TDP N 木

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");
            WillReturn.Add("1 2");
            WillReturn.Add("2 3");
            WillReturn.Add("3 4");
            //4
        }
        else if (InputPattern == "Input2") {
            WillReturn.Add("8");
            WillReturn.Add("1 2");
            WillReturn.Add("4 6");
            WillReturn.Add("6 7");
            WillReturn.Add("3 2");
            WillReturn.Add("2 4");
            WillReturn.Add("4 5");
            WillReturn.Add("8 6");
            //752
        }
        else {
            string wkStr;
            while ((wkStr = Console.ReadLine()) != null) WillReturn.Add(wkStr);
        }
        return WillReturn;
    }

    const long Hou = 1000000007;

    static long mN;

    // 隣接リスト
    static Dictionary<long, List<long>> mToNodeListDict = new Dictionary<long, List<long>>();

    static ChooseMod mInsChooseMod;

    // 木DPとReRootingの設定01 配られてる値が無い時の値
    static NodeInfo DeriveDefaultNodeVal()
    {
        NodeInfo DefaultNodeVal;
        DefaultNodeVal.PatternCnt = 1;
        DefaultNodeVal.NodeCnt = 0;
        return DefaultNodeVal;
    }

    // 木DPとReRootingの設定02 ノード値を引数として、配る値を返す
    static NodeInfo DeriveSendVal(NodeInfo pNodeInfo)
    {
        NodeInfo Copied = pNodeInfo;
        Copied.NodeCnt++;
        return Copied;
    }

    // 木DPとReRootingの設定03 ノード値と、配られた値を、引数とし、新しいノード値を返す
    static NodeInfo DeriveNewNodeVal(NodeInfo pNodeVal, NodeInfo pSendVal)
    {
        pNodeVal.PatternCnt *= pSendVal.PatternCnt;
        pNodeVal.PatternCnt %= Hou;

        // nCrを掛ける
        long n = pSendVal.NodeCnt + pNodeVal.NodeCnt;
        long r = pNodeVal.NodeCnt;
        pNodeVal.PatternCnt *= mInsChooseMod.DeriveChoose(n, r);
        pNodeVal.PatternCnt %= Hou;

        pNodeVal.NodeCnt += pSendVal.NodeCnt;

        return pNodeVal;
    }

    // 木DPとReRootingの設定04 配る値のListを引数として、最終的な配る値を返す
    static NodeInfo DeriveMergedSendVal(List<NodeInfo> pSendValList)
    {
        NodeInfo SendVal = DeriveDefaultNodeVal();
        foreach (NodeInfo EachSendVal in pSendValList) {
            SendVal = DeriveNewNodeVal(SendVal, EachSendVal);
        }
        return SendVal;
    }

    struct NodeInfo
    {
        internal long PatternCnt;
        internal long NodeCnt;
    }

    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]);
        mN = wkArr[0];

        mInsChooseMod = new ChooseMod(mN, Hou);

        foreach (string EachStr in InputList.Skip(1)) {
            SplitAct(EachStr);
            long S = wkArr[0];
            long T = wkArr[1];

            if (mToNodeListDict.ContainsKey(S) == false) {
                mToNodeListDict[S] = new List<long>();
            }
            if (mToNodeListDict.ContainsKey(T) == false) {
                mToNodeListDict[T] = new List<long>();
            }
            mToNodeListDict[S].Add(T);
            mToNodeListDict[T].Add(S);
        }

        // 前後に累積値を使うので、ノード番号の昇順にソート
        foreach (var EachPair in mToNodeListDict) {
            EachPair.Value.Sort();
        }

        // 状態[ノード]なDP表
        var TreeDPDict = new Dictionary<long, NodeInfo>();

        // 根から葉に向かってDFSしてノードのListを返す
        List<JyoutaiDef1> DFSResult = ExecDFS1(1);
        DFSResult.Reverse(); // 葉から根への順番にする

        foreach (JyoutaiDef1 EachJyoutai in DFSResult) {
            // 親ノードに配る木DP
            long Node = EachJyoutai.CurrNode;
            if (TreeDPDict.ContainsKey(Node) == false) {
                TreeDPDict[Node] = DeriveDefaultNodeVal();
            }

            long ParentNode = EachJyoutai.ParentNode;

            // 根ノードの場合
            if (ParentNode == -1) continue;

            if (TreeDPDict.ContainsKey(ParentNode) == false) {
                TreeDPDict[ParentNode] = DeriveDefaultNodeVal();
            }
            TreeDPDict[ParentNode] =
                DeriveNewNodeVal(TreeDPDict[ParentNode], DeriveSendVal(TreeDPDict[Node]));
        }

        // 木DPの結果
        //foreach (var EachPair in TreeDPDict.OrderBy(pX => pX.Key)) {
        //    Console.WriteLine("TreeDPDict[{0}]=(NodeCnt={1},PatternCnt={2})",
        //        EachPair.Key,
        //        EachPair.Value.NodeCnt,
        //        EachPair.Value.PatternCnt);
        //}

        // ReRootingなDFSを行う
        long Answer = 0;
        foreach (NodeInfo EachNodeInfo in ExecDFS2(1, TreeDPDict)) {
            Answer += EachNodeInfo.PatternCnt;
            Answer %= Hou;
        }
        Answer *= DeriveGyakugen(2);
        Answer %= Hou;
        Console.WriteLine(Answer);
    }

    // 根から葉に向かってDFSしてノードのListを返す
    struct JyoutaiDef1
    {
        internal long CurrNode;
        internal long ParentNode;
    }
    static List<JyoutaiDef1> ExecDFS1(long pRootNode)
    {
        var WillReturn = new List<JyoutaiDef1>();

        var Stk = new Stack<JyoutaiDef1>();
        JyoutaiDef1 WillPush;
        WillPush.CurrNode = pRootNode;
        WillPush.ParentNode = -1;
        Stk.Push(WillPush);

        var VisitedSet = new HashSet<long>();
        VisitedSet.Add(pRootNode);

        while (Stk.Count > 0) {
            JyoutaiDef1 Popped = Stk.Pop();
            WillReturn.Add(Popped);
            if (mToNodeListDict.ContainsKey(Popped.CurrNode) == false) {
                continue;
            }
            foreach (int EachToNode in mToNodeListDict[Popped.CurrNode]) {
                if (VisitedSet.Add(EachToNode)) {
                    WillPush.CurrNode = EachToNode;
                    WillPush.ParentNode = Popped.CurrNode;
                    Stk.Push(WillPush);
                }
            }
        }
        return WillReturn;
    }

    // ReRootingなDFSを行う
    struct JyoutaiDef2
    {
        internal long PrevRootNode;
        internal NodeInfo PrevRootVal;
        internal long CurrRootNode;
        internal NodeInfo CurrRootVal;
    }
    static IEnumerable<NodeInfo> ExecDFS2(long pRootNode, Dictionary<long, NodeInfo> pTreeDPDict)
    {
        var WillReturn = new List<long>();

        var Stk = new Stack<JyoutaiDef2>();
        JyoutaiDef2 WillPush;
        WillPush.PrevRootNode = -1;
        WillPush.PrevRootVal = new NodeInfo();
        WillPush.CurrRootNode = pRootNode;
        WillPush.CurrRootVal = pTreeDPDict[pRootNode];
        Stk.Push(WillPush);

        var VisitedSet = new HashSet<long>();
        VisitedSet.Add(pRootNode);

        var AnswerDict = new Dictionary<long, NodeInfo>();
        while (Stk.Count > 0) {
            JyoutaiDef2 Popped = Stk.Pop();
            AnswerDict[Popped.CurrRootNode] = Popped.CurrRootVal;

            // 累積値を管理する
            int ChildCnt = 0;
            if (mToNodeListDict.ContainsKey(Popped.CurrRootNode)) {
                mToNodeListDict[Popped.CurrRootNode].Remove(Popped.PrevRootNode);
                ChildCnt = mToNodeListDict[Popped.CurrRootNode].Count;
            }
            if (ChildCnt == 0) continue;

            NodeInfo[] RunValSei = new NodeInfo[ChildCnt]; // 正方向の累積値
            NodeInfo[] RunValRev = new NodeInfo[ChildCnt]; // 逆方向の累積値

            long StaNode = mToNodeListDict[Popped.CurrRootNode][0];
            long EndNode = mToNodeListDict[Popped.CurrRootNode][ChildCnt - 1];
            RunValSei[0] = DeriveSendVal(pTreeDPDict[StaNode]);
            RunValRev[RunValRev.GetUpperBound(0)] = DeriveSendVal(pTreeDPDict[EndNode]);

            for (int I = 1; I <= ChildCnt - 1; I++) {
                long CurrChildNode = mToNodeListDict[Popped.CurrRootNode][I];
                RunValSei[I] =
                    DeriveNewNodeVal(RunValSei[I - 1], DeriveSendVal(pTreeDPDict[CurrChildNode]));
            }
            for (int I = ChildCnt - 2; 0 <= I; I--) {
                long CurrChildNode = mToNodeListDict[Popped.CurrRootNode][I];
                RunValRev[I] = DeriveNewNodeVal(RunValRev[I + 1], DeriveSendVal(pTreeDPDict[CurrChildNode]));
            }

            // 状態の遷移
            for (int I = 0; I <= mToNodeListDict[Popped.CurrRootNode].Count - 1; I++) {
                long CurrRootNode = mToNodeListDict[Popped.CurrRootNode][I];
                if (VisitedSet.Add(CurrRootNode) == false) {
                    continue;
                }

                // 左方向の累積値
                NodeInfo LeftRunVal = DeriveDefaultNodeVal();
                bool HasLeftRunVal = false;
                if (I > 0) {
                    LeftRunVal = RunValSei[I - 1];
                    HasLeftRunVal = true;
                }

                // 右方向の累積値
                NodeInfo RightRunVal = DeriveDefaultNodeVal();
                bool HasRightRunVal = false;
                if (I < mToNodeListDict[Popped.CurrRootNode].Count - 1) {
                    RightRunVal = RunValRev[I + 1];
                    HasRightRunVal = true;
                }

                WillPush.PrevRootNode = Popped.CurrRootNode;
                var PrevRootValKouhoList = new List<NodeInfo>();
                if (HasLeftRunVal) PrevRootValKouhoList.Add(LeftRunVal);
                if (HasRightRunVal) PrevRootValKouhoList.Add(RightRunVal);
                if (Popped.PrevRootNode > -1) {
                    PrevRootValKouhoList.Add(DeriveSendVal(Popped.PrevRootVal));
                }
                WillPush.PrevRootVal = DeriveMergedSendVal(PrevRootValKouhoList);

                WillPush.CurrRootNode = CurrRootNode;
                var CurrRootValKouhoList = new List<NodeInfo>();
                CurrRootValKouhoList.Add(pTreeDPDict[CurrRootNode]);
                CurrRootValKouhoList.Add(DeriveSendVal(WillPush.PrevRootVal));
                WillPush.CurrRootVal = DeriveMergedSendVal(CurrRootValKouhoList);

                Stk.Push(WillPush);
            }
        }

        foreach (var EachPair in AnswerDict.OrderBy(pX => pX.Key)) {
            yield return EachPair.Value;
        }
    }

    // 引数の逆元を求める
    static long DeriveGyakugen(long pLong)
    {
        return DeriveBekijyou(pLong, Hou - 2, Hou);
    }

    // 繰り返し2乗法で、(NのP乗) Mod Mを求める
    static long DeriveBekijyou(long pN, long pP, long pM)
    {
        long CurrJyousuu = pN % pM;
        long CurrShisuu = 1;
        long WillReturn = 1;

        while (true) {
            // 対象ビットが立っている場合
            if ((pP & CurrShisuu) > 0) {
                WillReturn = (WillReturn * CurrJyousuu) % pM;
            }

            CurrShisuu *= 2;
            if (CurrShisuu > pP) return WillReturn;
            CurrJyousuu = (CurrJyousuu * CurrJyousuu) % pM;
        }
    }
}

#region ChooseMod
// 二項係数クラス
internal class ChooseMod
{
    private long mHou;

    private long[] mFacArr;
    private long[] mFacInvArr;
    private long[] mInvArr;

    // コンストラクタ
    internal ChooseMod(long pCnt, long pHou)
    {
        mHou = pHou;
        mFacArr = new long[pCnt + 1];
        mFacInvArr = new long[pCnt + 1];
        mInvArr = new long[pCnt + 1];

        mFacArr[0] = mFacArr[1] = 1;
        mFacInvArr[0] = mFacInvArr[1] = 1;
        mInvArr[1] = 1;
        for (int I = 2; I <= pCnt; I++) {
            mFacArr[I] = mFacArr[I - 1] * I % mHou;
            mInvArr[I] = mHou - mInvArr[mHou % I] * (mHou / I) % mHou;
            mFacInvArr[I] = mFacInvArr[I - 1] * mInvArr[I] % mHou;
        }
    }

    // nCrを返す
    internal long DeriveChoose(long pN, long pR)
    {
        if (pN < pR) return 0;
        if (pN < 0 || pR < 0) return 0;
        return mFacArr[pN] * (mFacInvArr[pR] * mFacInvArr[pN - pR] % mHou) % mHou;
    }
}
#endregion

解説

考察すると
根ごとに、木の訪問の場合の数の総和を求め、
2で割った値が解と分かるので、
木DPとReRootingで解いてます。

類題

ABC160-F Distributing Integers