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‎DSA/Notes.md

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@@ -1,4 +1,4 @@
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并查集 Disjoint-set
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### 并查集 Disjoint-set
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- [oi-wiki](https://oi-wiki.org/ds/dsu/)
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- 每个集合用一棵树来表示, 树根的编号就是整个集合的编号
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- 每个结点存储其父结点 `p[x]`
@@ -20,9 +20,9 @@
2020
[Almost Union-Find](https://onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=229&page=show_problem&problem=3138)
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### 树状数组 Binary Indexed Tree
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[oi-wiki: Binary Indexed Tree](https://oi-wiki.org/ds/fenwick/)
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[wikipedia: Fenwick_tree](https://en.wikipedia.org/wiki/Fenwick_tree)
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[五分钟丝滑动画讲解 | 树状数组](https://www.bilibili.com/video/BV1ce411u7qP/)
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- [oi-wiki: Binary Indexed Tree](https://oi-wiki.org/ds/fenwick/)
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- [wikipedia: Fenwick_tree](https://en.wikipedia.org/wiki/Fenwick_tree)
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- [五分钟丝滑动画讲解 | 树状数组](https://www.bilibili.com/video/BV1ce411u7qP/)
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- 单点修改, 区间查询 (基础)
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- 区间修改, 单点查询: 查分数组
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- 离线处理: 读取所有查询, 统一处理, 统一输出
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### 线段树 Segment Tree
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- [wikipedia:Segment Tree](https://en.wikipedia.org/wiki/Segment_tree)
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- [CF: SegTree](https://codeforces.com/blog/entry/57319)
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- [usaco](https://usaco.guide/adv/segtree-beats?lang=cpp)
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- [hackerearth](https://www.hackerearth.com/practice/data-structures/advanced-data-structures/segment-trees/tutorial/)
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- 分治 + 二叉树 + Lazy-Tag
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- 大区间的解可通过小区间的解合并而得
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- 单点修改 + 区间查询 (Base)

‎DSA/SegmentTree/AW1275.cpp

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/* 最大数
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*/
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#include <bits/stdc++.h>
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using namespace std;
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const size_t N = 2e5 + 10;
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size_t m;
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int p;
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int a;
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int main() {
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cin >> m >> p;
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}

‎DSA/SegmentTree/AW1275_noted.cpp

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/*
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作者：random_srand
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链接：https://www.acwing.com/solution/content/33974/
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来源：AcWing
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著作权归作者所有。商业转载请联系作者获得授权，非商业转载请注明出处。
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*/
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#include <iostream>
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using namespace std;
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const int N = 2e5 + 5;
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typedef long long LL;
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// 线段树的结点, 最大空间开4倍
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struct Node {
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int l, r;
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int v; // 最大值
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} tr[N * 4];
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int m, p;
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// u为当前线段树的结点编号
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void build(int u, int l, int r) {
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tr[u] = { l, r };
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if(l == r) return;
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int mid = l + r >> 1;
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build(u << 1, l, mid), build(u << 1 | 1, mid + 1, r);
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}
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// 查询以u为根节点，区间[l, r]中的最大值
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int query(int u, int l, int r) {
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// Tl-----Tr
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// L-------------R
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// 1.不必分治，直接返回
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if(tr[u].l >= l && tr[u].r <= r) return tr[u].v;
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int mid = tr[u].l + tr[u].r >> 1;
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int v = 0;
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// Tl----m----Tr
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// L-------------R
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// 2.需要在tr的左区间[Tl, m]继续分治
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if(l <= mid) v = query(u << 1, l, r);
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// Tl----m----Tr
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// L---------R
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// 3.需要在tr的右区间(m, Tr]继续分治
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if(r > mid) v = max(v, query(u << 1 | 1, l, r));
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// Tl----m----Tr
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// L-----R
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// 2.3涵盖了这种情况
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return v;
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}
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// u为结点编号，更新该结点的区间最大值
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void modify(int u, int x, int v) {
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if(tr[u].l == tr[u].r)
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tr[u].v = v; // 叶节点，递归出口
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else {
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int mid = tr[u].l + tr[u].r >> 1;
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// 分治处理左右子树, 寻找x所在的子树
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if(x <= mid)
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modify(u << 1, x, v);
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else
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modify(u << 1 | 1, x, v);
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// 回溯，拿子结点的信息更新父节点, 即pushup操作
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tr[u].v = max(tr[u << 1].v, tr[u << 1 | 1].v);
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}
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}
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int main() {
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// n表示树中的结点个数, last保存上一次查询的结果
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int n = 0, last = 0;
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cin >> m >> p;
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// 初始化线段树, 结点的区间最多为[1, m]。
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build(1, 1, m);
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while(m--) {
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char op;
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cin >> op;
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if(op == 'A') // 添加结点
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{
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int t;
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cin >> t;
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// 在n + 1处插入
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modify(1, n + 1, ((LL)t + last) % p);
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// 结点个数+1
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n++;
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} else {
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int L;
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cin >> L;
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// 查询[n - L + 1, n]内的最大值，u = 1，即从根节点开始查询
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last = query(1, n - L + 1, n);
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cout << last << endl;
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}
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}
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return 0;
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}

‎DSA/SegmentTree/LuoGu3372.cpp

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}
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}
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/*
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5 5
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1 5 4 2 3
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2 2 4
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1 2 3 2
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2 3 4
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1 1 5 1
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2 1 4
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‎DSA/SegmentTree/LuoGu3372_New.cpp

Whitespace-only changes.

‎DSA/SegmentTree/SegTreeTemplate.cpp

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#include <bits/stdc++.h>
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using namespace std;
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#define ls(x) ((x) << 1)
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#define rs(x) (((x) << 1) | 1)
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#define mid(l, r) (((l) + (r)) >> 1)
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8+
typedef pair<size_t, size_t> Interval;
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bool isInterSect(Interval r1, Interval r2) {
11+
if(r1.first > r2.first) swap(r1, r2);
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/* r1.first <= r2.first */
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if(r1.second >= r2.first) return true;
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return false;
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}
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template <typename T>
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class SegTree {
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private:
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vector<T> tree, tag;
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// int n, n4;
22+
23+
const vector<T>& arrRef;
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const size_t mRoot = 1, mRootStart = 1, mRootEnd;
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void build(const size_t root, const size_t start, const size_t end);
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void pushup(const size_t root);
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void pushdown(const size_t root, const size_t curStart,
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const size_t curEnd);
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void addtag(const size_t root, const size_t curStart, const size_t curEnd,
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const T val);
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void intervalUpdateImpl(const size_t start, const size_t end, const T val,
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const size_t root, const size_t curStart,
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const size_t curEnd);
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T getIntervalSumImpl(const size_t start, const size_t end,
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const size_t root, const size_t curStart,
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const size_t curEnd);
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public:
40+
SegTree(const vector<T>& arr)
41+
: mRootEnd(arr.size()), arrRef(arr), tree(arr.size() << 2, 0),
42+
tag(arr.size() << 2, 0) {
43+
/* idx start from 1*/
44+
assert(arr.size() >= 1 and arr.at(0) == 0);
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build(mRoot, mRootStart, mRootEnd);
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}
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48+
void intervalUpdate(const size_t start, const size_t end, const T val);
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T getIntervalSum(const size_t start, const size_t end);
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};
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template <typename T>
53+
void SegTree<T>::build(const size_t root, const size_t start,
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const size_t end) {
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/* leaf node */
56+
if(start == end) {
57+
tree[root] = arrRef[start];
58+
return;
59+
}
60+
/* divide */
61+
const size_t Mid = mid(start, end);
62+
63+
build(ls(root), start, Mid);
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build(rs(root), Mid + 1, end);
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/* calc root */
66+
pushup(root);
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}
68+
template <typename T>
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void SegTree<T>::pushup(const size_t root) {
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/* gather children value, push up to root */
71+
/* interval sum */
72+
tree[root] = tree[ls(root)] + tree[rs(root)];
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/* max/min */
74+
// tree[root] = min(tree[ls(root)], tree[rs(root)]);
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}
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template <typename T>
77+
void SegTree<T>::pushdown(const size_t root, const size_t curStart,
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const size_t curEnd) {
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/* push root tag to children,
80+
because we need to update child interval of root,
81+
so the tag of root will be invalid,
82+
but its children must record root's tag */
83+
if(tag[root]) {
84+
const size_t curMid = mid(curStart, curEnd);
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addtag(ls(root), curStart, curMid, tag[root]);
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addtag(rs(root), curMid + 1, curEnd, tag[root]);
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tag[root] = 0;
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}
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}
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template <typename T>
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void SegTree<T>::addtag(const size_t root, const size_t curStart,
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const size_t curEnd, const T val) {
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assert(curStart <= curEnd);
95+
tag[root] += val;
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tree[root] += val * (curEnd - curStart + 1);
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}
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template <typename T>
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void SegTree<T>::intervalUpdate(const size_t start, const size_t end,
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const T val) {
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intervalUpdateImpl(start, end, val, mRoot, mRootStart, mRootEnd);
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}
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template <typename T>
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void SegTree<T>::intervalUpdateImpl(const size_t start, const size_t end,
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const T val, const size_t root,
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const size_t curStart,
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const size_t curEnd) {
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assert(curStart <= curEnd);
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/* assert Intersect */
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assert(isInterSect({ start, end }, { curStart, curEnd }));
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/* query interval complete cover cur node interval
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** start <= curStart ~ curEnd <= end */
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if(start <= curStart and curEnd <= end) {
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addtag(root, curStart, curEnd, val);
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return;
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}
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/* else: cur Interval of root is not in [start, end],
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** need divide and conquer */
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/* firstly, pushdown root tag to children */
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pushdown(root, curStart, curEnd);
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/* then recursively divide and conquer */
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const size_t curMid = mid(curStart, curEnd);
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/* left child tree */
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if(curMid >= start)
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intervalUpdateImpl(start, end, val, ls(root), curStart, curMid);
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/* right child tree */
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if(curMid + 1 <= end)
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intervalUpdateImpl(start, end, val, rs(root), curMid + 1, curEnd);
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/* update root */
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pushup(root);
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}
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template <typename T>
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T SegTree<T>::getIntervalSumImpl(const size_t start, const size_t end,
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const size_t root, const size_t curStart,
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const size_t curEnd) {
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assert(curStart <= curEnd);
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/* query interval complete cover cur node interval, return tree[root]
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start <= curStart and curEnd <= end */
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if(start <= curStart and curEnd <= end) { return tree[root]; }
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/* not complete cover */
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pushdown(root, curStart, curEnd);
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T res = 0;
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const size_t curMid = mid(curStart, curEnd);
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if(curMid >= start)
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res += getIntervalSumImpl(start, end, ls(root), curStart, curMid);
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if(curMid + 1 <= end)
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res += getIntervalSumImpl(start, end, rs(root), curMid + 1, curEnd);
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return res;
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}
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template <typename T>
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T SegTree<T>::getIntervalSum(const size_t start, const size_t end) {
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return getIntervalSumImpl(start, end, mRoot, mRootStart, mRootEnd);
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}
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bool debug = true;
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void testSegTree() {
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vector<int> arr = { 0, 1, 2, 3, 4, 5, 6 };
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SegTree<int> segtree(arr);
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function<void(const size_t start, const size_t end, bool debug)>
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testIntervalSum;
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function<void(const size_t start, const size_t end, const int val)>
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testIntervalUpdate;
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testIntervalSum = [&arr, &segtree](const size_t start, const size_t end,
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bool debug) {
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auto stdAns = accumulate(arr.begin() + start, arr.begin() + end + 1, 0);
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auto stAns = segtree.getIntervalSum(start, end);
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if(stdAns != stAns) { cout << "wrong ans!" << endl; }
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if(debug) {
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cout << "test Interval Sum: [" << start << ", " << end << "]"
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<< endl;
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cout << "Standerd Ans: " << stdAns << endl;
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cout << "SegTree Ans " << stdAns << endl;
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}
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};
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for(int s = 1; s <= arr.size(); s++) {
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for(int e = s; e <= arr.size(); e++) { testIntervalSum(s, e, false); }
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}
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testIntervalUpdate = [&arr, &segtree, testIntervalSum](const size_t start,
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const size_t end,
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const int val) {
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//
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for(int i = start; i <= end; i++) { arr[i] += val; }
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cout << "arr: " << endl;
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for(int i = 0; i < arr.size(); i++) { cout << arr[i] << " "; }
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cout << endl;
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segtree.intervalUpdate(start, end, val);
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testIntervalSum(start, end, true);
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};
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testIntervalUpdate(1, 2, 5);
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testIntervalUpdate(1, 3, -2);
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testIntervalUpdate(2, 2, 1);
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testIntervalUpdate(5, 5, -3);
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}
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typedef uint64_t ull;
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typedef int64_t ll;
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const int N = 1e5 + 10;
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size_t n, m;
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vector<ll> arr(N);
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int main() {
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cin >> n >> m;
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for(int i = 1; i <= n; i++) { cin >> arr[i]; }
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arr.resize(n + 1);
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SegTree<ll> segtree(arr);
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while(m--) {
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int q, x, y;
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cin >> q;
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if(q == 1) {
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/* 1 x y k */
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ll k;
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cin >> x >> y >> k;
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segtree.intervalUpdate(x, y, k);
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} else if(q == 2) {
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cin >> x >> y;
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cout << segtree.getIntervalSum(x, y) << endl;
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} else {
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assert(false);
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}
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}
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return 0;
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}

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