CF896C Willem, Chtholly and Seniorious 题解

题目链接：CF 或者 洛谷

比较经典的题目

看到存在随机数据以及区间赋值先别急，我们发现第四个操作是很难办的，第四个操作貌似只有暴力才好做。这个时候我们可以考虑使用珂朵莉树来做，这题也是珂朵莉树的出处。使用平衡树去写珂朵莉树的话，那么随机数据下，连续段的期望为 \(\log{n}\) 个，所以使用平衡树进行二分查找分裂的复杂度为 \(\log{\log{n}}\)。加和覆盖都是基本的珂朵莉树操作，直接暴力遍历期望的 \(\log{n}\) 个段就行，复杂度为 \(O(\log{n})\)。查询区间第 \(k\) 大，由于段数并不多，我们可以进行排序以后然后逐段寻找答案，类似可持久化线段树二分的过程，排序复杂度为 \(O(\log{\log{n}})\)，然后再暴力逐段寻找答案 \(O(\log{n})\)。最后一个操作就是较为复杂的，对于一个段的贡献来说，显然为 \(cnt \times (val^x \bmod y)\)。而 \(cnt=r-l+1\)，直接快速幂即可。复杂度显然为：\(O(\log{n} \times \log{x}) \sim O(\log^2{n})\)。

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 MOD = 1e9 + 7;
ll n, m, seed, vMax;

inline ll rnd()
{
    const ll ans = seed;
    seed = (seed * 7 + 13) % MOD;
    return ans;
}

struct ODT
{
    int l, r;
    mutable ll val;

    bool operator<(const ODT& other) const
    {
        return l < other.l;
    }
};

set<ODT> node;
typedef set<ODT>::iterator iter;

inline iter split(const int pos)
{
    auto it = node.lower_bound(ODT(pos, 0, 0));
    if (it != node.end() and it->l == pos)return it;
    --it;
    if (it->r < pos)return node.end();
    auto [l,r,val] = *it;
    node.erase(it), node.insert(ODT(l, pos - 1, val));
    return node.insert(ODT(pos, r, val)).first;
}

inline void add(const int l, const int r, const int val)
{
    auto itr = split(r + 1), itl = split(l);
    for (auto it = itl; it != itr; ++it)it->val += val;
}

inline void assign(const int l, const int r, const int val)
{
    auto itr = split(r + 1), itl = split(l);
    node.erase(itl, itr), node.insert(ODT(l, r, val));
}

inline ll queryKth(const int l, const int r, int k)
{
    auto itr = split(r + 1), itl = split(l);
    vector<pll> curr;
    for (auto it = itl; it != itr; ++it)curr.emplace_back(it->val, it->r - it->l + 1);
    sort(all(curr));
    for (const auto [v,siz] : curr)
    {
        if (siz < k)k -= siz;
        else return v;
    }
    return 0;
}

inline ll querySum(const int l, const int r, const int x, const int y)
{
    auto itr = split(r + 1), itl = split(l);
    ll ans = 0;
    for (auto it = itl; it != itr; ++it)
    {
        auto [l,r,val] = *it;
        ans = (ans + (r - l + 1) * qPow(val, x, y)) % y;
    }
    return ans;
}

inline void solve()
{
    cin >> n >> m >> seed >> vMax;
    forn(i, 1, n)node.insert(ODT(i, i, rnd() % vMax + 1));
    while (m--)
    {
        const int op = rnd() % 4 + 1;
        int l = rnd() % n + 1;
        int r = rnd() % n + 1;
        if (l > r)swap(l, r);
        int x, y = 0;
        if (op == 3)x = rnd() % (r - l + 1) + 1;
        else x = rnd() % vMax + 1;
        if (op == 4)y = rnd() % vMax + 1;
        if (op == 1)add(l, r, x);
        else if (op == 2)assign(l, r, x);
        else if (op == 3)cout << queryKth(l, r, x) << endl;
        else cout << querySum(l, r, x, y) << endl;
    }
}

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;
}