DPL_1_G: 個数制限付きナップザック問題

問題へのリンク

C#のソース

using System;
using System.Collections.Generic;
using System.Linq;

class Program
{
    static string InputPattern = "InputX";

    static List<string> GetInputList()
    {
        var WillReturn = new List<string>();

        if (InputPattern == "Input1") {
            WillReturn.Add("4 8");
            WillReturn.Add("4 3 2");
            WillReturn.Add("2 1 1");
            WillReturn.Add("1 2 4");
            WillReturn.Add("3 2 2");
            //12
        }
        else if (InputPattern == "Input2") {
            WillReturn.Add("2 100");
            WillReturn.Add("1 1 100");
            WillReturn.Add("2 1 50");
            //150
        }
        else {
            string wkStr;
            while ((wkStr = Console.ReadLine()) != null) WillReturn.Add(wkStr);
        }
        return WillReturn;
    }

    struct ItemInfoDef
    {
        internal long Value;
        internal long Weight;
    }
    static List<ItemInfoDef> mItemInfoList = new List<ItemInfoDef>();

    static void Main()
    {
        List<string> InputList = GetInputList();

        long[] wkArr = { };
        Action<string> SplitAct = pStr =>
            wkArr = pStr.Split(' ').Select(pX => long.Parse(pX)).ToArray();

        SplitAct(InputList[0]);
        long WeightLimit = wkArr[1];

        foreach (string EachStr in InputList.Skip(1)) {
            SplitAct(EachStr);

            long Value = wkArr[0];
            long Weight = wkArr[1];
            long Cnt = wkArr[2];

            var Beki2List = new List<long>();
            while (Cnt > 0) {
                var ResultBeki2List = DeriveBeki2List(ref Cnt);
                Beki2List.AddRange(ResultBeki2List);
            }

            foreach (long EachCnt in Beki2List) {
                ItemInfoDef WillAdd;
                WillAdd.Value = Value * EachCnt;
                WillAdd.Weight = Weight * EachCnt;
                mItemInfoList.Add(WillAdd);
            }
        }

        // 最大価値[重さ合計]なインラインDP表
        long UB = WeightLimit;
        long?[] DPArr = new long?[UB + 1];
        DPArr[0] = 0;

        foreach (ItemInfoDef EachItem in mItemInfoList) {
            for (long I = UB; 0 <= I; I--) {
                if (DPArr[I].HasValue == false) continue;
                long NewI = I + EachItem.Weight;

                if (NewI > UB) continue;

                long NewVal = DPArr[I].Value + EachItem.Value;

                if (DPArr[NewI].HasValue) {
                    if (DPArr[NewI].Value >= NewVal) {
                        continue;
                    }
                }
                DPArr[NewI] = NewVal;
            }
        }
        Console.WriteLine(DPArr.Max());
    }

    static List<long> DeriveBeki2List(ref long pVal)
    {
        var Beki2List = new List<long>();

        long Beki2 = 1;
        while (true) {
            if (pVal < Beki2) break;

            Beki2List.Add(Beki2);
            pVal -= Beki2;
            Beki2 *= 2;
        }
        return Beki2List;
    }
}

解説

下記のように、ダブリングの考え方を使用してます。

品物が100個使える場合
1+2+4+8+16+32 = 63

100 - 63 = 37
1+2+4+8+16 = 31

37 - 31 = 6
1+2 = 3

6 - 3 = 3
1+2 = 3

で、1,2,4,8,16,32,1,2,4,8,16,1,2,1,2
ずつ使用できるとして、
DPの遷移回数を減らしてます。