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Problem 11 「格子内の最大の積」 †

20×20 の格子のうち連続した4数の積が最大になる部分を探す問題。
詳しくはリンク先を参照のこと。



解法
この問題は配列のない言語ではちょっとした難問です。
擬似的な配列としてリストのリストとしてnth0を2回適用してアクセスするのが一番楽です。
処理に失敗した部分はなかったことになるPrologの特性を利用して全部の地点で縦横斜めを検証し全ての4マス掛け算のリストを得その中で最大のものを答えとします。


もしも問題が10万行*10万列になった場合でも動く処理を書くとしたら、4行単位で一行ずつずらしながら処理してその4行で計算できる最大値と今まで見つかった最大値を検証するなど一工夫が必要です。


betweenを使った解法、

calc(_,_,_,[],[],Mult,Mult):-!.
calc(Map,Y,X,[DY|DYs],[DX|DXs],Mult,Result):-
	Y1 is Y+DY,
	X1 is X+DX,
	nth0(Y1,Map,Row),
	nth0(X1,Row,Num),
	Mult1 is Mult*Num,
	calc(Map,Y,X,DYs,DXs,Mult1,Result).

searchArrow(Map,Y,X,Result):-
 	member([DYs,DXs],[[[ 0, 0, 0, 0],[ 0, 1, 2, 3]],
			  [[ 0, 1, 2, 3],[ 0, 0, 0, 0]],
			  [[ 0, 1, 2, 3],[ 0,-1,-2,-3]],
			  [[ 0, 1, 2, 3],[ 0, 1, 2, 3]]]),
	calc(Map,Y,X,DYs,DXs,1,Result).

search(Map,Result):-
	between(0,19,X),
	between(0,19,Y),
	searchArrow(Map,Y,X,Result).


max(A,B,A):-A>B,!.
max(_,B,B):-!.

searchListMax([X],X):-!.
searchListMax([X|Xs],Result):-searchListMax(Xs,Re),max(Re,X,Result).
main11:-
	Map=[[08,02,22,97,38,15,00,40,00,75,04,05,07,78,52,12,50,77,91,08],
	     [49,49,99,40,17,81,18,57,60,87,17,40,98,43,69,48,04,56,62,00],
	     [81,49,31,73,55,79,14,29,93,71,40,67,53,88,30,03,49,13,36,65],
	     [52,70,95,23,04,60,11,42,69,24,68,56,01,32,56,71,37,02,36,91],
	     [22,31,16,71,51,67,63,89,41,92,36,54,22,40,40,28,66,33,13,80],
	     [24,47,32,60,99,03,45,02,44,75,33,53,78,36,84,20,35,17,12,50],
	     [32,98,81,28,64,23,67,10,26,38,40,67,59,54,70,66,18,38,64,70],
	     [67,26,20,68,02,62,12,20,95,63,94,39,63,08,40,91,66,49,94,21],
 	     [24,55,58,05,66,73,99,26,97,17,78,78,96,83,14,88,34,89,63,72],
	     [21,36,23,09,75,00,76,44,20,45,35,14,00,61,33,97,34,31,33,95],
	     [78,17,53,28,22,75,31,67,15,94,03,80,04,62,16,14,09,53,56,92],
	     [16,39,05,42,96,35,31,47,55,58,88,24,00,17,54,24,36,29,85,57],
	     [86,56,00,48,35,71,89,07,05,44,44,37,44,60,21,58,51,54,17,58],
	     [19,80,81,68,05,94,47,69,28,73,92,13,86,52,17,77,04,89,55,40],
	     [04,52,08,83,97,35,99,16,07,97,57,32,16,26,26,79,33,27,98,66],
 	     [88,36,68,87,57,62,20,72,03,46,33,67,46,55,12,32,63,93,53,69],
	     [04,42,16,73,38,25,39,11,24,94,72,18,08,46,29,32,40,62,76,36],
	     [20,69,36,41,72,30,23,88,34,62,99,69,82,67,59,85,74,04,36,16],
	     [20,73,35,29,78,31,90,01,74,31,49,71,48,86,81,16,23,57,05,54],
 	     [01,70,54,71,83,51,54,69,16,92,33,48,61,43,52,01,89,19,67,48]],
	findall(Mult,search(Map,Mult),Mults),
	searchListMax(Mults,Ans),
 	write(Ans).



原理主義的に再帰を使った解法。
calc(_,_,_,[],[],Mult,Mult):-!.
calc(Map,Y,X,[DY|DYs],[DX|DXs],Mult,Result):-
	Y1 is Y+DY,
	X1 is X+DX,
	nth0(Y1,Map,Row),
	nth0(X1,Row,Num),
	Mult1 is Mult*Num,
	calc(Map,Y,X,DYs,DXs,Mult1,Result).

searchArrow(Map,Y,X,Result):-
	member([DYs,DXs],[[[ 0, 0, 0, 0],[ 0, 1, 2, 3]],
			  [[ 0, 1, 2, 3],[ 0, 0, 0, 0]],
			  [[ 0, 1, 2, 3],[ 0,-1,-2,-3]],
			  [[ 0, 1, 2, 3],[ 0, 1, 2, 3]]]),
	calc(Map,Y,X,DYs,DXs,1,Result).
searchX(_,_,X,_):-
	19<X,
	!,
	fail.
searchX(Map,Y,X,Result):-
	searchArrow(Map,Y,X,Result).
searchX(Map,Y,X,Result):-
	X1 is X+1,
	searchX(Map,Y,X1,Result).

searchY(_,Y,_):-
	19<Y,
	!,
	fail.
searchY(Map,Y,Result):-
	searchX(Map,Y,0,Result).
searchY(Map,Y,Result):-
	Y1 is Y+1,
	searchY(Map,Y1,Result).

max(A,B,A):-A>B,!.
max(_,B,B):-!.

searchListMax([X],X):-!.
searchListMax([X|Xs],Result):-searchListMax(Xs,Re),max(Re,X,Result).
main11:-
	Map=[[08,02,22,97,38,15,00,40,00,75,04,05,07,78,52,12,50,77,91,08],
	     [49,49,99,40,17,81,18,57,60,87,17,40,98,43,69,48,04,56,62,00],
	     [81,49,31,73,55,79,14,29,93,71,40,67,53,88,30,03,49,13,36,65],
	     [52,70,95,23,04,60,11,42,69,24,68,56,01,32,56,71,37,02,36,91],
	     [22,31,16,71,51,67,63,89,41,92,36,54,22,40,40,28,66,33,13,80],
	     [24,47,32,60,99,03,45,02,44,75,33,53,78,36,84,20,35,17,12,50],
	     [32,98,81,28,64,23,67,10,26,38,40,67,59,54,70,66,18,38,64,70],
	     [67,26,20,68,02,62,12,20,95,63,94,39,63,08,40,91,66,49,94,21],
	     [24,55,58,05,66,73,99,26,97,17,78,78,96,83,14,88,34,89,63,72],
	     [21,36,23,09,75,00,76,44,20,45,35,14,00,61,33,97,34,31,33,95],
	     [78,17,53,28,22,75,31,67,15,94,03,80,04,62,16,14,09,53,56,92],
	     [16,39,05,42,96,35,31,47,55,58,88,24,00,17,54,24,36,29,85,57],
	     [86,56,00,48,35,71,89,07,05,44,44,37,44,60,21,58,51,54,17,58],
	     [19,80,81,68,05,94,47,69,28,73,92,13,86,52,17,77,04,89,55,40],
	     [04,52,08,83,97,35,99,16,07,97,57,32,16,26,26,79,33,27,98,66],
 	     [88,36,68,87,57,62,20,72,03,46,33,67,46,55,12,32,63,93,53,69],
	     [04,42,16,73,38,25,39,11,24,94,72,18,08,46,29,32,40,62,76,36],
	     [20,69,36,41,72,30,23,88,34,62,99,69,82,67,59,85,74,04,36,16],
	     [20,73,35,29,78,31,90,01,74,31,49,71,48,86,81,16,23,57,05,54],
	     [01,70,54,71,83,51,54,69,16,92,33,48,61,43,52,01,89,19,67,48]],
	findall(Mult,searchY(Map,0,Mult),Mults),
	searchListMax(Mults,Ans),
	write(Ans).