JAGSummer2014Day2H Distance Sum

• January 6, 2017
• aizu online judge

はい

問題概要(原文)

$N$ 頂点の重み付き無向辺からなる根付き木がある． 頂点 $1$ から頂点 $N$ までを順番に黒く塗る． 現在頂点 $i(1 \leq i \leq N)$ までを塗った時点である． このとき，ある頂点 $v$ を選んで，$v$ と色が塗られた頂点たちからの距離の和が最小になるような $v$ を考え， このときの距離の和を求めよ．

考察

とりあえず，色が塗られた頂点たちから頂点 $v$ へ移動するときの距離の和を求めよ，という問題を考えよう． まず，色が塗られた頂点たちから根に移動して，それから目的とする頂点 $v$ へ移動することにする． すると，ある頂点 $v$ へ移動するコストは以下の図のように 根に集めるコスト + 塗った頂点たちを $v$ に移動させるコスト となるが，簡単に $O(1)$ で求められる．

しかし実際は図のようなコストを求めたい．

そのために以下の青い線で表せる分を $2$ 回分差し引いてやればいい． これはHL分解したときに，うまく差し引く係数を持たせればいい．

さて，$v$ としてどのような頂点を選ぶのが最適だろうか？ $i-1$ まで塗った時点で $v$ だったやつを $u$ としよう． 冷静に考えると， $u,i$ 間に $v$ は存在するはずである． なぜなら，そのようなパスから離れると明らかに距離が増えるためである．

$u, \mathrm{lca}(u,i)$ 間と $i,\mathrm{lca}(u,i)$ 間の両方を二分探索しながら極値を探そう．

計算量は $O(N \log^{3} N)$ だが HL分解もFenwick Treeも軽いのでなんとか間に合う． 二分探索の回数ももしかするとなにかで抑えられているのかもしれない(?)

ソースコード

using System;
using System.Linq;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
//using System.Numerics;
//using Point = System.Numerics.Complex;
using Number = System.Int64;
namespace Program
{
    public class Solver
    {
        public void Solve()
        {
            var n = sc.Integer();
            var T = new HLTreeGraph(n);
            var G = Enumerate(n, x => new List<KeyValuePair<int, long>>());
            var dist = new long[n];
            T.dist = dist;
            for (int i = 1; i < n; i++)
            {
                var p = sc.Integer() - 1;
                var d = sc.Integer();
                T.AddEdge(p, i, d);
                G[p].Add(new KeyValuePair<int, Number>(i, d));
            }
            T.Build(0);
            var k = 0L;
            var sum = 0L;
            Func<int, long> f = v =>
                {
                    if (v == -1) return 1L << 60;
                    var ret = sum + k * dist[v];
                    ret += T.Query(v);
                    return ret;
                };
            Func<int, int, KeyValuePair<int, long>> g = (v, p) =>
                  {
                      var l = T.d[p] - 1; var r = T.d[v];
                      while (l + 1 < r)
                      {
                          var m = (l + r) / 2;
                          var ll = f(T.Get(v, m)); var rr = f(T.Get(v, m + 1));
                          if (ll < rr) r = m;
                          else l = m;
                      }
                      v = T.Get(v, r);
                      return new KeyValuePair<int, Number>(v, f(v));
                  };
            var c = 0;


            for (int i = 0; i < n; i++)
            {
                k++;
                sum += dist[i];
                T.Update(i);

                var lca = T.GetLCA(c, i);
                var x = g(c, lca);
                var y = g(i, lca);
                var z = -1L;
                if (x.Value <= y.Value)
                { c = x.Key; z = x.Value; }
                else { c = y.Key; z = y.Value; }
                IO.Printer.Out.WriteLine(z);
            }
        }

        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());
        static T[] Enumerate<T>(int n, Func<int, T> f) { var a = new T[n]; for (int i = 0; i < n; ++i) a[i] = f(i); return a; }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
    }
}

#region main
static class Ex
{
    //static public string AsString(this IEnumerable<char> ie) { return new string(System.Linq.Enumerable.ToArray(ie)); }
    //static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") { return string.Join(st, ie); }
    static public void Main()
    {
        var solver = new Program.Solver();
        solver.Solve();
        Program.IO.Printer.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO
{
    using System.IO;
    using System.Text;
    using System.Globalization;
    public class Printer: StreamWriter
    {
        static Printer() { Out = new Printer(Console.OpenStandardOutput()) { AutoFlush = false }; }
        public static Printer Out { get; set; }
        public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
        public Printer(System.IO.Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
        public Printer(System.IO.Stream stream, Encoding encoding) : base(stream, encoding) { }
        public void Write<T>(string format, T[] source) { base.Write(format, source.OfType<object>().ToArray()); }
        public void WriteLine<T>(string format, T[] source) { base.WriteLine(format, source.OfType<object>().ToArray()); }
    }
    public class StreamScanner
    {
        public StreamScanner(Stream stream) { str = stream; }
        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }
        private byte read()
        {
            if (isEof) return 0;
            if (ptr >= len) { ptr = 0; if ((len = str.Read(buf, 0, 1024)) <= 0) { isEof = true; return 0; } }
            return buf[ptr++];
        }
        public char Char() { byte b = 0; do b = read(); while ((b < 33 || 126 < b) && !isEof); return (char)b; }

        public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n'; b = (char)read())
                if (b == 0) break;
                else if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long()
        {
            if (isEof) return long.MinValue;
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != 0 && b != '-' && (b < '0' || '9' < b));
            if (b == 0) return long.MinValue;
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        public int Integer() { return (isEof) ? int.MinValue : (int)Long(); }
        public double Double() { var s = Scan(); return s != "" ? double.Parse(s, CultureInfo.InvariantCulture) : double.NaN; }
        private T[] enumerate<T>(int n, Func<T> f)
        {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f();
            return a;
        }

        public char[] Char(int n) { return enumerate(n, Char); }
        public string[] Scan(int n) { return enumerate(n, Scan); }
        public double[] Double(int n) { return enumerate(n, Double); }
        public int[] Integer(int n) { return enumerate(n, Integer); }
        public long[] Long(int n) { return enumerate(n, Long); }
    }
}
#endregion

#region Edge
public struct Edge
{
    public int from, to, cost;
    public long add;
    public Edge(int f, int t, int x) : this() { from = f; to = t; cost = x; }
}
#endregion
#region HLTreeGraph

public class HLTreeGraph
{
    public long[] dist;
    /// <summary>
    /// 縮約前の頂点の数
    /// </summary>
    int N;
    /// <summary>
    /// 縮約前のグラフ
    /// </summary>
    List<Edge>[] G;
    /// <summary>
    /// チェインの集合
    /// </summary>
    List<Chain> H = new List<Chain>();

    int[] subTreeSize;
    int[] par;
    int[] pos;

    /// <summary>
    /// 元の木上の深さ
    /// </summary>
    public int[] d;


    /// <summary>
    ///uが属するチェイン
    /// </summary>
    Chain[] go;
    public HLTreeGraph(int n)
    {
        N = n;
        G = Enumerate(n, x => new List<Edge>());
        subTreeSize = new int[n];
        pos = new int[n];
        d = new int[n];
        par = new int[n];
        go = new Chain[n];
    }
    public void AddEdge(int f, int t, int x)
    {
        G[f].Add(new Edge(f, t, x));
        G[t].Add(new Edge(t, f, x));
    }
    #region impl
    public void Build(int root)
    {
        ComputeSubTreeSize(root);
        Decomposite(new Edge(-1, root, 0), -1, 0);
    }
    public void ComputeSubTreeSize(int root)
    {
        const long X = 1000000000;
        var stack = new FastStack<long>(N + 1);
        stack.Push(root * X);
        par[root] = -1;
        d[root] = 0;
        dist[root] = 0;
        while (stack.Any())
        {
            var val = stack.Peek();
            var u = (int)(val / X);
            var it = (int)(val % X);

            if (it == G[u].Count)
            {
                stack.Pop();
                subTreeSize[u]++;
                if (par[u] >= 0) subTreeSize[par[u]] += subTreeSize[u];
            }
            else
            {
                var to = G[u][it].to;
                stack.Last++;
                if (to == par[u]) continue;

                par[to] = u;
                d[to] = d[u] + 1;
                dist[to] = dist[u] + G[u][it].cost;
                stack.Push(to * X);
            }
        }

    }
    public void Decomposite(Edge light, int prevId, int lv)
    {
        var chain = new Chain() { light = light, id = H.Count, parId = prevId, level = lv };
        H.Add(chain);
        var prev = light.from;
        var cur = light.to;
        while (cur != prev)
        {
            var next = cur;
            var max = 0;
            go[cur] = chain;
            pos[cur] = chain.heavy.Count;
            foreach (var to in G[cur])
            {
                var t = to.to;
                if (t != prev) max = Math.Max(max, subTreeSize[t]);
            }
            foreach (var to in G[cur])
            {
                var t = to.to;
                if (t == prev) continue;
                if (max == subTreeSize[t])
                {
                    //Debug.WriteLine("{0}->{1}", cur, t);
                    max = 1 << 30;
                    next = t;
                    chain.heavy.Add(to);
                }
                else Decomposite(to, chain.id, lv + 1);
            }
            prev = cur;
            cur = next;
        }
        chain.init(this);

    }
    #endregion

    public void Update(int v)
    {
        while (v != -1)
        {
            go[v].sub.Add(pos[v] + 1, -2 * dist[v]);
            go[v].coef.Add(1, -2); go[v].coef.Add(pos[v] + 1, 2);
            if (go[v].light.from != -1)
                go[v].light.add += 2 * dist[go[v].light.from];
            v = go[v].light.from;
        }
    }
    /// <summary>
    /// (u,v)に関するクエリを処理する
    /// </summary>
    public long Query(int v)
    {
        long ans = 0;
        while (v != -1)
        {
            ans += go[v].coef[pos[v] + 1] * dist[v];
            ans += go[v].sub[pos[v] + 1];
            if (go[v].light.from != -1)
                ans += go[v].light.add;
            v = go[v].light.from;
        }
        return ans;
    }
    public int Get(int v, int x)
    {
        if (x < 0) return -1;
        if (d[v] < x) return -1;
        for (;;)
        {
            var p = go[v].light.to;
            if (d[p] == x) return p;
            else if (d[p] < x)
                return go[v].heavy[x - d[p] - 1].to;
            v = go[v].light.from;
        }
    }
    /// <summary>
    /// LCA(u,v)を返す
    /// </summary>
    public int GetLCA(int u, int v)
    {
        while (go[u].id != go[v].id)
        {
            if (go[u].level < go[v].level) v = go[v].light.from;
            else u = go[u].light.from;
        }
        if (d[u] <= d[v]) return u;
        else return v;
    }

    static T[] Enumerate<T>(int n, Func<int, T> f) { var a = new T[n]; for (int i = 0; i < n; ++i) a[i] = f(i); return a; }

}
public class Chain
{
    /// <summary>
    /// light edge
    /// </summary>
    public Edge light;
    /// <summary>
    /// heavy edgeの集合
    /// </summary>
    public List<Edge> heavy = new List<Edge>();
    /// <summary>
    /// 親のチェインの番号
    /// </summary>
    public int parId;
    /// <summary>
    /// 縮約後の木での深さ
    /// </summary>
    public int level;
    public int id;
    public FenwickTree coef;
    public FenwickTree sub;
    public void init(HLTreeGraph G)
    {
        coef = new FenwickTree(heavy.Count + 2);
        sub = new FenwickTree(heavy.Count + 2);

    }

}
#endregion
#region FenwickTree
public class FenwickTree
{
    int n;
    Number[] bit;
    int max = 1;
    public FenwickTree(int size)
    {
        n = size; bit = new Number[n + 1];
        while ((max << 1) <= n) max <<= 1;
    }
    /// <summary>sum[a,b]</summary>
    public Number this[int i, int j] { get { return this[j] - this[i - 1]; } }
    /// <summary>sum[0,i]</summary>
    public Number this[int i] { get { Number s = 0; for (; i > 0; i -= i & -i) s += bit[i]; return s; } }
    public int LowerBound(Number w)
    {
        if (w <= 0) return 0;
        int x = 0;
        for (int k = max; k > 0; k >>= 1)
            if (x + k <= n && bit[x + k] < w)
            {
                w -= bit[x + k];
                x += k;
            }
        return x + 1;
    }
    /// <summary>add v to bit[i]</summary>
    public void Add(int i, Number v)
    {
        if (i == 0) System.Diagnostics.Debug.Fail("BIT is 1 indexed");
        for (; i <= n; i += i & -i) bit[i] += v;
    }
    public Number[] Items
    {
        get
        {
            var ret = new Number[n + 1];
            for (int i = 0; i < ret.Length; i++)
                ret[i] = this[i, i];
            return ret;
        }
    }
}
#endregion

#region Stack<T>
public class FastStack<T>
{
    T[] data;
    int ptr;
    public FastStack(int size) { data = new T[size]; }
    public void Push(T item) { data[ptr++] = item; }
    public T Pop() { return data[--ptr]; }
    public T Peek() { return data[ptr - 1]; }
    public bool Any() { return ptr != 0; }
    public T Last { get { return data[ptr - 1]; } set { data[ptr - 1] = value; } }
    public int Count { get { return ptr; } }
}
#endregion

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