2024 jscpc E题 Divide题解
题目链接：Divide
分析题意，区间要取最大值，然后除以 \(2\)，向下取整，不断执行 \(k\) 次这样的操作，最后问你区间最大值。看一眼 \(k \le 1e9\)，看眼 \(n,val \le 1e5\)，再看眼时限：\(6s\)，当时赛场上刚看到时想到的是 \(根号/大常数双\log?\)，这题出发点其实还是很显然的：均摊结构。
常见的均摊结构，对元素大小为 \(n\) 的数进行一系列操作：
-
对一个数不断除以 \(x\)，最终为 \(0\) 以后不变，次数为 \(\log\) 级别。
-
对一个数不断开方，最终为 \(1\) 以后不变，根据 \(master\) 主定理，类似 \(vb\) 树的分析，至多为 \(\log\log{n}\) 次。
-
对一个数不断做 \(a_i=\gcd{(a_i,x)}\) 操作，最终为 \(1\)，如果每次操作是有效的，那么至多为 \(\log\) 级别。证明：对于一个数来说，如果这个操作 \(x\) 恰好满足 \(x\) 恰好为 \(a_i\) 的倍数，显然他们的 \(\gcd\) 恰好为 \(a_i\)，并不会发生改变，是无效操作。那么有效操作，显然一定会减小。容易知道除了 \(1\) 以外，\(2\) 可能是 \(a_i\) 的最小因数，那么反之最大因数 \(mx \le \dfrac{a_i}{2}\)，所以每次操作变为最大公因数不会小于各自的最大因数，即至少减小为原来的一半，即情况 \(1\)。
那么一个很好想的东西，就是这 \(n\) 个数全部变为最终的 \(0\)，至多有 \(n\log{n}\) 个数，当然去重后显然也不会超过 \(n\)，但到此你还是并不清楚查询问题怎么解决。
考虑暴力
赛场时队友给了一个很好的对拍的暴力思路，就是我们把 \([l,r]\) 上的数全部丢入到堆当中，然后我们执行 \(k\) 次这样的操作，每次弹出堆顶，然后除以 \(2\) 以后加入到堆当中，执行 \(k\) 次后的堆顶即为答案，其实也就是模拟题目过程。考虑这个模型简化下，我们如果让所有数及其它们的操作数全部放入堆中，那么我们发现，操作就变为了：每次弹出堆顶，最后堆顶即为答案，这个很好思考，手玩一下就懂了。那么这个堆肯定不能要了，我们考虑更快的一些数据结构去维护一个区间 \([l,r]\) 上的所有真实操作数。考虑询问的问题是不断弹出最大的数，即从最大的数开始数，弹出 \(1\) 个数以后的最大数是第 \(2\) 大的数，弹出第 \(k\) 个最大数剩下的堆顶的数是第 \(k+1\) 大的数。
最终问题解决方案：求一个区间 \([l,r]\) 从大到小的第 \(k\) 大操作数。
解法
赛场上没注意不带修改。。直接码了个树套树被卡空间，赛后又被卡时间了，挂了几发换主席树过了。
本题实际可以看成 \(t=n\log{n}\) 个数的规模问题，实际上，常见的树套树解决第 \(k\) 大有三 \(\log\) 的解法，即 二分答案 \(+ (区间+值域限制的\ check)\)，这样的话，对 \(t\le 2e6\) 来说显然不可接受。对于线段树套权值树来说，我们一样可以做树上二分，具体的我们需要把所有区间树对应的权值树的节点全部取出，这样可加性问题在算 \(siz\) 的时候，就可以直接累计各个权值树的贡献了，区间限制下的节点数量显然是 \(\log{n}\) 的，树上二分计算右子树的贡献时，显然要算 \(\log{n}\) 个节点 \((注意\ n是区间限制,t是元素个数限制)\)，这种算法的瓶颈点在预处理原数组上，这种做法很难过。
线段树套权值线段树参照代码
#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;
#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;
constexpr int MX = 1e5;
int n, q, x;
struct
{
struct Out
{
int cnt;
int left, right;
} node[N << 9];
int cnt;
#define left(x) node[x].left
#define right(x) node[x].right
#define cnt(x) node[x].cnt
void add(int& curr, const int pos, const int l = 0, const int r = MX)
{
if (!curr) curr = ++cnt;
cnt(curr)++;
const int mid = l + r >> 1;
if (l == r) return;
if (pos <= mid) add(left(curr), pos, l, mid);
else add(right(curr), pos, mid + 1, r);
}
int root[N << 2];
void Add(const int curr, const int pos, const int val, const int l = 1, const int r = n)
{
add(root[curr], val);
if (l == r) return;
const int mid = l + r >> 1;
if (pos <= mid) Add(ls(curr), pos, val, l, mid);
else Add(rs(curr), pos, val, mid + 1, r);
}
void getNode(vector<int>& ans, const int curr, const int l, const int r, const int s = 1, const int e = n)
{
if (l <= s and e <= r)
{
ans.push_back(root[curr]);
return;
}
const int mid = s + e >> 1;
if (l <= mid) getNode(ans,ls(curr), l, r, s, mid);
if (r > mid) getNode(ans,rs(curr), l, r, mid + 1, e);
}
int query(vector<int>& rt, const int k, const int l = 0, const int r = MX)
{
if (l == r) return l;
int siz = 0;
for (const int x : rt) siz += cnt(right(x));
const int mid = l + r >> 1;
if (siz >= k)
{
for (int& x : rt) x = right(x);
return query(rt, k, mid + 1, r);
}
for (int& x : rt) x = left(x);
return query(rt, k - siz, l, mid);
}
int query(const int l, const int r, const int k)
{
vector<int> ans;
getNode(ans, 1, l, r);
return query(ans, k);
}
} seg;
inline void solve()
{
cin >> n >> q;
forn(i, 1, n)
{
cin >> x;
while (x) seg.Add(1, i, x), x >>= 1;
}
while (q--)
{
int l, r, k;
cin >> l >> r >> k;
cout << seg.query(l, r, k + 1) << 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;
}
考虑用树状数组优化外层树空间和时间常数：
树状数组套权值线段树参照代码
#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;
#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 n, q, x;
constexpr int MX = 1e5;
struct
{
struct Out
{
int cnt;
int left, right;
} node[N << 9];
int cnt;
#define left(x) node[x].left
#define right(x) node[x].right
#define cnt(x) node[x].cnt
void add(int& curr, const int pos, const int l = 0, const int r = MX)
{
if (!curr) curr = ++cnt;
cnt(curr)++;
const int mid = l + r >> 1;
if (l == r) return;
if (pos <= mid) add(left(curr), pos, l, mid);
else add(right(curr), pos, mid + 1, r);
}
int root[N];
void Add(int x, const int val)
{
while (x <= n) add(root[x], val), x += lowBit(x);
}
void getRoot(vector<int>& ans, int x) const
{
while (x) ans.push_back(root[x]), x -= lowBit(x);
}
int query(vector<int>& L, vector<int>& R, const int k, const int l = 0, const int r = MX)
{
if (l == r) return l;
const int mid = l + r >> 1;
int siz = 0;
for (const int x : R) siz += cnt(right(x));
for (const int x : L) siz -= cnt(right(x));
if (siz >= k)
{
for (int& x : R) x = right(x);
for (int& x : L) x = right(x);
return query(L, R, k, mid + 1, r);
}
for (int& x : R) x = left(x);
for (int& x : L) x = left(x);
return query(L, R, k - siz, l, mid);
}
int query(const int l, const int r, const int k)
{
vector<int> L, R;
getRoot(L, l - 1);
getRoot(R, r);
return query(L, R, k);
}
} seg;
inline void solve()
{
cin >> n >> q;
forn(i, 1, n)
{
cin >> x;
while (x) seg.Add(i, x), x >>= 1;
}
while (q--)
{
int l, r, k;
cin >> l >> r >> k;
cout << seg.query(l, r, k + 1) << 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;
}
跑得飞快，复杂度大概是 \(O(t\log{n}\log{n}+q\log{v}\log{n})\)。
考虑不带修，直接主席树：
主席树参照代码
#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;
#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 MX = 1e5;
constexpr int N = MX * log2(MX) + 10;
struct Node
{
int left, right, cnt;
} node[N << 5];
int n, q, x, cnt;
#define left(x) node[x].left
#define right(x) node[x].right
#define cnt(x) node[x].cnt
inline void add(const int pre, int& curr, const int pos, const int l = 0, const int r = MX)
{
(node[curr = ++cnt] = node[pre]).cnt++;
if (l == r) return;
const int mid = l + r >> 1;
if (pos <= mid) add(left(pre),left(curr), pos, l, mid);
else add(right(pre),right(curr), pos, mid + 1, r);
}
inline int query(const int L, const int R, const int k, const int l = 0, const int r = MX)
{
if (l == r) return l;
const int mid = l + r >> 1;
const int rightSize = cnt(right(R)) - cnt(right(L));
if (rightSize >= k) return query(right(L),right(R), k, mid + 1, r);
return query(left(L),left(R), k - rightSize, l, mid);
}
int root[N];
inline void solve()
{
cin >> n >> q;
forn(i, 1, n)
{
cin >> x;
add(root[i - 1], root[i], x);
while (x >>= 1) add(root[i], root[i], x);
}
while (q--)
{
int l, r, k;
cin >> l >> r >> k;
cout << query(root[l - 1], root[r], k + 1) << 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;
}
复杂度大概是 \(O(t\log{V}+q\log{V})\)。
不带修，那么我们可以用归并树去做，即为每个值处理一个有序下标桶，可以利用二分判断一个值被包含的区间贡献，那么搬到树上，即为权值树套有序数组，因为不带修，所以可以使用归并排序合并两个有序数组进行预处理。然后我们可以处理出一个前缀和数组，用于算出一个区间上的操作数有多少，这样当 \(k\) 大于等于它，显然可以直接输出 \(0\)，减少常数。
归并树参照代码
#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;
#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;
constexpr int MX = 1e5;
vector<int> init[N];
int n, x, q;
vector<int> idx[N << 2];
ll pre[N];
inline void pushUp(const int curr)
{
auto &L = idx[ls(curr)], &R = idx[rs(curr)], &val = idx[curr];
int idx1 = 0, idx2 = 0;
while (idx1 < L.size() and idx2 < R.size()) val.push_back(L[idx1] <= R[idx2] ? L[idx1++] : R[idx2++]);
while (idx1 < L.size()) val.push_back(L[idx1++]);
while (idx2 < R.size()) val.push_back(R[idx2++]);
}
inline void build(const int curr = 1, const int l = 0, const int r = MX)
{
if (l == r)
{
idx[curr] = init[l];
return;
}
const int mid = l + r >> 1;
build(ls(curr), l, mid), build(rs(curr), mid + 1, r);
pushUp(curr);
}
inline int query(const int curr, const int l, const int r, const int k, const int s = 0, const int e = MX)
{
if (s == e) return s;
const int rightSize = ranges::upper_bound(idx[rs(curr)], r) - ranges::upper_bound(idx[rs(curr)], l - 1);
const int mid = s + e >> 1;
if (rightSize >= k) return query(rs(curr), l, r, k, mid + 1, e);
return query(ls(curr), l, r, k - rightSize, s, mid);
}
inline void solve()
{
cin >> n >> q;
forn(i, 1, n)
{
cin >> x;
while (x) pre[i]++, init[x].push_back(i), x >>= 1;
}
forn(i, 1, n) pre[i] += pre[i - 1];
build();
while (q--)
{
int l, r, k;
cin >> l >> r >> k;
cout << (pre[r] - pre[l - 1] <= k ? 0 : query(1, l, r, k + 1)) << 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;
}
当然这类问题显然还有更好写的整体二分做法：
整体二分
#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;
#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;
struct Query
{
int l, r, k, id;
bool isQuery;
} qu[N << 5], qL[N << 5], qR[N << 5];
int ans[N], bit[N];
int n, q, idx, x;
vector<int> child[N];
inline void add(int x, const int val)
{
while (x <= n) bit[x] += val, x += lowBit(x);
}
inline int query(int x)
{
int ans = 0;
while (x) ans += bit[x], x -= lowBit(x);
return ans;
}
inline void binary(const int L, const int R, const int idxL, const int idxR)
{
if (L == R)
{
forn(i, idxL, idxR) ans[qu[i].id] = L;
return;
}
const int mid = L + R >> 1;
int cntL = 0, cntR = 0;
forn(i, idxL, idxR)
{
auto& [l,r,v,id,isQuery] = qu[i];
if (isQuery)
{
const int siz = query(r) - query(l - 1);
if (v <= siz) qR[++cntR] = qu[i];
else v -= siz, qL[++cntL] = qu[i];
continue;
}
if (v > mid) add(l, 1), qR[++cntR] = qu[i];
else qL[++cntL] = qu[i];
}
forn(i, 1, cntR) if (!qR[i].isQuery) add(qR[i].l, -1);
forn(i, 1, cntL) qu[i + idxL - 1] = qL[i];
forn(i, 1, cntR) qu[i + idxL + cntL - 1] = qR[i];
binary(L, mid, idxL, idxL + cntL - 1);
binary(mid + 1, R, idxL + cntL, idxR);
}
inline void solve()
{
cin >> n >> q;
forn(i, 1, n)
{
cin >> x;
while (x) child[i].push_back(x), x /= 2;
}
forn(i, 1, n) for (const int x : child[i]) qu[++idx] = Query(i, i, x, 0, false);
forn(i, 1, q)
{
int l, r, k;
cin >> l >> r >> k;
qu[++idx] = Query(l, r, k + 1, i, true);
}
binary(0, 1e5, 1, idx);
forn(i, 1, q) cout << ans[i] << 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;
}
复杂度大概是 \(O((t+q)\log^2{n})\)
根号算法显然不太行，\(2e6\) 的数据量如果用值域分块也看上去很不可行，如果平衡根号复杂度大概有序块能做到类似 \(q\sqrt{t\log{t}}\) 这种东西。
有序块的参照做法
#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;
#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;
constexpr int MX = 1e5;
vector<int> pos[N];
int n, x, q;
vector<int> bol[N];
int idx[N], s[N], e[N];
inline int querySize(const vector<int>& curr, const int x)
{
return curr.size() - (ranges::lower_bound(all(curr), x) - curr.begin());
}
inline int queryBolck(const int l, const int r, const int x)
{
const int L = idx[l], R = idx[r];
int ans = 0;
if (L == R)
{
forn(i, l, r) ans += querySize(pos[i], x);
return ans;
}
forn(i, l, e[L]) ans += querySize(pos[i], x);
forn(i, s[R], r) ans += querySize(pos[i], x);
forn(i, L+1, R-1) ans += querySize(bol[i], x);
return ans;
}
inline int query(const int l, const int r, const int k)
{
int L = 0, R = MX;
while (L < R)
{
const int mid = L + R + 1 >> 1;
if (queryBolck(l, r, mid) >= k) L = mid;
else R = mid - 1;
}
return L;
}
inline void solve()
{
cin >> n >> q;
const int siz = sqrt(n);
const int cnt = (n + siz - 1) / siz;
forn(i, 1, n)
{
cin >> x;
while (x) pos[i].push_back(x), x >>= 1;
reverse(all(pos[i]));
idx[i] = (i - 1) / siz + 1;
}
forn(i, 1, cnt)
{
s[i] = (i - 1) * siz + 1;
e[i] = min(n, i * siz);
forn(j, s[i], e[i]) bol[i].insert(bol[i].end(),all(pos[j]));
}
while (q--)
{
int l, r, k;
cin >> l >> r >> k;
cout << query(l, r, k + 1) << 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;
}
最后
这题官解给的最优做法，如果没理解错大概是 线段树上的分散层叠算法，然后可以用 \(bitset\) 优化之类的，等后续理解了再进行补充。
