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Problem 18
三角形を頂点から下まで移動するときの最大の和を求めよ
以下の三角形の頂点から下まで移動するとき, その数値の和の最大値は23になる.
3
7 4
2 4 6
8 5 9 3
この例では .
以下の三角形を頂点から下まで移動するとき, その最大の和を求めよ.
75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88 02 77 73 07 63 67
99 65 04 28 06 16 70 92
41 41 26 56 83 40 80 70 33
41 48 72 33 47 32 37 16 94 29
53 71 44 65 25 43 91 52 97 51 14
70 11 33 28 77 73 17 78 39 68 17 57
91 71 52 38 17 14 91 43 58 50 27 29 48
63 66 04 68 89 53 67 30 73 16 69 87 40 31
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23
注: ここではたかだか 16384 通りのルートしかないので, すべてのパターンを試すこともできる. Problem 67 は同じ問題だが 100行あるので, 総当りでは解けない. もっと賢い方法が必要である.
Maximum path sum I
By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.
3
7 4
2 4 6
8 5 9 3
That is, .
Find the maximum total from top to bottom of the triangle below:
75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88 02 77 73 07 63 67
99 65 04 28 06 16 70 92
41 41 26 56 83 40 80 70 33
41 48 72 33 47 32 37 16 94 29
53 71 44 65 25 43 91 52 97 51 14
70 11 33 28 77 73 17 78 39 68 17 57
91 71 52 38 17 14 91 43 58 50 27 29 48
63 66 04 68 89 53 67 30 73 16 69 87 40 31
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23
NOTE: As there are only 16384 routes, it is possible to solve this problem by trying every route. However, Problem 67, is the same challenge with a triangle containing one-hundred rows; it cannot be solved by brute force, and requires a clever method! ;o)
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Scala
更新日 : 2013.02.01
scala> var str = """
| 75
| 95 64
| 17 47 82
| 18 35 87 10
| 20 04 82 47 65
| 19 01 23 75 03 34
| 88 02 77 73 07 63 67
| 99 65 04 28 06 16 70 92
| 41 41 26 56 83 40 80 70 33
| 41 48 72 33 47 32 37 16 94 29
| 53 71 44 65 25 43 91 52 97 51 14
| 70 11 33 28 77 73 17 78 39 68 17 57
| 91 71 52 38 17 14 91 43 58 50 27 29 48
| 63 66 04 68 89 53 67 30 73 16 69 87 40 31
| 04 62 98 27 23 09 70 98 73 93 38 53 60 04 23
| """
str: java.lang.String =
"
75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88 02 77 73 07 63 67
99 65 04 28 06 16 70 92
41 41 26 56 83 40 80 70 33
41 48 72 33 47 32 37 16 94 29
53 71 44 65 25 43 91 52 97 51 14
70 11 33 28 77 73 17 78 39 68 17 57
91 71 52 38 17 14 91 43 58 50 27 29 48
63 66 04 68 89 53 67 30 73 16 69 87 40 31
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23
"
scala> val ary = str.trim().split("\n").toList
ary: List[java.lang.String] = List(75, 95 64, 17 47 82, 18 35 87 10, 20 04 82 47 65, 19 01 23 75 03 34, 88 02 77 73 07 63 67, 99 65 04 28 06 16 70 92, 41 41 26 56 83 40 80 70 33, 41 48 72 33 47 32 37 16 94 29, 53 71 44 65 25 43 91 52 97 51 14, 70 11 33 28 77 73 17 78 39 68 17 57, 91 71 52 38 17 14 91 43 58 50 27 29 48, 63 66 04 68 89 53 67 30 73 16 69 87 40 31, 04 62 98 27 23 09 70 98 73 93 38 53 60 04 23)
scala> val tri = ary.map(str => str.split(" ").toList)
tri: List[List[java.lang.String]] = List(List(75), List(95, 64), List(17, 47, 82), List(18, 35, 87, 10), List(20, 04, 82, 47, 65), List(19, 01, 23, 75, 03, 34), List(88, 02, 77, 73, 07, 63, 67), List(99, 65, 04, 28, 06, 16, 70, 92), List(41, 41, 26, 56, 83, 40, 80, 70, 33), List(41, 48, 72, 33, 47, 32, 37, 16, 94, 29), List(53, 71, 44, 65, 25, 43, 91, 52, 97, 51, 14), List(70, 11, 33, 28, 77, 73, 17, 78, 39, 68, 17, 57), List(91, 71, 52, 38, 17, 14, 91, 43, 58, 50, 27, 29, 48), List(63, 66, 04, 68, 89, 53, 67, 30, 73, 16, 69, 87, 40, 31), List(04, 62, 98, 27, 23, 09, 70, 98, 73, 93, 38, 53, 60, 04, 23))
scala> val rev = tri.map(lst => lst.map(str => str.toInt)).reverse rev: List[List[Int]] = List(List(4, 62, 98, 27, 23, 9, 70, 98, 73, 93, 38, 53, 60, 4, 23), List(63, 66, 4, 68, 89, 53, 67, 30, 73, 16, 69, 87, 40, 31), List(91, 71, 52, 38, 17, 14, 91, 43, 58, 50, 27, 29, 48), List(70, 11, 33, 28, 77, 73, 17, 78, 39, 68, 17, 57), List(53, 71, 44, 65, 25, 43, 91, 52, 97, 51, 14), List(41, 48, 72, 33, 47, 32, 37, 16, 94, 29), List(41, 41, 26, 56, 83, 40, 80, 70, 33), List(99, 65, 4, 28, 6, 16, 70, 92), List(88, 2, 77, 73, 7, 63, 67), List(19, 1, 23, 75, 3, 34), List(20, 4, 82, 47, 65), List(18, 35, 87, 10), List(17, 47, 82), List(95, 64), List(75))
scala> def large(list: List[Int]):List[Int] = list match {
| case x :: xs => if (xs.isEmpty) List() else {math.max(x, xs.head) :: large(xs)}
| case Nil => List()
| }
large: (list: List[Int])List[Int]
scala> var large_ary = large(rev.head)
large_ary: List[Int] = List(62, 98, 98, 27, 23, 70, 98, 98, 93, 93, 53, 60, 60, 23)
scala> for (ary <- rev) {
| if (ary.length == large_ary.length) {
| val lst = (ary zip large_ary).map(t => t match {case (x,y) => x + y})
| for (x <- lst) print(x + ",")
| println
| large_ary = large(lst)
| }
| }
125,164,102,95,112,123,165,128,166,109,122,147,100,54,
255,235,154,150,140,179,256,209,224,172,174,176,148,
325,246,187,178,256,329,273,302,263,242,193,233,
378,317,231,321,354,372,393,354,360,293,247,
419,365,393,387,419,425,430,376,454,322,
460,434,419,475,508,470,510,524,487,
559,499,479,536,514,526,594,616,
647,501,613,609,533,657,683,
666,614,636,684,660,717,
686,640,766,731,782,
704,801,853,792,
818,900,935,
995,999,
1074,
F#
更新日 : 2013.02.01
> let str = " - 75 - 95 64 - 17 47 82 - 18 35 87 10 - 20 04 82 47 65 - 19 01 23 75 03 34 - 88 02 77 73 07 63 67 - 99 65 04 28 06 16 70 92 - 41 41 26 56 83 40 80 70 33 - 41 48 72 33 47 32 37 16 94 29 - 53 71 44 65 25 43 91 52 97 51 14 - 70 11 33 28 77 73 17 78 39 68 17 57 - 91 71 52 38 17 14 91 43 58 50 27 29 48 - 63 66 04 68 89 53 67 30 73 16 69 87 40 31 - 04 62 98 27 23 09 70 98 73 93 38 53 60 04 23 - " - ;; val str : string = " 75 95 64 17 47 82 18 35 87 10 20 04 82 47 65 19 01 23 75 03 "+[300 chars]
> let ary = str.Trim().Split('\n')
- ;;
val ary : string [] =
[|"75"; "95 64"; "17 47 82"; "18 35 87 10"; "20 04 82 47 65";
"19 01 23 75 03 34"; "88 02 77 73 07 63 67"; "99 65 04 28 06 16 70 92";
"41 41 26 56 83 40 80 70 33"; "41 48 72 33 47 32 37 16 94 29";
"53 71 44 65 25 43 91 52 97 51 14"; "70 11 33 28 77 73 17 78 39 68 17 57";
"91 71 52 38 17 14 91 43 58 50 27 29 48";
"63 66 04 68 89 53 67 30 73 16 69 87 40 31";
"04 62 98 27 23 09 70 98 73 93 38 53 60 04 23"|]
> let tri = ary |> Array.map(fun str -> str.Split(' '))
- ;;
val tri : string [] [] =
[|[|"75"|]; [|"95"; "64"|]; [|"17"; "47"; "82"|]; [|"18"; "35"; "87"; "10"|];
[|"20"; "04"; "82"; "47"; "65"|]; [|"19"; "01"; "23"; "75"; "03"; "34"|];
[|"88"; "02"; "77"; "73"; "07"; "63"; "67"|];
[|"99"; "65"; "04"; "28"; "06"; "16"; "70"; "92"|];
[|"41"; "41"; "26"; "56"; "83"; "40"; "80"; "70"; "33"|];
[|"41"; "48"; "72"; "33"; "47"; "32"; "37"; "16"; "94"; "29"|];
[|"53"; "71"; "44"; "65"; "25"; "43"; "91"; "52"; "97"; "51"; "14"|];
[|"70"; "11"; "33"; "28"; "77"; "73"; "17"; "78"; "39"; "68"; "17"; "57"|];
[|"91"; "71"; "52"; "38"; "17"; "14"; "91"; "43"; "58"; "50"; "27"; "29";
"48"|];
[|"63"; "66"; "04"; "68"; "89"; "53"; "67"; "30"; "73"; "16"; "69"; "87";
"40"; "31"|];
[|"04"; "62"; "98"; "27"; "23"; "09"; "70"; "98"; "73"; "93"; "38"; "53";
"60"; "04"; "23"|]|]
> let rev = tri |> Array.rev
- ;;
val rev : string [] [] =
[|[|"04"; "62"; "98"; "27"; "23"; "09"; "70"; "98"; "73"; "93"; "38"; "53";
"60"; "04"; "23"|];
[|"63"; "66"; "04"; "68"; "89"; "53"; "67"; "30"; "73"; "16"; "69"; "87";
"40"; "31"|];
[|"91"; "71"; "52"; "38"; "17"; "14"; "91"; "43"; "58"; "50"; "27"; "29";
"48"|];
[|"70"; "11"; "33"; "28"; "77"; "73"; "17"; "78"; "39"; "68"; "17"; "57"|];
[|"53"; "71"; "44"; "65"; "25"; "43"; "91"; "52"; "97"; "51"; "14"|];
[|"41"; "48"; "72"; "33"; "47"; "32"; "37"; "16"; "94"; "29"|];
[|"41"; "41"; "26"; "56"; "83"; "40"; "80"; "70"; "33"|];
[|"99"; "65"; "04"; "28"; "06"; "16"; "70"; "92"|];
[|"88"; "02"; "77"; "73"; "07"; "63"; "67"|];
[|"19"; "01"; "23"; "75"; "03"; "34"|]; [|"20"; "04"; "82"; "47"; "65"|];
[|"18"; "35"; "87"; "10"|]; [|"17"; "47"; "82"|]; [|"95"; "64"|]; [|"75"|]|]
> let rev = rev |> Array.map(fun ary -> ary |> Array.map(fun str -> int32 str) |> Array.toList) |> Array.toList - ;; val rev : int32 list list = [[4; 62; 98; 27; 23; 9; 70; 98; 73; 93; 38; 53; 60; 4; 23]; [63; 66; 4; 68; 89; 53; 67; 30; 73; 16; 69; 87; 40; 31]; [91; 71; 52; 38; 17; 14; 91; 43; 58; 50; 27; 29; 48]; [70; 11; 33; 28; 77; 73; 17; 78; 39; 68; 17; 57]; [53; 71; 44; 65; 25; 43; 91; 52; 97; 51; 14]; [41; 48; 72; 33; 47; 32; 37; 16; 94; 29]; [41; 41; 26; 56; 83; 40; 80; 70; 33]; [99; 65; 4; 28; 6; 16; 70; 92]; [88; 2; 77; 73; 7; 63; 67]; [19; 1; 23; 75; 3; 34]; [20; 4; 82; 47; 65]; [18; 35; 87; 10]; [17; 47; 82]; [95; 64]; [75]]
> let rec large lst = - match lst with - | x::xs -> if xs.IsEmpty then [] else max x xs.Head :: large xs - | [] -> [] - ;; val large : 'a list -> 'a list when 'a : comparison
> let mutable large_ary = large rev.Head - ;; val mutable large_ary : int32 list = [62; 98; 98; 27; 23; 70; 98; 98; 93; 93; 53; 60; 60; 23] > for ary in rev do - if ary.Length = large_ary.Length then - let lst = - (List.zip ary large_ary) - |> List.map(fun (x, y) -> x + y) - - for x in lst do printf "%d," x - printf "\n" - - large_ary <- large lst - ;; 125,164,102,95,112,123,165,128,166,109,122,147,100,54, 255,235,154,150,140,179,256,209,224,172,174,176,148, 325,246,187,178,256,329,273,302,263,242,193,233, 378,317,231,321,354,372,393,354,360,293,247, 419,365,393,387,419,425,430,376,454,322, 460,434,419,475,508,470,510,524,487, 559,499,479,536,514,526,594,616, 647,501,613,609,533,657,683, 666,614,636,684,660,717, 686,640,766,731,782, 704,801,853,792, 818,900,935, 995,999, 1074, val it : unit = ()
