CF696B Puzzles 题解
题目链接：CF 或者 洛谷
注意一些计算即可
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\(期望=概率\times 出现次数\)。
-
\(dfs\) 一个点，会 \(dfs\) 完以这个点为根的子树，再 \(dfs\) 下一个兄弟节点为根的子树。
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对于两个兄弟节点而言，它们的相对 \(dfs\) 序仅有两种，谁前谁后。
对于本题而言，优先关注时间戳的计算：\(时间戳=dfs中出现在自己之前的点数\)。现在直接去算过于困难，我们使用贡献法去思考，哪些信息会影响到它的时间戳。
如图所示，红色为当前计算点，显然的是，\(fa\) 的期望值假如已经知道了，那么待算点的期望值是由 \(fa\) 的期望值 \(+其余兄弟节点的贡献+1\) 计算得到。期望在本题当中也可以理解为平均出现次数。
考虑树形 \(dp[i]\)，表示 \(i\) 节点的期望，那么与 \(dp[i]=dp[fa]+兄弟子树贡献+1\)，最后加 \(1\) 是因为当前子树的平均期望时间戳，等于在它之前 \(dfs\) 中出现的点期望个数 \(+1\)，即为当前时间戳。考虑兄弟节点贡献，基于第二点，如果我们选择 \(dfs\) 了一个兄弟节点，它将会产生 \(size[son]\) 的贡献，因为会 \(dfs\) 完这棵子树才能选择 \(dfs\) 下一个点，这就是它的出现次数。基于第三点，我们知道，只有 \(dfs\) 序在当前节点之前的才会有贡献，在它之后 \(dfs\) 的并无贡献，在当前节点之前 \(dfs\) 的概率显然为 \(\frac{1}{2}\)，所以每个兄弟节点的贡献为 \(\frac{1}{2} \times size[son]\)，它们的总贡献即为：\(\frac{1}{2}\sum size[son]\)，而 \(\sum size[son]=size[fa]-size[curr]-1\)，即以父节点为根的子树大小去掉当前需要计算的子树大小，再去掉父节点这个根的单点，即为所求。
即，紫色减去红色即为绿色部分。所以我们预处理出子树大小，直接做 \(dp\) 即可：
\[dp[curr]=dp[fa]+\frac{size[fa]-size[curr]-1}{2}+1
\]
注意到本题特殊限制：
\(fa[i]<i\)，这也意味着，回忆我们 \(dfs\) 处理子树大小，显然是从叶子节点往上统计，所以我们也可以直接倒序遍历，不需要 \(dfs\) 也是满足无后效性，以叶子节点往上统计。同理，\(dp\) 更新，正序即可。
C++参照代码
#include <bits/stdc++.h>
// #pragma GCC optimize(2)
// #pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math")
// #pragma GCC target("sse,sse2,sse3,ssse3,sse4.1,sse4.2,avx,avx2,popcnt,tune=native")
// #define isPbdsFile
#ifdef isPbdsFile
#include <bits/extc++.h>
#else
#include <ext/pb_ds/priority_queue.hpp>
#include <ext/pb_ds/hash_policy.hpp>
#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/trie_policy.hpp>
#include <ext/pb_ds/tag_and_trait.hpp>
#include <ext/pb_ds/hash_policy.hpp>
#include <ext/pb_ds/list_update_policy.hpp>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/exception.hpp>
#include <ext/rope>
#endif
using namespace std;
using namespace __gnu_cxx;
using namespace __gnu_pbds;
typedef long long ll;
typedef long double ld;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
typedef tuple<int, int, int> tii;
typedef tuple<ll, ll, ll> tll;
typedef unsigned int ui;
typedef unsigned long long ull;
typedef __int128 i128;
#define hash1 unordered_map
#define hash2 gp_hash_table
#define hash3 cc_hash_table
#define stdHeap std::priority_queue
#define pbdsHeap __gnu_pbds::priority_queue
#define sortArr(a, n) sort(a+1,a+n+1)
#define all(v) v.begin(),v.end()
#define yes cout<<"YES"
#define no cout<<"NO"
#define Spider ios_base::sync_with_stdio(false);cin.tie(nullptr);cout.tie(nullptr);
#define MyFile freopen("..\\input.txt", "r", stdin),freopen("..\\output.txt", "w", stdout);
#define forn(i, a, b) for(int i = a; i <= b; i++)
#define forv(i, a, b) for(int i=a;i>=b;i--)
#define ls(x) (x<<1)
#define rs(x) (x<<1|1)
#define endl '\n'
//用于Miller-Rabin
[[maybe_unused]] static int Prime_Number[13] = {0, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37};
template <typename T>
int disc(T* a, int n)
{
return unique(a + 1, a + n + 1) - (a + 1);
}
template <typename T>
T lowBit(T x)
{
return x & -x;
}
template <typename T>
T Rand(T l, T r)
{
static mt19937 Rand(time(nullptr));
uniform_int_distribution<T> dis(l, r);
return dis(Rand);
}
template <typename T1, typename T2>
T1 modt(T1 a, T2 b)
{
return (a % b + b) % b;
}
template <typename T1, typename T2, typename T3>
T1 qPow(T1 a, T2 b, T3 c)
{
a %= c;
T1 ans = 1;
for (; b; b >>= 1, (a *= a) %= c)if (b & 1)(ans *= a) %= c;
return modt(ans, c);
}
template <typename T>
void read(T& x)
{
x = 0;
T sign = 1;
char ch = getchar();
while (!isdigit(ch))
{
if (ch == '-')sign = -1;
ch = getchar();
}
while (isdigit(ch))
{
x = (x << 3) + (x << 1) + (ch ^ 48);
ch = getchar();
}
x *= sign;
}
template <typename T, typename... U>
void read(T& x, U&... y)
{
read(x);
read(y...);
}
template <typename T>
void write(T x)
{
if (typeid(x) == typeid(char))return;
if (x < 0)x = -x, putchar('-');
if (x > 9)write(x / 10);
putchar(x % 10 ^ 48);
}
template <typename C, typename T, typename... U>
void write(C c, T x, U... y)
{
write(x), putchar(c);
write(c, y...);
}
template <typename T11, typename T22, typename T33>
struct T3
{
T11 one;
T22 tow;
T33 three;
bool operator<(const T3 other) const
{
if (one == other.one)
{
if (tow == other.tow)return three < other.three;
return tow < other.tow;
}
return one < other.one;
}
T3() { one = tow = three = 0; }
T3(T11 one, T22 tow, T33 three) : one(one), tow(tow), three(three)
{
}
};
template <typename T1, typename T2>
void uMax(T1& x, T2 y)
{
if (x < y)x = y;
}
template <typename T1, typename T2>
void uMin(T1& x, T2 y)
{
if (x > y)x = y;
}
constexpr int N = 1e5 + 10;
int fa[N], siz[N];
float dp[N];
int n;
inline void solve()
{
cout << fixed << setprecision(1);
cin >> n, siz[0] = siz[1] = 1;
forn(i, 2, n)cin >> fa[i], siz[i] = 1;
forv(i, n, 1)siz[fa[i]] += siz[i];
forn(i, 1, n)cout << (dp[i] = dp[fa[i]] + (siz[fa[i]] - siz[i] - 1) * 0.5 + 1) << ' ';
}
signed int main()
{
// MyFile
Spider
//------------------------------------------------------
// clock_t start = clock();
int test = 1;
// read(test);
// cin >> test;
forn(i, 1, test)solve();
// while (cin >> n, n)solve();
// while (cin >> test)solve();
// clock_t end = clock();
// cerr << "time = " << double(end - start) / CLOCKS_PER_SEC << "s" << endl;
}
Go 参照代码
package main
import (
"bufio"
"fmt"
"io"
"os"
"runtime/debug"
)
func run(_r io.Reader, _w io.Writer) {
in := bufio.NewScanner(_r)
in.Split(bufio.ScanWords)
out := bufio.NewWriter(_w)
defer func(out *bufio.Writer) {
_ = out.Flush()
}(out)
read := func() (x int) {
in.Scan()
tmp := in.Bytes()
if tmp[0] == '-' {
for _, b := range tmp[1:] {
x = x*10 + int(b&15)
}
return -x
} else {
for _, b := range in.Bytes() {
x = x*10 + int(b&15)
}
}
return
}
//-------------------------------------------------------------
var n = read()
var fa = make([]int, n+1)
var siz = make([]int, n+1)
siz[1] = 1
for i := 2; i <= n; i++ {
fa[i] = read()
siz[i] = 1
}
var dp = make([]float64, n+1)
for i := n; i >= 1; i-- {
siz[fa[i]] += siz[i]
}
dp[1] = 1
for i := 2; i <= n; i++ {
dp[i] = dp[fa[i]] + float64(siz[fa[i]]-siz[i]-1)/2 + 1
}
for i := 1; i <= n; i++ {
_, _ = fmt.Fprintf(out, "%.1f ", dp[i])
}
}
func main() {
debug.SetGCPercent(-1)
run(os.Stdin, os.Stdout)
}
C# 参照代码
using CompLib.Util;
using static System.Console;
namespace CompLib.Util
{
using System;
using System.Linq;
internal class Scanner
{
private string[] _line;
private int _index;
private const char Separator = ' ';
public Scanner()
{
_line = System.Array.Empty<string>();
_index = 0;
}
private string Next()
{
if (_index < _line.Length) return _line[_index++];
string s;
do
{
s = Console.ReadLine() ?? string.Empty;
} while (s.Length == 0);
_line = s.Split(Separator);
_index = 0;
return _line[_index++];
}
public string ReadLine()
{
_index = _line.Length;
return Console.ReadLine() ?? string.Empty;
}
public int NextInt() => int.Parse(Next());
public long NextLong() => long.Parse(Next());
public double NextDouble() => double.Parse(Next());
public decimal NextDecimal() => decimal.Parse(Next());
public char NextChar() => Next()[0];
public char[] NextCharArray() => Next().ToCharArray();
private IEnumerable<string> Array()
{
var s = Console.ReadLine() ?? string.Empty;
_line = s.Length == 0 ? System.Array.Empty<string>() : s.Split(Separator);
_index = _line.Length;
return _line;
}
public int[] IntArray() => Array().AsParallel().Select(int.Parse).ToArray();
public long[] LongArray() => Array().AsParallel().Select(long.Parse).ToArray();
public double[] DoubleArray() => Array().AsParallel().Select(double.Parse).ToArray();
public decimal[] DecimalArray() => Array().AsParallel().Select(decimal.Parse).ToArray();
}
}
internal abstract class Program
{
private static void Solve()
{
var sc = new Scanner();
var t = 1;
// t = sc.NextInt();
for (var i = 0; i < t; i++) Query(sc);
Out.Flush();
}
private const int Inf = 1000_000_007;
private static void Query(Scanner sc)
{
var n = sc.NextInt();
var fa = new int[n + 1];
for (var i = 2; i <= n; i++) fa[i] = sc.NextInt();
var siz = new int[n + 1];
Array.Fill(siz, 1);
var dp = new double[n + 1];
dp[1] = 1;
for (var i = n; i >= 1; i--) siz[fa[i]] += siz[i];
for (var i = 2; i <= n; i++) dp[i] = dp[fa[i]] + (siz[fa[i]] - siz[i] - 1) * 0.5 + 1;
for (var i = 1; i <= n; i++) Write($"{dp[i]:F1} ");
}
public static void Main() => Solve();
// public static void Main() => new Thread(Solve, 1 << 27).Start();
}
\[时间复杂度为:\ O(n)
\]
