTS:
: 4737096935

For part 2, we begin by doing some preparation.
We collect corners and edges.
#+begin_src emacs-lisp :results none
  (setq data-prev (-rotate 1 data)
        data-next (-rotate -1 data)
        data-edges (-zip-lists data data-next)
        data-pv (-zip-lists data data-prev data-next))

  (defun normalize (x)
    (if (= x 0) 0
      (/ x (abs x))))

  (defun cornerize (a b)
    (list (normalize (- (car b) (car a)))
          (normalize (- (cadr b) (cadr a)))))

  (defun hor-or-ver (a b)
    (if (= (car a) (car b)) 'vertical
      'horizontal))

  (setq data-corners
        (--map (cons (car it)
                     (cons (cornerize (cadr it) (car it))
                           (cornerize (car it) (caddr it))))
               data-pv))

  (setq data-horizontal-edges (--filter (eq 'horizontal (apply #'hor-or-ver it)) data-edges)
        data-vertical-edges (--filter (eq 'vertical (apply #'hor-or-ver it)) data-edges))

  ;; Let us sort the edges
  (setq data-horizontal-edges (--sort (< (cadar it) (cadar other)) data-horizontal-edges)
        data-vertical-edges (--sort (< (caar it) (caar other)) data-vertical-edges))
#+end_src

We do some sanity check: is there any vertex that is not a corner?
#+begin_src emacs-lisp
  (--remove (= 0 (advent/dot (cadr it) (cddr it))) data-corners)
#+end_src

#+RESULTS:

No, that's good.  Now check if all vertices are distinct.
#+begin_src emacs-lisp
  (= (length (-distinct data)) (length data))
#+end_src

#+RESULTS:
: t

That's good. This eliminates some corner cases.  Now examine the
orientation; the result can be 1 or -1
if it is 1, then the domain is always to the right of the edges
if it is -1, then the domain is always to the left of the edges
#+begin_src emacs-lisp
  (setq orientation (normalize (apply '- (-map #'cadr (car data-vertical-edges)))))
#+end_src

#+RESULTS:
: 1

Now we cook up a function to check if a given point X Y is inside or outside
#+begin_src emacs-lisp :results none
  ;; The next two functions could be sped up by using bisection.
  ;; See if it is necessary

    (defun vertical-edges-up-to (x)
      (--take-while (<= (caar it) x) data-vertical-edges))

    (defun horizontal-edges-up-to (y)
      (--take-while (<= (cadar it) y) data-horizontal-edges))

    (defun vertical-edge-goes-through (edge y)
      (let ((miny (min (cadar edge) (cadadr edge)))
            (maxy (max (cadar edge) (cadadr edge))))
        (and (< miny y) (< y maxy))))

    (defun horizontal-edge-goes-through (edge x)
      (let ((minx (min (caar edge) (caadr edge)))
            (maxx (max (caar edge) (caadr edge))))
        (and (< minx x) (< x maxx))))

    (defun odd-p (n)
      (= 1 (logand n 1)))

    (defun even-p (n)
      (= 0 (logand n 1)))

    (defun inside-p (x y)
      (let* ((horizontal-head (horizontal-edges-up-to y))
             (vertical-head (vertical-edges-up-to x))
             (x (+ x 0.5))
             (y (+ y 0.5))
             (filtered-horizontal-head (--filter (horizontal-edge-goes-through it x) horizontal-head))
             (filtered-vertical-head (--filter (vertical-edge-goes-through it y) vertical-head)))
        (and (odd-p (length filtered-horizontal-head))
             (odd-p (length filtered-vertical-head)))))
#+end_src


 #+RESULTS:

Of course it would be too costly to check for every square in the
candidate rects, so we first remove rectangles that cannot work for a
number of reasons; since vertices are unique, if a vertex sits
strictly inside a rectangle, the rectangle must have some bad tiles
inside.  We call a rect _reasonable_ if such a thing does not happen.

#+begin_src emacs-lisp
  (setq rects (symmetric-pairs data))

  (defun strictly-contains-p (rect p)
    (let ((minx (min (caar rect) (cadr rect)))
          (maxx (max (caar rect) (cadr rect)))
          (miny (min (cadar rect) (caddr rect)))
          (maxy (max (cadar rect) (caddr rect)))
          (px (car p))
          (py (cadr p)))
      (and (< minx px) (< px maxx)
           (< miny py) (< py maxy))))

  (setq reasonable-rects
        (-remove (lambda (rect) (--any (strictly-contains-p rect it) data)) rects))

  (length reasonable-rects)
#+end_src

Nice, that is an OK number.
Since a rectangle is reasonable, possible edges that cut through it
would have to do through all of it.  We can thus check along the edges
only
#+begin_src emacs-lisp
  (defun whole-rect-inside-p (rect)
  (let ((minx (min (caar rect) (cadr rect)))
        (maxx (max (caar rect) (cadr rect)))
        (miny (min (cadar rect) (caddr rect)))
        (maxy (max (cadar rect) (caddr rect))))
    ;; Check along the minx column and miny row
    (and (--every (inside-p minx it) (-iota (- maxy miny) miny))
         (--every (inside-p it miny) (-iota (- maxx minx) minx)))))

  (setq sorted-reasonable-rects (--sort (> (area it) (area other)) reasonable-rects))
#+end_src

Grab some popcorn, this will take a while
#+begin_src emacs-lisp
  (let ((num 0))
    (setq largest-good-rect (--first (progn
                                       (message (format "%d" (setq num (1+ num))))
                                       (whole-rect-inside-p it))
                                     sorted-reasonable-rects)))
(area largest-good-rect)
#+end_src

#+RESULTS:
1644094530

what follows is an alternative way... first realize what the shape
looks like
   #+begin_src emacs-lisp
     (setq svg-preamble "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>
     <svg
        version=\"1.1\"
        id=\"svg1\"
        width=\"1000\"
        height=\"1000\"
        sodipodi:docname=\"tmp.svg\"
        xmlns=\"http://www.w3.org/2000/svg\"
        xmlns:svg=\"http://www.w3.org/2000/svg\">
       <defs
          id=\"defs1\" />
       <sodipodi:namedview
          id=\"namedview1\"
          pagecolor=\"#ffffff\"
          bordercolor=\"#666666\"
          borderopacity=\"1.0\"
          inkscape:showpageshadow=\"2\"
          inkscape:pageopacity=\"0.0\"
          inkscape:pagecheckerboard=\"0\"
          inkscape:deskcolor=\"#d1d1d1\"
          showgrid=\"false\"
          inkscape:zoom=\"1\"
          inkscape:cx=\"150\"
          inkscape:cy=\"75.067024\"
          inkscape:window-width=\"1000000\"
          inkscape:window-height=\"1000000\"
          inkscape:window-x=\"0\"
          inkscape:window-y=\"0\"
          inkscape:window-maximized=\"1\"
          inkscape:current-layer=\"svg1\" />")

     (defun scale-coordinate  (x)
       (* .01 x))

     (defun draw-svg ()
       (with-temp-buffer
        (let ((minx (-min (-map #'car data)))
              (maxx (-max (-map #'car data)))
              (miny (-min (-map #'cadr data)))
              (maxy (-max (-map #'cadr data))) )
          (insert svg-preamble) ; (format "<svg height=%d width=%d xmlns=\"http://www.w3.org/2000/svg\">" (+ maxy miny) (+ maxx minx))
          (--each data-edges (insert (format "<line x1=\"%f\" y1=\"%f\" x2=\"%f\" y2=\"%f\" style=\"stroke:#000;stroke-width:0.1\"/>\n"
                                            (scale-coordinate (caar it))
                                            (scale-coordinate (cadar it))
                                            (scale-coordinate (caadr it))
                                            (scale-coordinate (cadadr it)))))
          (insert "</svg>")
          )
        (write-file "tmp.svg")))
     (draw-svg)
   #+end_src

Oh motherfucker;

Find the two possible vertices
#+begin_src emacs-lisp
      ;; find the two longest horizontal edges
      (defun edge-width (it)
        (abs (- (caadr it) (caar it))))

      (setq special-two-edges (-take 2 (--sort (> (edge-width it) (edge-width other)) data-horizontal-edges))
            special-two-vertices (--map (list (max (caar it) (caadr it)) (cadar it)) special-two-edges))

      ;; the maximal rectangle will have one of these two as its vertex

      ;; let us now find the height at which the x cuts the outer shape
      (defun vertical-edge-goes-through (edge y)
          (let ((miny (min (cadar edge) (cadadr edge)))
                (maxy (max (cadar edge) (cadadr edge))))
            (and (<= miny y) (<= y maxy))))

        (defun horizontal-edge-goes-through (edge x)
          (let ((minx (min (caar edge) (caadr edge)))
                (maxx (max (caar edge) (caadr edge))))
            (and (<= minx x) (<= x maxx))))

      ;; this only gives two edges. We are lucky.  These are the two y-coordinates
      (setq y-range (-map #'cadar (-difference (--filter (horizontal-edge-goes-through it (caar special-two-vertices)) data-horizontal-edges) special-two-edges)))

      ;; now we filter the other vertices
      (setq candidates-top (--filter (and (>= (cadr it) (car y-range)) (< (cadr it) 50000) (< (car it) 50000)) data)
            candidates-bottom (--filter (and (<= (cadr it) (cadr y-range)) (> (cadr it) 50000) (< (car it) 50000)) data))


      (car (-sort '> (-map #'area (--map (cons (cadr special-two-vertices) it) candidates-bottom))))
      1643916742

      (-sort '> (-map #'area (--map (cons (car special-two-vertices) it) candidates-top)))
    candidates-bottom

  (cadr (--map (cons (cadr special-two-vertices) it) candidates-bottom))
#+end_src

#+RESULTS:
| (94532 50249) | 5639 | 68743 |

Then we can go from here...