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;;; Google Code Jam 2020, Round 1B, Problem 1: Expogo
;;
;; This is suboptimal but works for all cases.
;;
;; We do a recursive search with memoization (caching), and prune
;; impossible regions.
;;
;; Apparently there is also a constructive method, which I suspected
;; but couldn't find.
(defun solve (&optional (in *standard-input*))
(dotimes (caseno (the (integer 0 1000) (read in)))
(format t "Case #~D: " (+ caseno 1))
(solve-case in)))
(defun solve-case (in)
(solve-case-1 (read in) (read in)))
(defun solve-case-1 (x y)
(let ((min-jumps (floor (log (+ (abs x) (abs y)) 2))))
(let ((memo (make-hash-table :test #'equal)))
(do ((k (- min-jumps 1) (+ k 1)))
((> k (+ min-jumps 1))
(format t "IMPOSSIBLE~%"))
(let ((sol (solve-case-2 (ash 1 k) x y memo)))
(unless (eq sol 'false)
(dolist (move (nreverse sol))
(format t "~C" move))
(format t "~%")
(return-from solve-case-1 t)))))))
(defun solve-case-2 (jump x y memo)
(let ((result (gethash (list jump x y) memo)))
(or result
(setf (gethash (list jump x y) memo)
(cond ((= jump 0)
(if (= x y 0) '() 'false))
((>= (+ (abs x) (abs y)) (+ jump jump)) 'false)
(t (dolist (dir (list #\W #\E #\N #\S) 'false)
(multiple-value-bind (xinc yinc)
(ecase dir
((#\W) (values (- jump) 0))
((#\E) (values jump 0))
((#\N) (values 0 jump))
((#\S) (values 0 (- jump))))
(let ((sol (solve-case-2
(ash jump -1)
(- x xinc)
(- y yinc)
memo)))
(unless (eq sol 'false)
(return (cons dir sol))))))))))))
(solve)