AGC005E Sugigma: The Showdown
かなり好き
問題概要(原文)
$N$ 頂点のグラフが与えられる． ここに $N-1$ 本の赤い辺と $N-1$ 本の青い辺を追加する． 赤い辺だけの連結成分も青い辺だけの連結成分も木となる．
A君とB君がこのグラフ上でゲームをする． A君はできる限り逃げ回る．B君はA君を捕まえたい．
交互にターンが来る．
- 奇数ターンではA君は赤い辺を辿って1つ隣に移動してもよい．
- 偶数ターンではB君は青い辺を辿って1つ隣に移動してもよい．
A君は最大何ターン生き残れるか？
考察
図のようなグラフでA君が頂点 $1$ にいるとき，$1,4$ 間を移動することでずっと逃げ回れる．
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より一般には赤い辺の両端であるような頂点 $u,v$ について，青い辺を辿って $u$ から $v$ に移動する距離が $3$ 以上なら逃げ回れる．このような頂点は必勝頂点を呼ぼう．
またこのような辺を考えないと，赤い辺はB君のグラフ上で長さ $2$ 以下の辺しか残らない．
A君の戦略は以下の 2 種類しかない．
- 必勝頂点に可能な限り早く向かう
- B君から最も遠い点に向かう
B君の戦略は 1 種類しかない
- A君に向かって移動する
つまり，B君のいる点を根とする根付き木上でA君が赤い辺を辿って移動したとき，B君が着くまでに必勝頂点に着けるかどうかの判定と，そうでないときB君が着くまでにたどり着ける，B君から最も遠い点はどこかが分かればよい．
これはB君が着くまでの締切みたいなものがついた状態でBFSをするだけ．
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 a = sc.Integer() - 1;
var b = sc.Integer() - 1;
var F = new int[n - 1];
var T = new int[n - 1];
for (int i = 0; i < n - 1; i++)
{
F[i] = sc.Integer() - 1;
T[i] = sc.Integer() - 1;
}
var H = new HLTreeGraph(n);
for (int i = 0; i < n - 1; i++)
H.AddEdge(sc.Integer() - 1, sc.Integer() - 1);
H.Build(b);
var G = Enumerate(n, x => new List<int>());
var win = new bool[n];
for (int i = 0; i < n - 1; i++)
{
var f = F[i];
var t = T[i];
var d = H.Query(f, t);
if (d >= 3) win[f] = win[t] = true;
//else
{
G[f].Add(t);
G[t].Add(f);
}
}
var D = Enumerate(n, x => H.Query(b, x) * 2);
var E = Enumerate(n, x => 1000000000);
E[a] = 0;
var q = new Queue<int>();
q.Enqueue(a);
while (q.Count > 0)
{
var p = q.Dequeue();
if (win[p])
{
IO.Printer.Out.WriteLine(-1);
return;
}
foreach (var to in G[p])
{
var d = E[p] + 2;
if (D[to] > d && E[to] > d)
{
E[to] = d;
q.Enqueue(to);
}
}
}
Debug.WriteLine(D.AsJoinedString());
Debug.WriteLine(E.AsJoinedString());
var max = 0L;
for (int i = 0; i < n; i++)
if (E[i] < 1000000) max = Math.Max(max, D[i]);
IO.Printer.Out.WriteLine(max);
}
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;
public Edge(int f, int t) : this() { from = f; to = t; }
}
#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
#region HLTreeGraph
public class HLTreeGraph
{
/// <summary>
/// 縮約前の頂点の数
/// </summary>
int N;
/// <summary>
/// 縮約後の頂点の数
/// </summary>
int M;
/// <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)
{
G[f].Add(new Edge(f, t));
G[t].Add(new Edge(t, f));
}
#region impl
public void Build(int root)
{
ComputeSubTreeSize(root);
Decomposite(new Edge(-1, root), -1, 0);
M = H.Count;
}
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;
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;
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
/// <summary>
/// (u,v)に関するクエリを処理する
/// </summary>
public long Query(int u, int v)
{
long ans = 0;
ans += d[u];
ans += d[v];
while (go[u].id != go[v].id)
{
if (go[u].level > go[v].level)
u = go[u].light.from;
else
v = go[v].light.from;
}
if (pos[u] < pos[v])
ans += -2 * d[u];
else ans += -2 * d[v];
return ans;
}
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 RangeAddFenwickTree seg;
public void init(HLTreeGraph G)
{
//seg = new RangeAddFenwickTree(heavy.Count + 2);
}
public override string ToString()
{
var s = new List<int>();
s.Add(light.from);
s.Add(light.to);
foreach (var e in heavy)
s.Add(e.to);
return string.Join("->", s);
}
}
#endregion
コメント
- こっちは得意
