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势能分析线段树学习记录

news 2025/9/13 9:35:03

算法讲解111【扩展】线段树专题2-线段树的离散化、二分搜索、特别修改_哔哩哔哩_bilibili

对于gcd, 取log, 开根号, 取模, 等操作, 一个数的下降比较快, 可以考虑暴力操作势能分析

P4145 上帝造题的七分钟 2 / 花神游历各国 - 洛谷

对于每个数, 最多操作六次开根号操作就会降到1, 线段树修改操作直接修改到叶子节点, 但是如果当前区间的最大值是1, 则这个区间不再需要进行操作, 不进入这个区间, 这样的话, 每个数最多操作6次, 加上树高, 一共是6 * logn * n次

#include<bits/stdc++.h>
using namespace std;
using i64 = long long;//线段树，区间修改，区间查询
//https://www.luogu.com.cn/problem/P3372
template<typename Info, typename Tag>
struct SegmentTree {
#define ls (id<<1)
#define rs (id<<1|1)SegmentTree() = default;SegmentTree(int n) : n(n), info(n << 2), tag(n << 2), len(n << 2) {}SegmentTree(const SegmentTree<Info, Tag> &o) : n(o.n), info(o.info), tag(o.tag) {}template<typename T>SegmentTree(const std::vector<T> &init) : SegmentTree((int)init.size()) {auto build = [&](auto self, int id, int l, int r) ->void {len[id] = r - l + 1;if(l == r) {info[id] = init[l];return;}int mid = (l + r) / 2;self(self, ls, l, mid);self(self, rs, mid + 1, r);pushup(id);};build(build, 1, 0, n - 1);}void apply(int id, const Tag &dx) {info[id].apply(dx, len[id]);tag[id].apply(dx);}void pushup(int id) {info[id] = info[ls] + info[rs];}void pushdown(int id) {apply(ls, tag[id]);apply(rs, tag[id]);tag[id] = Tag();}void modify(int id, int l, int r, int pos, const Info &val) {if(l == r) {info[id] = val;return;}pushdown(id);int mid = (l + r) / 2;if(pos <= mid) {modify(ls, l, mid, pos, val);} else {modify(rs, mid + 1, r, pos, val);}pushup(id);}void rangeUpdate(int id, int l, int r, int x, int y, const Tag &dx) {if (info[id].maxx == 1) {return;}if(l == r) {info[id].sum = sqrt(info[id].sum);info[id].maxx = sqrt(info[id].maxx);return;}int mid = (l + r) / 2;pushdown(id);if(x <= mid) {rangeUpdate(ls, l, mid, x, y, dx);}if(y > mid) {rangeUpdate(rs, mid + 1, r, x, y, dx);}pushup(id);}Info rangeQuery(int id, int l, int r, int x, int y) {if(x <= l && r <= y) {return info[id];}int mid = (l + r) / 2;pushdown(id);Info res;if(x <= mid) {res = res + rangeQuery(ls, l, mid, x, y);}if(y > mid) {res = res + rangeQuery(rs, mid + 1, r, x, y);}return res;}void rangeUpdate(int l, int r, const Tag &dx) {rangeUpdate(1, 0, n - 1, l, r, dx);}void update(int pos, const Tag &dx) {rangeUpdate(pos, pos, dx);}Info rangeQuery(int l, int r) {return rangeQuery(1, 0, n - 1, l, r);}Info query(int pos) {return rangeQuery(pos, pos);}void modify(int pos, const Info &val) {return modify(1, 0, n - 1, pos, val);}
#undef ls
#undef rsint n;std::vector<Info> info;std::vector<Tag> tag;std::vector<int> len;
};constexpr i64 INF = 4E18;
i64 mod = 1e9 + 7;
struct Tag {i64 add = 0;i64 mul = 1;void apply(const Tag &dx) {// mul = (mul * dx.mul) % mod;// add = (add * dx.mul + dx.add) % mod;}
};struct Info {i64 sum = 0;i64 maxx = 0;Info() {}Info(i64 x) : sum(x), maxx(x) {};void apply(const Tag &dx, const int &len) {// sum = (sum * dx.mul + dx.add * len) % mod;}
};Info operator+(const Info &x, const Info &y) {Info res;res.sum = (x.sum + y.sum);res.maxx = max(x.maxx, y.maxx);return res;
}
void solve() {int n;cin >> n;vector<i64> a(n + 1);for (int i = 1; i <= n; i++) cin >> a[i];SegmentTree<Info, Tag> seg(a);int m;cin >> m;while(m--) {int k, l, r;cin >> k >> l >> r;if (r < l) swap(l, r);if (k == 0) {Tag tag;seg.rangeUpdate(l, r, tag);} else {cout << seg.rangeQuery(l, r).sum << '\n';}}}int main() {std::ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);int T = 1;//cin >> T;while (T--) solve();return 0;
}

Problem - D - Codeforces

每次取模操作, 如果mod小于x, 则x取模之后最少减半, 所以一个数最多进行log次取模操作

#include<bits/stdc++.h>
using namespace std;
using i64 = long long;
//线段树，区间修改，区间查询
//https://www.luogu.com.cn/problem/P3372
template<typename Info, typename Tag>
struct SegmentTree {
#define ls (id<<1)
#define rs (id<<1|1)SegmentTree() = default;SegmentTree(int n) : n(n), info(n << 2), tag(n << 2), len(n << 2) {}SegmentTree(const SegmentTree<Info, Tag> &o) : n(o.n), info(o.info), tag(o.tag) {}template<typename T>SegmentTree(const std::vector<T> &init) : SegmentTree((int)init.size()) {auto build = [&](auto self, int id, int l, int r) ->void {len[id] = r - l + 1;if(l == r) {info[id] = init[l];return;}int mid = (l + r) / 2;self(self, ls, l, mid);self(self, rs, mid + 1, r);pushup(id);};build(build, 1, 0, n - 1);}void apply(int id, const Tag &dx) {info[id].apply(dx, len[id]);tag[id].apply(dx);}void pushup(int id) {info[id] = info[ls] + info[rs];}void pushdown(int id) {apply(ls, tag[id]);apply(rs, tag[id]);tag[id] = Tag();}void modify(int id, int l, int r, int pos, const Info &val) {if(l == r) {info[id] = val;return;}pushdown(id);int mid = (l + r) / 2;if(pos <= mid) {modify(ls, l, mid, pos, val);} else {modify(rs, mid + 1, r, pos, val);}pushup(id);}void rangeUpdate(int id, int l, int r, int x, int y, const Tag &dx) {if (info[id].maxx < dx.mod) {return;}if (l == r) {info[id].maxx %= dx.mod;info[id].sum %= dx.mod;return;}int mid = (l + r) / 2;pushdown(id);if(x <= mid) {rangeUpdate(ls, l, mid, x, y, dx);}if(y > mid) {rangeUpdate(rs, mid + 1, r, x, y, dx);}pushup(id);}Info rangeQuery(int id, int l, int r, int x, int y) {if(x <= l && r <= y) {return info[id];}int mid = (l + r) / 2;pushdown(id);Info res;if(x <= mid) {res = res + rangeQuery(ls, l, mid, x, y);}if(y > mid) {res = res + rangeQuery(rs, mid + 1, r, x, y);}return res;}void rangeUpdate(int l, int r, const Tag &dx) {rangeUpdate(1, 0, n - 1, l, r, dx);}void update(int pos, const Tag &dx) {rangeUpdate(pos, pos, dx);}Info rangeQuery(int l, int r) {return rangeQuery(1, 0, n - 1, l, r);}Info query(int pos) {return rangeQuery(pos, pos);}void modify(int pos, const Info &val) {return modify(1, 0, n - 1, pos, val);}
#undef ls
#undef rsint n;std::vector<Info> info;std::vector<Tag> tag;std::vector<int> len;
};constexpr i64 INF = 4E18;
i64 mod = 1e9 + 7;
struct Tag {i64 mod = 0;void apply(const Tag &dx) {// mul = (mul * dx.mul) % mod;// add = (add * dx.mul + dx.add) % mod;}
};struct Info {i64 sum = 0;i64 maxx = 0;Info() {}Info(i64 x) : sum(x), maxx(x) {};void apply(const Tag &dx, const int &len) {// sum = (sum * dx.mul + dx.add * len) % mod;}
};Info operator+(const Info &x, const Info &y) {Info res;res.sum = (x.sum + y.sum);res.maxx = max(x.maxx, y.maxx);return res;
}void solve() {int n, m;cin >> n >> m;vector<int> a(n + 1);for (int i = 1; i <= n; i++) cin >> a[i];SegmentTree<Info, Tag> seg(a);while(m--) {int k;cin >> k;if (k == 1) {int l, r;cin >> l >> r;cout << seg.rangeQuery(l, r).sum << '\n';} else if (k == 2) {int l, r, mod;cin >> l >> r >> mod;Tag tag;tag.mod = mod;seg.rangeUpdate(l, r, tag);} else {int i, x;cin >> i >> x;Info info;info = x;seg.modify(i, info);}}}int main() {std::ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);int T = 1;//cin >> T;while (T--) solve();return 0;
}