整体上挺不错的一场。做得也比较顺，配合也挺好。

C Bernoulli’s Principle

高中物理题。所有水滴运动时间相同，水平运动距离和$\sqrt{H-h_{i}}$成正比，把这个值作为关键字排序即可。

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/*

This code template was updated by Yukii_P on 2025/8/2.

*/
#include <bits/stdc++.h>
#define FIO cin.tie(0); ios::sync_with_stdio(false)
#define all(x) (x).begin(), (x).end()
#define fi first
#define se second
#define TEST
#define TESTS int t = 1; cin >> t; while (t--)



template<typename T> void chkmin(T& a, T b) { a = std::min(a, b); }
template<typename T> void chkmax(T& a, T b) { a = std::max(a, b); }
template<typename T, const size_t S>
std::ostream& operator<<(std::ostream& o, const std::array<T, S>& x) { for (int i = (o << "[", 0); i < S; ++i) o << x[i] << (i + 1 < x.size() ? ", " : ""); return o << "]";}
template<typename T>
std::ostream& operator<<(std::ostream& o, const std::vector<T>& x) { for (int i = (o << "[", 0); i < x.size(); ++i) o << x[i] << (i + 1 < x.size() ? ", " : ""); return o << "]";}

#if 1
#define int i64
#define inf 0x3f3f3f3f3f3f3f3fLL
#else
#define inf 0x3f3f3f3f
#endif

#ifndef ONLINE_JUDGE
#define debug(...) std::cerr << #__VA_ARGS__ << " = ", dbg1(__VA_ARGS__)
void dbg2() { std::cerr << "\n"; }
template<typename T, typename... T2> 
void dbg2(T x, T2... args) { std::cerr << ", " << x; dbg2(args...); }
template<typename T, typename... T2> 
void dbg1(T x, T2... args) { std::cerr << x; dbg2(args...); }
template<typename T> 
void dbg1(std::vector<T> x, size_t s) { s = std::min<size_t>(s, x.size()); for (size_t i = (std::cerr << "[", 0); i < s; ++i) std::cerr << x[i] << (i + 1 < s ? ", " : "]\n"); }
#else
#define debug(...) 
#endif

using namespace std;
using i64 = long long;
using u32 = unsigned;
using u64 = unsigned long long;
using pii = std::pair<int, int>;

inline char nc() {
    static char buf[1 << 20], *p1, *p2;
    return p1 == p2 && (p2 = (p1 = buf) + fread(buf, 1, 1 << 20, stdin), p1 == p2) ? EOF : *p1++;
}
#ifndef ONLINE_JUDGE
#define nc getchar
#endif
void read() {}
template<typename T, typename... T2> 
inline void read(T &x, T2 &... oth) {
    x = 0; char c = nc(), up = c;
    while(!isdigit(c)) up = c, c = nc();
    while(isdigit(c)) x = x * 10 + c - '0', c = nc();
    up == '-' ? x = -x : 0;
    read(oth...);
}

constexpr int N = 2e5 + 10;
constexpr int MOD = 998244353;
constexpr double g = 9.8;

void solve() {
    int n, H;
    cin >> n >> H;
    vector<double> h(n);
    for (int i = 0; i < n; ++i) cin >> h[i];
    vector<double> t(n), d(n);
    for (int i = 0; i < n; ++i) {
        t[i] = sqrt(2 * h[i] / g);
        d[i] = sqrt(2 * g * (H - h[i])) * t[i]; 
    }
    vector<int> p(n);
    iota(all(p), 0);
    ranges::sort(p, [&](auto i, auto j) {
        return d[i] < d[j];
    });
    for (int i = 0; i < n; ++i) cout << p[i] + 1 << " \n"[i == n - 1];

}

signed main() {
    
    FIO;
    TESTS {
        solve();
    }

    return 0;
}

/*

  _  _                            _____             _               _               _    
 | \| |    ___    _ _      ___   |_   _|    _ _    (_)    __ __    (_)    __ _     | |   
 | .` |   / _ \  | ' \    |___|    | |     | '_|   | |    \ V /    | |   / _` |    | |   
 |_|\_|   \___/  |_||_|   _____   _|_|_   _|_|_   _|_|_   _\_/_   _|_|_  \__,_|   _|_|_  
_|"""""|_|"""""|_|"""""|_|     |_|"""""|_|"""""|_|"""""|_|"""""|_|"""""|_|"""""|_|"""""| 
"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-' 

(Font: ASCII - Train)

"El Psy Kongroo." -- Hououin Kyoma

"私は極色カスタ推しです...だが、狽音ウルシと夜鳴トバリも推す :)" -- Yukii_P

*/

A Insert One

还是有点没分析清楚就写了，前期交了一堆差点洪文以为这场又寄了。还好分析清楚过了。

修改给定数，在某个位置插入一个数字1，最大化该数。分正负考虑。对于正数，发现如果其中存在0，在0前插1必定必在其后插入要好。因此从高到低找0，没找到则插在最后。对于负数，如果其中存在大于1的数字，在其前插入必定比在其后插入要好。继续从高到低找，没找到就在最后插。

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/*

This code template was updated by Yukii_P on 2025/8/2.

*/
#include <bits/stdc++.h>
#define FIO cin.tie(0); ios::sync_with_stdio(false)
#define all(x) (x).begin(), (x).end()
#define fi first
#define se second
#define TEST
#define TESTS int t = 1; cin >> t; while (t--)

#if 0
#define int i64
#define inf 0x3f3f3f3f3f3f3f3fLL
#else
#define inf 0x3f3f3f3f
#endif

template<typename T> void chkmin(T& a, T b) { a = std::min(a, b); }
template<typename T> void chkmax(T& a, T b) { a = std::max(a, b); }
template<typename T, const size_t S>
std::ostream& operator<<(std::ostream& o, const std::array<T, S>& x) { for (int i = (o << "[", 0); i < S; ++i) o << x[i] << (i + 1 < x.size() ? ", " : ""); return o << "]";}
template<typename T>
std::ostream& operator<<(std::ostream& o, const std::vector<T>& x) { for (int i = (o << "[", 0); i < x.size(); ++i) o << x[i] << (i + 1 < x.size() ? ", " : ""); return o << "]";}

#ifndef ONLINE_JUDGE
#define debug(...) std::cerr << #__VA_ARGS__ << " = ", dbg1(__VA_ARGS__)
void dbg2() { std::cerr << "\n"; }
template<typename T, typename... T2> 
void dbg2(T x, T2... args) { std::cerr << ", " << x; dbg2(args...); }
template<typename T, typename... T2> 
void dbg1(T x, T2... args) { std::cerr << x; dbg2(args...); }
template<typename T> 
void dbg1(std::vector<T> x, int s) { s = std::min<int>(s, x.size()); for (int i = (std::cerr << "[", 0); i < s; ++i) std::cerr << x[i] << (i + 1 < s ? ", " : "]\n"); }
#else
#define debug(...) 
#endif

using namespace std;
using i64 = long long;
using u32 = unsigned;
using u64 = unsigned long long;
using pii = std::pair<int, int>;

inline char nc() {
    static char buf[1 << 20], *p1, *p2;
    return p1 == p2 && (p2 = (p1 = buf) + fread(buf, 1, 1 << 20, stdin), p1 == p2) ? EOF : *p1++;
}
#ifndef ONLINE_JUDGE
#define nc getchar
#endif
void read() {}
template<typename T, typename... T2> 
inline void read(T &x, T2 &... oth) {
    x = 0; char c = nc(), up = c;
    while(!isdigit(c)) up = c, c = nc();
    while(isdigit(c)) x = x * 10 + c - '0', c = nc();
    up == '-' ? x = -x : 0;
    read(oth...);
}

constexpr int N = 2e5 + 10;
constexpr int MOD = 998244353;

void solve() {
    string s;
    cin >> s;
    if (s[0] == '-') {
        int ok = 0;
        for (int i = 1; i < s.size(); ++i) {
            if (s[i] > '1') {
                ok = 1;
                s.insert(i, "1");
                break;
            }
        }
        if (!ok) 
            for (int i = s.size() - 1; i >= 1; --i) {
                if (s[i] <= '1') {
                    s.insert(i + 1, "1");
                    break;
                }
            }
    } else {
        int ok = 0;
        for (int i = 0; i < s.size(); ++i) {
            if (s[i] < '1') {
                s.insert(i, "1");
                ok = 1;
                break;
            }
        }
        if (!ok) {
            s += "1";
        } 
    }
    
    cout << s << "\n";
}

signed main() {
    
    FIO;
    TESTS {
        solve();
    }

    return 0;
}

/*

  _  _                            _____             _               _               _    
 | \| |    ___    _ _      ___   |_   _|    _ _    (_)    __ __    (_)    __ _     | |   
 | .` |   / _ \  | ' \    |___|    | |     | '_|   | |    \ V /    | |   / _` |    | |   
 |_|\_|   \___/  |_||_|   _____   _|_|_   _|_|_   _|_|_   _\_/_   _|_|_  \__,_|   _|_|_  
_|"""""|_|"""""|_|"""""|_|     |_|"""""|_|"""""|_|"""""|_|"""""|_|"""""|_|"""""|_|"""""| 
"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-'"`-0-0-' 

(Font: ASCII - Train)

"El Psy Kongroo." -- Hououin Kyoma

"私は極色カスタ推しです...だが、狽音ウルシと夜鳴トバリも推す :)" -- Yukii_P

*/

F Broken LED Lights

把每个灯管看做二进制亮或不亮，预处理所有灯管的掩码，二进制暴力枚举。直接做会T，按照掩码的1的位数分组，再剪剪枝就过了。

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#include<bits/stdc++.h>

using namespace std;

using ll = long long;
using ull = unsigned long long;
using i32 = int;
using i64 = long long;
using i128 = __int128;

template<typename T1,typename T2>
ostream& operator<<(ostream& o,const tuple<T1,T2>& t){
	return o<<"("<<get<0>(t)<<","<<get<1>(t)<<")";
}

template<typename T1,typename T2,typename T3>
ostream& operator<<(ostream& o,const tuple<T1,T2,T3>& t){
	return o<<"("<<get<0>(t)<<","<<get<1>(t)<<","<<get<2>(t)<<")";
}

template<typename T,const size_t S>
ostream& operator<<(ostream& o,const array<T,S>& arr){
	for(int i=(o<<"[",0);i<S;++i)o<<arr[i]<<(i+1<S?",":"");
	return o<<"]\n";
}

template<typename T>
ostream& operator<<(ostream& o,const vector<T>& vec){
	for(int i=(o<<"[",0);i<vec.size();++i)o<<vec[i]<<(i+1<vec.size()?",":"");
	return o<<"]\n";
}

constexpr int M = 1<<21, mod = 998244353;// 1e9+7;
constexpr double pi = acos(-1), eps = 1e-9;

int d[10]={
	0b1110111,
	0b0010010,
	0b1011101,
	0b1011011,
	0b0111010,
	0b1101011,
	0b1101111,
	0b1010010,
	0b1111111,
	0b1111011
};

vector<vector<int>> s(25);
void init(){
	for(int i=1;i<(1<<21);++i){
		s[__builtin_popcount(i)].emplace_back(i);
	}
	for(int i=1;i<=21;++i){
		ranges::sort(s[i]);
	}
	return;
}

bitset<M> vis;
int stk[105];
void solve(){
	int n,m;
	cin>>n>>m;
	vector<int> vec(n);
	for(int x;auto& v:vec){
		cin>>x;
		for(int i=0;i<m;++i){
			v<<=7;
			v|=d[x%10];
			x/=10;
		}
	}
	if(n==1){
		cout<<"0\n";
		return;
	}
	auto check=[&](int mask){
		bool res=true;
		int top=0;
		for(auto i:vec){
			i&=mask;
			if(vis[i]){
				res=false;
				break;
			}
			stk[top++]=i;
			vis[i]=1;
		}
		for(int i=0;i<top;++i){
			vis[stk[i]]=0;
		}
		return res;
	};
	for(int i=1;i<=m*7;++i){
		for(auto j:s[i]){
			if(j>=(1<<(m*7)))break;
			if(check(j)){
				cout<<i<<"\n";
				return;
			}
		}
	}
	
	return;
}

int main() {
	ios::sync_with_stdio(false);
	cin.tie(0);
	
	init();
	
	int t{1};
	cin>>t;
	while(t--){
		solve();
	}
	
	return 0;
}

B Inversion Number Parity

由于空间限制，原序列都存不下来。观察操作性质，发现逆序对奇偶性仅和操作区间奇偶和循环移动位数奇偶有关，简单维护就做完了。

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#include<bits/stdc++.h>
#define ll long long;
using namespace std;
int n,a,b,c,f[4],l,r,d;
int U=(1<<30)-1;
void init()
{
    f[0]=U&a;
    f[1]=U&b;
    f[2]=U&c;
}
int nextf()
{
    int g=f[0]^((f[0]<<16)&U);
    int h=g^(g>>5);
    f[3]=h^((h<<1)&U)^f[1]^f[2];
    f[0]=f[1],f[1]=f[2],f[2]=f[3];
    return f[3];
}
void solve()
{
    cin>>n>>a>>b>>c;
    int ans=0;
    init();
    for(int i=0;i<n;++i)
    {
        if(0!=nextf()%(n-i)) ans^=1;
    }
    cout<<ans;
    for(int i=1;i<n;++i)
    {
        l=nextf()%n;
        r=nextf()%n;
        if(l>r) swap(l,r);
        d=nextf()%n+1;
        ans^=(r-l)&d&1;
        cout<<ans;
    }
    cout<<"\n";
}

int main()
{   
    ios::sync_with_stdio(false);
    cin.tie(nullptr);cout.tie(nullptr);
    int TxT=1;
    cin>>TxT;
    while(TxT--) solve();
    return 0;
}

J Multiplication in Base the Square Root of -2

看成普通的二进制高精度乘法先乘，区别在于进位逻辑。每次先处理第i位和第i+2位，第i位的2倍和第i+2位相互抵消。做完后如果第i位还大于1，给i+2和i+4位各加第i位除以2的结果。

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#include <iostream>
#include <algorithm>
#include <array>
#include <cstdint>
#include <cassert>
#include <cstring>
#include <random>
#include <chrono>
#include <string>
#pragma GCC target("avx2")
#include <immintrin.h>
using i32=int32_t;
using i64=int64_t;
using u32=uint32_t;
using u64=uint64_t;
using idt=std::size_t;
using I256=__m256i;
using std::cin;
using std::cout;
template<class T,idt k>using arr=std::array<T,k>;
std::mt19937_64 rng(0xee0000+std::chrono::system_clock().now().time_since_epoch().count()+std::random_device{}());
#define fn constexpr auto
#define let const auto
#define def inline auto
#define idef [[gnu::always_inline]] def
def store(void*p,I256 x){
    _mm256_store_si256((I256*)p,x);
}
def load256(const void*p){
    return _mm256_load_si256((const I256*)p);
}
def loadu256(const void*p){
    return _mm256_loadu_si256((const __m256i_u*)p);
}
fn shrk(u32 x,u32 M){
    return std::min(x,x-M);
}
fn dilt(u32 x,u32 M){
    return std::min(x,x+M);
}
fn reduce(u64 x,u32 ninv,u32 M)->u32 {
    return (x+u64(u32(x)*ninv)*M)>>32;
}
fn mul(u32 x,u32 y,u32 ninv,u32 M){
    return reduce(u64(x)*y,ninv,M);
}
fn mul_b_fixed(u32 x,u32 y,u32 binv,u32 M)->u32 {
    return (u64(x)*y+u64(binv*x)*M)>>32;
}
fn mul_s(u32 x,u32 y,u32 ninv,u32 M){
    return shrk(reduce(u64(x)*y,ninv,M),M);
}
fn qpw(u32 a,u32 b,u32 ninv,u32 M,u32 r){
    for(;b;b>>=1,a=mul(a,a,ninv,M)){
        if(b&1){
            r=mul(r,a,ninv,M);
        }
    }
    return r;
}
fn qpw_s(u32 a,u32 b,u32 ninv,u32 M,u32 r){
    return shrk(qpw(a,b,ninv,M,r),M);
}
def shrk32(I256 x,I256 M){
    return _mm256_min_epu32(x,_mm256_sub_epi32(x,M));
}
def dilt32(I256 x,I256 M){
    return _mm256_min_epu32(x,_mm256_add_epi32(x,M));
}
def add32(I256 x,I256 y){
    return _mm256_add_epi32(x,y);
}
def Lsub32(I256 x,I256 y,I256 M){
    return _mm256_add_epi32(x,_mm256_sub_epi32(M,y));
}
def add32(I256 x,I256 y,I256 M){
    return shrk32(_mm256_add_epi32(x,y),M);
}
def sub32(I256 x,I256 y,I256 M){
    return dilt32(_mm256_sub_epi32(x,y),M);
}
def lmove(I256 x){
    return _mm256_bsrli_epi128(x,4);
}
def reduce(I256 a,I256 b,I256 ninv,I256 M){
    auto c=_mm256_mul_epu32(a,ninv),d=_mm256_mul_epu32(b,ninv);
    c=_mm256_mul_epu32(c,M),d=_mm256_mul_epu32(d,M);
    return _mm256_or_si256(lmove(_mm256_add_epi64(a,c)),_mm256_add_epi64(b,d));
}
template<int b_only_even=0>def mul(I256 a,I256 b,I256 ninv,I256 M){
    return reduce(_mm256_mul_epu32(a,b),_mm256_mul_epu32(lmove(a),b_only_even?b:lmove(b)),ninv,M);
}
def mul_s(I256 a,I256 b,I256 ninv,I256 M){
    return shrk32(mul(a,b,ninv,M),M);
}
template<int b_only_even=0>def mul_b_fixed(I256 a,I256 b,I256 bninv,I256 M){
    I256 cc=_mm256_mul_epu32(a,bninv),dd=_mm256_mul_epu32(lmove(a),b_only_even?bninv:lmove(bninv));
    I256 c=_mm256_mul_epu32(a,b),d=_mm256_mul_epu32(lmove(a),b_only_even?b:lmove(b));
    cc=_mm256_mul_epu32(cc,M),dd=_mm256_mul_epu32(dd,M);
    return _mm256_or_si256(lmove(_mm256_add_epi64(c,cc)),_mm256_add_epi64(d,dd));
}
template<int b_only_even=0>def mul_b_fixed_cross(I256 a,I256 b,I256 bninv,I256 M){
    I256 cc=_mm256_mul_epu32(a,bninv),dd=_mm256_mul_epu32(lmove(a),b_only_even?bninv:lmove(bninv));
    I256 c=_mm256_mul_epu32(a,b),d=_mm256_mul_epu32(lmove(a),b_only_even?b:lmove(b));
    cc=_mm256_mul_epu32(cc,M),dd=_mm256_mul_epu32(dd,M);
    return _mm256_or_si256(_mm256_add_epi64(c,cc),lmove(_mm256_add_epi64(d,dd)));
}
def mul_upd_rt(I256 a,I256 bu,I256 M){
    auto cc=_mm256_mul_epu32(a,bu),c=_mm256_mul_epu32(a,_mm256_srli_epi64(bu,32));
    cc=_mm256_mul_epu32(cc,M);
    return shrk32(_mm256_srli_epi64(_mm256_add_epi64(c,cc),32),M);
}
def mul_upd_rr(I256 a,I256 bu,I256 M){
    return mul_b_fixed<1>(a,bu,_mm256_srli_epi64(bu,32),M);
}
idef expand(u32 x){
    return _mm256_set1_epi32(x);
}
def select_load32(I256*f,idt p){
    return ((u32*)f)[p];
}
fn bcl(idt x){
    return x<2?1:idt(2)<<std::__lg(x-1);
}
#undef fn
#undef let
#undef def
#undef idef
#if defined(__linux__) || defined(__unix__)
#define _USE_MMAP 1
#include <sys/mman.h>
#include <sys/stat.h>
#else
#define _USE_MMAP 0
#endif
#define fn constexpr auto
#define let const auto
#define def inline auto
#define idef [[gnu::always_inline]] def
fn llimit=idt(1)<<20;
template<class T>concept trivialT=std::is_trivial_v<T>;
template<trivialT T,idt al=32>def _lalloc(idt n){
    return new(std::align_val_t(al))T[n];
}
template<trivialT T,idt al=32>def _lfree(T*p){
    ::operator delete[](p,std::align_val_t(al));
}
template<trivialT T,idt aln=32>def lalloc(idt n){
//#if _USE_MMAP
//  idt bytes=sizeof(T)*n;
//  if(bytes>llimit){
//      auto raw=mmap(nullptr,bytes,PROT_READ|PROT_WRITE,MAP_PRIVATE|MAP_ANONYMOUS,-1,0);
//      madvise(raw,bytes,MADV_POPULATE_WRITE);
//      return (T*)raw;
//  }
//#endif
    return _lalloc<T,aln>(n);
}
template<trivialT T,idt aln=32>def lfree(T*p,idt n){
#if _USE_MMAP
    idt bytes=sizeof(T)*n;
    if (bytes>llimit){
        munmap(p,bytes);
        return;
    }
#endif
    _lfree<T,aln>(p);
}
template<trivialT T>def copy(T*f,const T*g,idt n){
    return (T*)memcpy(f,g,n*sizeof(T));
}
template<trivialT T>def clear(T*f,idt n){
    return (T*)memset(f,0,n*sizeof(T));
}
namespace NTT_interal{
    fn _lgmax=26,_lg_iter_thresold=6;
    fn _iter_thresold=idt(1)<<_lg_iter_thresold;
    static_assert(_lg_iter_thresold%2==0);
//f[0,8) = fx * f[0,8) * g[0,8) (mod x^8 - ww)
    idef __conv8_4(u32*__restrict__ f,u32*__restrict__ g,std::array<u32,4> ww,I256 Ninv,I256 Mod,I256 Mod2){
        alignas(64) u32 awa[4][16];
        alignas(64) I256 res0[4]={},res1[4]={};
#pragma GCC unroll(4)
        for(auto i=0;i<4;++i){
            store(g+i*8,shrk32(shrk32(load256(g+i*8),Mod2),Mod));
            auto ff=load256(f+i*8);
            ff=shrk32(ff,Mod2);
            let ffw=shrk32(mul<1>(ff,expand(ww[i]),Ninv,Mod),Mod);
            ff=shrk32(ff,Mod);
            store(awa[i],ffw),store(awa[i]+8,ff);
        }
        for(auto i=0;i<8;++i){
#pragma GCC unroll(4)
            for(auto j=0;j<4;++j){
                let bi=expand(g[j*8+i]);
                let aj=loadu256(awa[j]+8-i);
                let aj2=lmove(aj);
                res0[j]=_mm256_add_epi64(res0[j],_mm256_mul_epu32(bi,aj));
                res1[j]=_mm256_add_epi64(res1[j],_mm256_mul_epu32(bi,aj2));
            }
        }
#pragma GCC unroll(4)
        for(auto i=0;i<4;++i){
            store(f+i*8,shrk32(reduce(res0[i],res1[i],Ninv,Mod),Mod2));
        }
    }
    struct NTT32_info{
        u32 mod,mod2,ninv,one,r2,r3,imag,imagninv,RT1[_lgmax],RT3[_lgmax];
        alignas(32) arr<u32,8> rt3[_lgmax-2],rt3i[_lgmax-2],bwb,bwbi;
        u64 rt4n[_lgmax-3],rt4n2[_lgmax-3],rt4ni[_lgmax-3],rt4n2i[_lgmax-3];
        alignas(32) arr<u32,8> rt4nr,rt4nr2,rt4nri,rt4nr2i;
        constexpr NTT32_info(u32 m):
        mod(m),mod2(m*2),ninv([&]{auto n=2+m;for(auto i=0;i<4;++i){n*=2+m*n;}return n;}()),
        one((-m)%m),r2((-u64(m))%m),r3(mul_s(r2,r2,ninv,m)),
        imag{},imagninv{},RT1{},RT3{},rt3{},rt3i{},bwb{},bwbi{},rt4n{},rt4n2{},rt4ni{},rt4n2i{},rt4nr{},rt4nr2{},rt4nri{},rt4nr2i{}{
            auto k=__builtin_ctz(m-1);
            auto _g=mul(3,r2,ninv,mod);
            for(;;++_g){
                if(qpw_s(_g,mod>>1,ninv,mod,one)!=one){
                    break;
                }
            }
            _g=qpw(_g,mod>>k,ninv,mod,one);
            u32 rt1[_lgmax-1]={},rt1i[_lgmax-1]={};
            rt1[k-2]=_g,rt1i[k-2]=qpw(_g,mod-2,ninv,mod,one);
            for(auto i=k-2;i>0;--i){
                rt1[i-1]=mul(rt1[i],rt1[i],ninv,mod);
                rt1i[i-1]=mul(rt1i[i],rt1i[i],ninv,mod);
            }
            RT1[k-1]=qpw_s(_g,3,ninv,mod,one);
            for(auto i=k-1;i>0;--i){
                RT1[i-1]=mul_s(RT1[i],RT1[i],ninv,mod);
            }
            imag=rt1[0],imagninv=imag*ninv;
            bwb={rt1[1],0,rt1[0],0,mod-mul_s(rt1[0],rt1[1],ninv,mod)};
            bwbi={rt1i[1],0,rt1i[0],0,mul_s(rt1i[0],rt1i[1],ninv,mod)};
            auto pr=one,pri=one;
            for(auto i=0;i<k-2;++i){
                let r=mul_s(pr,rt1[i+1],ninv,mod),ri=mul_s(pri,rt1i[i+1],ninv,mod);
                let r2=mul_s(r,r,ninv,mod),r2i=mul_s(ri,ri,ninv,mod);
                let r3=mul_s(r,r2,ninv,mod),r3i=mul_s(ri,r2i,ninv,mod);
                rt3[i]={r*ninv,r,r2*ninv,r2,r3*ninv,r3};
                RT3[i+2]=rt3[i][1];
                rt3i[i]={ri*ninv,ri,r2i*ninv,r2i,r3i*ninv,r3i};
                pr=mul(pr,rt1i[i+1],ninv,mod),pri=mul(pri,rt1[i+1],ninv,mod);
            }
            u32 w[8]={},wi[8]={};
            w[0]=one,wi[0]=one;
            for(auto i=0;i<3;++i){
                pr=rt1[i],pri=rt1i[i];
                for(auto j=1<<i,k=0;k<j;++k){
                    w[j+k]=mul_s(w[k],pr,ninv,mod);
                    wi[j+k]=mul_s(wi[k],pri,ninv,mod);
                }
            }
            rt4nr={w[7],w[0],w[6],w[1],w[5],w[2],w[4],w[3]};
            rt4nr2={w[1],w[3],w[1],w[0],w[0],w[2],w[0],w[1]};
            rt4nri={wi[7],wi[0],wi[6],wi[1],wi[5],wi[2],wi[4],wi[3]};
            rt4nr2i={wi[1],wi[3],wi[1],wi[0],wi[0],wi[2],wi[0],wi[1]};
            pr=one,pri=one;
            for(auto i=1;i<k-3;++i){
                let r=mul_s(pr,rt1[i+2],ninv,mod),ri=mul_s(pri,rt1i[i+2],ninv,mod);
                let r2=mul_s(r,r,ninv,mod),r2i=mul_s(ri,ri,ninv,mod);
                let r4=mul_s(r2,r2,ninv,mod),r4i=mul_s(r2i,r2i,ninv,mod);
                rt4n[i]=u64(r*ninv)<<32|r,rt4ni[i]=u64(ri*ninv)<<32|ri;
                rt4n2[i]=u64(r2)<<32|r4,rt4n2i[i]=u64(r2i)<<32|r4i;
                pr=mul(pr,rt1i[i+2],ninv,mod),pri=mul(pri,rt1[i+2],ninv,mod);
            }
        }
        def _vec_dif(I256*const f,idt n)const{
            alignas(32) arr<u32,8> st_1[_lgmax>>1];
            let Mod=expand(mod),Mod2=expand(mod2),Ninv=expand(ninv),Imag=expand(imag),ImagNinv=expand(imagninv);
            let id24=_mm256_set_epi32(4,0,2,0,4,0,2,0);
            let lgn=__builtin_ctzll(n);
            std::fill(st_1,st_1+(lgn>>1),bwb);
            let nn=n>>(lgn&1),m=std::min(n,_iter_thresold);
            if(nn!=n){
                for(idt i=0;i<nn;++i){
                    auto p0=f+i,p1=f+nn+i;
                    let f0=load256(p0),f1=load256(p1);
                    let g0=add32(f0,f1,Mod2),g1=Lsub32(f0,f1,Mod2);
                    store(p0,g0),store(p1,g1);
                }
            }
            for(auto L=nn>>2;L>0;L>>=2){
                for(idt i=0;i<L;++i){
                    auto p0=f+i,p1=p0+L,p2=p1+L,p3=p2+L;
                    let f1=load256(p1),f3=load256(p3),f2=load256(p2),f0=load256(p0);
                    let g3=mul_b_fixed<1>(Lsub32(f1,f3,Mod2),Imag,ImagNinv,Mod),g1=add32(f1,f3,Mod2);
                    let g0=add32(f0,f2,Mod2),g2=sub32(f0,f2,Mod2);
                    let h0=add32(g0,g1,Mod2),h1=Lsub32(g0,g1,Mod2);
                    let h2=add32(g2,g3),h3=Lsub32(g2,g3,Mod2);
                    store(p0,h0),store(p1,h1),store(p2,h2),store(p3,h3);
                }
            }
            int t=std::min(_lg_iter_thresold,lgn)&-2,p=(t-2)>>1;
            for(idt j=0;j<n;j+=m,t=__builtin_ctzll(j)&-2,p=(t-2)>>1){
                auto const g=f+j;
                for(idt l=(idt(1)<<t),L=l>>2;L>1;l=L,L>>=2,t-=2,--p){
                    auto rt=load256(st_1+p);
                    for(idt i=(j==0?l:0),k=(j+i)>>t;i<m;i+=l,++k){
                        let r1=_mm256_permutevar8x32_epi32(rt,id24);
                        let r1Ninv=_mm256_permutevar8x32_epi32(_mm256_mul_epu32(rt,Ninv),id24);
                        rt=mul_upd_rt(rt,load256(rt3+__builtin_ctzll(~k)),Mod);
                        let r2=_mm256_shuffle_epi32(r1,_MM_PERM_BBBB),nr3=_mm256_shuffle_epi32(r1,_MM_PERM_DDDD);
                        let r2Ninv=_mm256_shuffle_epi32(r1Ninv,_MM_PERM_BBBB),nr3Ninv=_mm256_shuffle_epi32(r1Ninv,_MM_PERM_DDDD);
                        for(idt j=0;j<L;++j){
                            auto p0=g+i+j,p1=p0+L,p2=p1+L,p3=p2+L;
                            let f1=load256(p1),f3=load256(p3),f2=load256(p2),f0=load256(p0);
                            let g1=mul_b_fixed<1>(f1,r1,r1Ninv,Mod),ng3=mul_b_fixed<1>(f3,nr3,nr3Ninv,Mod);
                            let g2=mul_b_fixed<1>(f2,r2,r2Ninv,Mod),g0=shrk32(f0,Mod2);
                            let h3=mul_b_fixed<1>(add32(g1,ng3),Imag,ImagNinv,Mod),h1=sub32(g1,ng3,Mod2);
                            let h0=add32(g0,g2,Mod2),h2=sub32(g0,g2,Mod2);
                            let o0=add32(h0,h1),o1=Lsub32(h0,h1,Mod2);
                            let o2=add32(h2,h3),o3=Lsub32(h2,h3,Mod2);
                            store(p0,o0),store(p1,o1),store(p2,o2),store(p3,o3);
                        }
                    }
                    store(st_1+p,rt);
                }
                {//L == 1
                    auto rt=load256(st_1);
                    for(idt i=j+(j==0)*4;i<j+m;i+=4){
                        let r1=_mm256_permutevar8x32_epi32(rt,id24);
                        rt=mul_upd_rt(rt,load256(rt3+__builtin_ctzll(~i>>2)),Mod);
                        auto p0=f+i,p1=p0+1,p2=p0+2,p3=p0+3;
                        let f1=load256(p1),f3=load256(p3),f2=load256(p2),f0=load256(p0);
                        let r2=_mm256_shuffle_epi32(r1,_MM_PERM_BBBB),nr3=_mm256_shuffle_epi32(r1,_MM_PERM_DDDD);
                        let g1=mul<1>(f1,r1,Ninv,Mod),ng3=mul<1>(f3,nr3,Ninv,Mod);
                        let g2=mul<1>(f2,r2,Ninv,Mod),g0=shrk32(f0,Mod2);
                        let h3=mul_b_fixed<1>(add32(g1,ng3),Imag,ImagNinv,Mod),h1=sub32(g1,ng3,Mod2);
                        let h0=add32(g0,g2,Mod2),h2=sub32(g0,g2,Mod2);
                        let o0=add32(h0,h1),o1=Lsub32(h0,h1,Mod2);
                        let o2=add32(h2,h3),o3=Lsub32(h2,h3,Mod2);
                        store(p0,o0),store(p1,o1),store(p2,o2),store(p3,o3);
                    }
                    store(st_1,rt);
                }
            }
        }
        def _vec_dit(I256*const f,idt n)const{
            alignas(32) arr<u32,8> st_1[_lgmax>>1];
            let Mod=expand(mod),Mod2=expand(mod2),Ninv=expand(ninv),Imag=expand(imag),ImagNinv=expand(imagninv);
            let id24=_mm256_set_epi32(4,0,2,0,4,0,2,0);
            let lgn=__builtin_ctzll(n);
            std::fill(st_1,st_1+(lgn>>1),bwbi);
            let nn=n>>(lgn&1),m=std::min(n,_iter_thresold);
            let fx=mul_s(mod-((mod-1)>>lgn),r3,ninv,mod);
            let Fx=expand(fx),FxNinv=expand(fx*ninv);
            store(st_1,Fx);
            for(idt j=0;j<n;j+=m){
                auto tt=__builtin_ctzll(j+m),t=4,p=1;
                {//L == 1, append coefficient
                    auto rt=load256(st_1);
                    for(idt i=j;i<j+m;i+=4){
                        let r1=_mm256_permutevar8x32_epi32(rt,id24);
                        rt=mul_upd_rt(rt,load256(rt3i+__builtin_ctzll(~i>>2)),Mod);
                        auto const p0=f+i,p1=p0+1,p2=p1+1,p3=p2+1;
                        let f2=load256(p2),f3=load256(p3),f0=load256(p0),f1=load256(p1);
                        let r2=_mm256_shuffle_epi32(r1,_MM_PERM_BBBB),r3=_mm256_shuffle_epi32(r1,_MM_PERM_DDDD);
                        let g3=mul_b_fixed<1>(Lsub32(f3,f2,Mod2),Imag,ImagNinv,Mod),g2=add32(f2,f3,Mod2);
                        let g0=add32(f0,f1,Mod2),g1=sub32(f0,f1,Mod2);
                        let h2=Lsub32(g0,g2,Mod2),h3=Lsub32(g1,g3,Mod2);
                        let h0=add32(g0,g2),h1=add32(g1,g3);
                        let o2=mul<1>(h2,r2,Ninv,Mod),o0=mul_b_fixed<1>(h0,Fx,FxNinv,Mod);
                        let o1=mul<1>(h1,r1,Ninv,Mod),o3=mul<1>(h3,r3,Ninv,Mod);
                        store(p0,o0),store(p1,o1),store(p2,o2),store(p3,o3);
                    }
                    store(st_1,rt);
                }
                for(idt l=16,L=4;t<=tt;L=l,l<<=2,t+=2,++p){
                    idt diff=j+m-std::max(l,m),i=0;
                    I256 rt=load256(st_1+p),*const g=f+diff;
                    if(diff==0){
                        if(l==n){
                            for(idt i=0;i<L;++i){
                                auto const p0=f+i,p1=p0+L,p2=p1+L,p3=p2+L;
                                let f2=load256(p2),f3=load256(p3),f0=load256(p0),f1=load256(p1);
                                let g3=mul_b_fixed<1>(Lsub32(f3,f2,Mod2),Imag,ImagNinv,Mod),g2=add32(f2,f3,Mod2);
                                let g0=add32(f0,f1,Mod2),g1=sub32(f0,f1,Mod2);
                                let h0=add32(g0,g2,Mod2),h1=add32(g1,g3,Mod2);
                                let h2=sub32(g0,g2,Mod2),h3=sub32(g1,g3,Mod2);
                                let o0=shrk32(h0,Mod),o1=shrk32(h1,Mod);
                                let o2=shrk32(h2,Mod),o3=shrk32(h3,Mod);
                                store(p0,o0),store(p1,o1),store(p2,o2),store(p3,o3);
                            }
                        }
                        else{
                            for(idt i=0;i<L;++i){
                                auto const p0=f+i,p1=p0+L,p2=p1+L,p3=p2+L;
                                let f2=load256(p2),f3=load256(p3),f0=load256(p0),f1=load256(p1);
                                let g3=mul_b_fixed<1>(Lsub32(f3,f2,Mod2),Imag,ImagNinv,Mod),g2=add32(f2,f3,Mod2);
                                let g0=add32(f0,f1,Mod2),g1=sub32(f0,f1,Mod2);
                                let h0=add32(g0,g2,Mod2),h1=add32(g1,g3,Mod2);
                                let h2=sub32(g0,g2,Mod2),h3=sub32(g1,g3,Mod2);
                                store(p0,h0),store(p1,h1),store(p2,h2),store(p3,h3);
                            }
                        }
                        i=l;
                    }
                    for(idt k=(j+i)>>t;i<m;i+=l,++k){
                        let r1=_mm256_permutevar8x32_epi32(rt,id24);
                        let r1Ninv=_mm256_permutevar8x32_epi32(_mm256_mul_epu32(rt,Ninv),id24);
                        rt=mul_upd_rt(rt,load256(rt3i+__builtin_ctzll(~k)),Mod);
                        let r2=_mm256_shuffle_epi32(r1,_MM_PERM_BBBB),r3=_mm256_shuffle_epi32(r1,_MM_PERM_DDDD);
                        let r2Ninv=_mm256_shuffle_epi32(r1Ninv,_MM_PERM_BBBB),r3Ninv=_mm256_shuffle_epi32(r1Ninv,_MM_PERM_DDDD);
                        for(idt j=0;j<L;++j){
                            auto const p0=g+i+j,p1=p0+L,p2=p1+L,p3=p2+L;
                            let f2=load256(p2),f3=load256(p3),f0=load256(p0),f1=load256(p1);
                            let g3=mul_b_fixed<1>(Lsub32(f3,f2,Mod2),Imag,ImagNinv,Mod),g2=add32(f2,f3,Mod2);
                            let g0=add32(f0,f1,Mod2),g1=sub32(f0,f1,Mod2);
                            let h2=Lsub32(g0,g2,Mod2),h3=Lsub32(g1,g3,Mod2);
                            let h0=add32(g0,g2),h1=add32(g1,g3);
                            let o2=mul_b_fixed<1>(h2,r2,r2Ninv,Mod),o0=shrk32(h0,Mod2);
                            let o1=mul_b_fixed<1>(h1,r1,r1Ninv,Mod),o3=mul_b_fixed<1>(h3,r3,r3Ninv,Mod);
                            store(p0,o0),store(p1,o1),store(p2,o2),store(p3,o3);
                        }
                    }
                    store(st_1+p,rt);
                }
            }   
            if(nn!=n){
                for(idt i=0;i<nn;++i){
                    auto const p0=f+i,p1=f+nn+i;
                    let f0=load256(p0),f1=load256(p1);
                    let g0=add32(f0,f1,Mod2),g1=sub32(f0,f1,Mod2);
                    let h0=shrk32(g0,Mod),h1=shrk32(g1,Mod);
                    store(p0,h0),store(p1,h1);
                }
            }
        }
        def _vec_cvdt8(I256*__restrict__ f,I256*__restrict__ g,idt lm)const{
            auto RR=one;
            let Ninv=expand(ninv),Mod=expand(mod),Mod2=expand(mod2);
            for(idt i=0;i<lm;i+=4){
                let RRi=mul_b_fixed(RR,imag,imagninv,mod);
                __conv8_4((u32*)(f+i),(u32*)(g+i),{RR,mod2-RR,RRi,mod2-RRi},Ninv,Mod,Mod2);
                RR=mul(RR,RT3[__builtin_ctzll(i+4)],ninv,mod);
            }
        }
    };
    
}
struct auto_timer{
    std::chrono::system_clock::time_point klj;
    auto_timer():klj(std::chrono::system_clock::now()){
        
    }
    ~auto_timer(){
        std::chrono::duration<long double,std::milli> ioke=std::chrono::system_clock::now()-klj;
        std::clog<<ioke.count()<<"ms"<<std::endl;
    }
};
#undef fn
#undef let
#undef def
#undef idef
using namespace std;
vector<int> NTT(const string& s1,const string& s2){
    constexpr NTT_interal::NTT32_info fntt(998244353);
    idt n,m,lm;
    n=s1.length();
    m=s2.length();
    lm=bcl(std::max<idt>(64,n+m-1));
    auto f=lalloc<u32>(lm),g=lalloc<u32>(lm);
    for(idt i=0;i<n;++i){
        f[i]=s1[i]-'0';
    }
    clear(f+n,lm-n);
    for(idt i=0;i<m;++i){
        g[i]=s2[i]-'0';
    }
    clear(g+m,lm-m);
    {
//      auto_timer ot;
        fntt._vec_dif((I256*)f,lm>>3);
        fntt._vec_dif((I256*)g,lm>>3);
        fntt._vec_cvdt8((I256*)f,(I256*)g,lm>>3);
        fntt._vec_dit((I256*)f,lm>>3);
    }
    vector<int> res(n+m-1);
    for(idt i=0;i<n+m-1;++i){
        res[i]=f[i];
    }
    lfree(f,lm),lfree(g,lm);
    return res;
}
int ans[1<<21]{};
void solve(){
    string a,b;
    cin>>a>>b;
    ranges::reverse(a);
    ranges::reverse(b);
    auto f=NTT(a,b);
    int n=f.size()-1;
    for(int i=0;i<=n;++i){
        if(i<f.size()){
            ans[i]+=f[i];
        }
    }
    for(int i=0;i<=n;++i){
        int t = min(ans[i] / 2, ans[i + 2]);
        ans[i] -= t * 2;
        ans[i + 2] -= t;
        if(ans[i]>1){
            n=max(n,i+4);
            ans[i+4]+=ans[i]/2;
            ans[i+2]+=ans[i]/2;
            ans[i]&=1;
        }
    }
    bool flag=false;
    for(int i=n;i>=0;--i){
        if(!flag&&!ans[i])continue;
        flag=true;
        cout<<ans[i];
    }
    if(!flag){
        cout<<0;
    }
    cout<<"\n";
    for(int i=0;i<=n;++i){
        ans[i]=0;
    }
    return;
}

int main(){
    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr);
    int t;
    cin>>t;
    while(t--){
        solve();
    }
    return 0;
}

G Changing Minimum Spanning Tree

基于Kruskal，按照边权给边分组，一个区间的边权可以任意给未加边的点对添边，这样这个新边必定在最小生成树中。

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#include<bits/stdc++.h>

using namespace std;

using ll = long long;
using ull = unsigned long long;
using i32 = int;
using i64 = long long;
using i128 = __int128;

template<typename T1,typename T2>
ostream& operator<<(ostream& o,const tuple<T1,T2>& t){
	return o<<"("<<get<0>(t)<<","<<get<1>(t)<<")";
}

template<typename T1,typename T2,typename T3>
ostream& operator<<(ostream& o,const tuple<T1,T2,T3>& t){
	return o<<"("<<get<0>(t)<<","<<get<1>(t)<<","<<get<2>(t)<<")";
}

template<typename T,const size_t S>
ostream& operator<<(ostream& o,const array<T,S>& arr){
	for(int i=(o<<"[",0);i<S;++i)o<<arr[i]<<(i+1<S?",":"");
	return o<<"]\n";
}

template<typename T>
ostream& operator<<(ostream& o,const vector<T>& vec){
	for(int i=(o<<"[",0);i<vec.size();++i)o<<vec[i]<<(i+1<vec.size()?",":"");
	return o<<"]\n";
}

constexpr int M = 2e5+5, mod = 1e9+7;
constexpr double pi = acos(-1), eps = 1e-9;

void solve(){
	int n,m,k;
	cin>>n>>m>>k;
	vector<array<int,3>> edges(m);
	vector<vector<int>> G(n+1);
	for(auto& [u,v,w]:edges){
		cin>>u>>v>>w;
		G[u].emplace_back(v);
		G[v].emplace_back(u);
	}
	ranges::sort(edges,[&](auto& a,auto& b){return a[2]<b[2];});
	ll ans=0,cur=1ll*n*(n-1)/2,e=m;
	vector<int> f(n+1),cnt(n+1,1);
	vector<vector<int>> s(n+1);
	for(int i=1;i<=n;++i)s[i].emplace_back(i);
	iota(f.begin(),f.end(),0);
	auto merge=[&](int x,int y){
		x=f[x],y=f[y];
		if(x==y)return 0;
		if(cnt[x]>cnt[y])swap(x,y);
		cur-=1ll*cnt[x]*cnt[y];
		cnt[y]+=cnt[x];
		for(auto u:s[x]){
			for(auto v:G[u]){
				if(f[v]==y)--e;
			}
		}
		for(auto u:s[x]){
			f[u]=y;
			s[y].emplace_back(u);
		}
		return 1;
	};
	int flag=0,curw=1;
	for(auto [u,v,w]:edges){
		if(w>curw){
			(ans+=(cur-e)*(w-curw))%=mod;
			curw=w;
		}
		flag+=merge(u,v);
//		cout<<curw<<" "<<cur<<" "<<e<<" "<<ans<<" . ";
	}
	if(flag<n-2){
		ans=0;
	}
	else if(flag==n-2){
		ans=(cur-e)*k%mod;
	}
	cout<<ans<<"\n";
	
	return;
}

int main() {
	ios::sync_with_stdio(false);
	cin.tie(0);
	
	int t{1};
	cin>>t;
	while(t--){
		solve();
	}
	
	return 0;
}