题目链接：符卡对决
视频讲解待上传
经典的题目，对于这个 \([l,r]\) 询问，我们先关注期望怎么算。
考虑方案总数和有效的和，方案总数显然有 \(\dfrac{n\times (n-1)}{2}\)，现在还需要关注有效和，我们关注对于若干个有效的关系用一个比较形象的数据结构表示-----并查集，那么两个卡牌之间有效就可以看做并查集的 \(merge\)，现在我们来考虑合并两个并查集会有多少贡献：
对于并查集 \(x\) 和 \(y\) 合并，枚举 \(x_i\) 与所有的 \(y_i\) 的贡献即为：\(x_i\times \sum y_i\)，然后我们再将所有的 \(x_i\) 贡献加起来，那么它们合并的总贡献就为：\(\sum x_i \times \sum y_i\)，所以并查集带权维护和即可算单个 \(merge\) 操作激活的贡献。
考虑是否好删除操作，不好删除，那么显然就是最好只加不减，那么我们如果对 \(m\) 个操作做回滚莫队维护查询即可。
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 = 2e5 + 10;
constexpr int MOD = 1e9 + 7;
pii op[N];
ll curr;
int a[N], pos[N];
ll n;
int m, q, last;
stack<tuple<int, int, int, int, int>> del;
struct
{
int fa[N], siz[N];
ll sum[N];
void init()
{
forn(i, 1, n) fa[i] = i, sum[i] = a[i], siz[i] = 1;
}
int find(const int x)
{
return x == fa[x] ? x : find(fa[x]);
}
ll merge(int x, int y, const bool isBack = false)
{
x = find(x), y = find(y);
if (x == y) return 0;
if (siz[x] > siz[y]) swap(x, y);
if (isBack) del.emplace(x, fa[x], y, siz[y], sum[y]);
fa[x] = y;
siz[y] += siz[x];
const ll res = sum[x] * sum[y] % MOD;
sum[y] = (sum[x] + sum[y]) % MOD;
return res;
}
} dsu;
struct
{
int fa[N];
ll sum[N];
void init(const int l, const int r)
{
forn(i, l, r)
{
const auto [u,v] = op[i];
fa[u] = u, fa[v] = v;
sum[u] = a[u], sum[v] = a[v];
}
}
int find(const int x)
{
return x == fa[x] ? x : fa[x] = find(fa[x]);
}
ll merge(int x, int y)
{
x = find(x), y = find(y);
if (x == y) return 0;
ll res = sum[x] * sum[y] % MOD;
fa[x] = y, sum[y] = (sum[y] + sum[x]) % MOD;
return res;
}
} tmp;
struct Mo
{
int l, r, id;
bool operator<(const Mo& other) const
{
return pos[l] ^ pos[other.l] ? pos[l] < pos[other.l] : r < other.r;
}
} node[N];
ll ans[N];
ll tmpAns;
inline void add(const int id, const bool isTmp, const bool isBack)
{
const auto [u,v] = op[id];
if (isTmp) tmpAns = (tmpAns + tmp.merge(u, v)) % MOD;
else curr = (curr + dsu.merge(u, v, isBack)) % MOD;
}
inline void solve()
{
read(n, m, q);
const ll inv = qPow(n * (n - 1) / 2 % MOD, MOD - 2, MOD) % MOD;
const int siz = m / sqrt(q);
forn(i, 1, n) read(a[i]);
forn(i, 1, m) read(op[i].first, op[i].second), pos[i] = (i - 1) / siz + 1;
forn(i, 1, q)
{
auto& [l,r,id] = node[i];
read(l, r), id = i;
}
sortArr(node, q);
int l = 1, r = 0;
forn(i, 1, n) tmp.fa[i] = i, tmp.sum[i] = a[i];
forn(i, 1, q)
{
const auto [L,R,id] = node[i];
if (pos[L] == pos[R])
{
forn(i, L, R) add(i, true, false);
ans[id] = tmpAns * inv % MOD;
tmp.init(L, R);
tmpAns = 0;
continue;
}
if (last != pos[L])
{
dsu.init();
curr = 0;
l = min(m, pos[L] * siz);
r = l - 1;
last = pos[L];
}
while (r < R) add(++r, false, false);
ll preAns = curr;
int tmpL = l;
while (tmpL > L) add(--tmpL, false, true);
ans[id] = curr * inv % MOD;
curr = preAns;
while (!del.empty())
{
const auto [x,faX,y,sizY,sumY] = del.top();
dsu.fa[x] = x, dsu.siz[y] = sizY, dsu.sum[y] = sumY;
del.pop();
}
}
forn(i, 1, q) write(endl, ans[i]);
}
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;
}