C#で Project Euler P11
問題
上の 20×20 の格子のうち, 斜めに並んだ4つの数字が赤くマークされている.
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
それらの数字の積は 26 × 63 × 78 × 14 = 1788696 となる.上の 20×20 の格子のうち, 上下左右斜めのいずれかの方向で連続する4つの数字の積のうち最大のものはいくつか?
Problem 11 - PukiWiki
方針
C#で Project Euler P8 - さんぽみちが拡張された問題。
よって、素直にこれを2次元へ拡張する。
解答
using System; using System.Collections.Generic; using System.Linq; using System.IO; public static class P11 { public static void Main() { Console.WriteLine(Calc()); } private const int MultiDepth = 4; private static Dictionary<Tuple<int, int, int>, Tuple<ulong, ulong, ulong, ulong>> multiMap = new Dictionary<Tuple<int, int, int>, Tuple<ulong, ulong, ulong, ulong>>(); private static ulong Calc() { string rawSquare; using(var sr = new StreamReader("P11_source.txt")) { rawSquare = sr.ReadToEnd(); } var square = rawSquare .Replace("\r\n", "\n") .Split('\n') .Select(line => line .Split(' ') .Select(elem => Convert.ToUInt64(elem)) .ToArray() ).ToArray(); MakeMap(square, MultiDepth); ulong max = 0; for(var y = 0; y < square.Length - MultiDepth; y++) { for(var x = MultiDepth; x < square[0].Length - MultiDepth; x++) { if(multiMap.ContainsKey(Tuple.Create(MultiDepth, x, y))) { var val = multiMap[Tuple.Create(MultiDepth, x, y)]; max = val.Item1 > max ? val.Item1 : max; max = val.Item2 > max ? val.Item2 : max; max = val.Item3 > max ? val.Item3 : max; max = val.Item4 > max ? val.Item4 : max; } } } return max; } private static void MakeMap(ulong[][] square, int depth) { MakeMapD1(square); for(int depthWork = depth >> 1, depthIndex = 2; depthWork > 0; depthWork >>= 1, depthIndex <<= 1) { var beforeDepth = depthIndex >> 1; for(var y = 0; y < square.Length - 1 - depthIndex; y++) { for(var x = 0; x < depthIndex; x++) { var before = multiMap[Tuple.Create(beforeDepth, x, y)]; SetMap( depthIndex, x, y, before.Item1 * multiMap[Tuple.Create(beforeDepth, x + beforeDepth, y)].Item1, before.Item2 * multiMap[Tuple.Create(beforeDepth, x, y + beforeDepth)].Item2, before.Item3 * multiMap[Tuple.Create(beforeDepth, x + beforeDepth, y + beforeDepth)].Item3, 0 ); } for(var x = depthIndex; x < square[0].Length - 1 - depthIndex; x++) { var before = multiMap[Tuple.Create(beforeDepth, x, y)]; SetMap( depthIndex, x, y, before.Item1 * multiMap[Tuple.Create(beforeDepth, x + beforeDepth, y)].Item1, before.Item2 * multiMap[Tuple.Create(beforeDepth, x, y + beforeDepth)].Item2, before.Item3 * multiMap[Tuple.Create(beforeDepth, x + beforeDepth, y + beforeDepth)].Item3, before.Item4 * multiMap[Tuple.Create(beforeDepth, x - beforeDepth, y + beforeDepth)].Item4 ); } for(var x = square[0].Length - 1 - depthIndex; x < square[0].Length; x++) { var before = multiMap[Tuple.Create(beforeDepth, x, y)]; SetMap( depthIndex, x, y, 0, before.Item2 * multiMap[Tuple.Create(beforeDepth, x, y + beforeDepth)].Item2, 0, before.Item4 * multiMap[Tuple.Create(beforeDepth, x - beforeDepth, y + beforeDepth)].Item4 ); } } for(var y = square.Length - 1 - depthIndex; y < square.Length; y++) { for(var x = 0; x < square[0].Length - 1 - depthIndex; x++) { var before = multiMap[Tuple.Create(beforeDepth, x, y)]; SetMap( depthIndex, x, y, before.Item1 * multiMap[Tuple.Create(beforeDepth, x + beforeDepth, y)].Item1, 0, 0, 0 ); } } } } private static void MakeMapD1(ulong[][] square) { for(var y = 0; y < square.Length - 1; y++) { SetMap(1, 0, y, square[y][0], square[y][0], square[y][0], 0); for(var x = 1; x < square[0].Length - 1; x++) { SetMap(1, x, y, square[y][x], square[y][x], square[y][x], square[y][x]); } SetMap(1, square[0].Length - 1, y, 0, square[y][square[0].Length - 1], 0, square[y][square[0].Length - 1]); } for(var x = 0; x < square[0].Length - 1; x++) { SetMap(1, x, square.Length - 1, square[square.Length - 1][x], 0, 0, 0); } } private static void SetMap(int depth, int x, int y, ulong h, ulong v, ulong s, ulong b) { multiMap.Add(Tuple.Create(depth, x, y), Tuple.Create(h, b, s, b)); } }
計算量
本プログラム：$O(n^2 \log m)$
総当たり：$O(n^2m)$
$ n $：格子の一辺あたりの長さ
$ m $：乗じる数字の長さ
一言だけ言いたいのが、オーダー記法では反映されなくて悲しいが、1つの層を処理するたびに計算する必要のある数字の一辺の長さは $2^{(階層)}$ ずつ減っていくんだよ！
$ \log n $ まで計算すると、斜面が対数関数の形をしたピラミッドの頂上となって数字は一つしか残っていないのを考えるとわかりやすいよ！
