From 5badcf4225e1d894edcdf154f922c0270dafbdc4 Mon Sep 17 00:00:00 2001 From: flykhan Date: Mon, 3 Apr 2023 16:59:32 +0800 Subject: [PATCH] =?UTF-8?q?=E7=AE=97=E6=B3=95=E6=A8=A1=E6=9D=BF=E6=95=B4?= =?UTF-8?q?=E7=90=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 代码模板/基础算法.md | 313 +++++++++++++++++++++++ 代码模板/基础算法.pdf | Bin 0 -> 155843 bytes 代码模板/搜索与图论.md | 538 ++++++++++++++++++++++++++++++++++++++++ 代码模板/搜索与图论.pdf | Bin 0 -> 261503 bytes 代码模板/数学知识.md | 510 +++++++++++++++++++++++++++++++++++++ 代码模板/数学知识.pdf | Bin 0 -> 331838 bytes 代码模板/数据结构.md | 537 +++++++++++++++++++++++++++++++++++++++ 代码模板/数据结构.pdf | Bin 0 -> 191999 bytes 8 files changed, 1898 insertions(+) create mode 100644 代码模板/基础算法.md create mode 100644 代码模板/基础算法.pdf create mode 100644 代码模板/搜索与图论.md create mode 100644 代码模板/搜索与图论.pdf create mode 100644 代码模板/数学知识.md create mode 100644 代码模板/数学知识.pdf create mode 100644 代码模板/数据结构.md create mode 100644 代码模板/数据结构.pdf diff --git a/代码模板/基础算法.md b/代码模板/基础算法.md new file mode 100644 index 0000000..aa70278 --- /dev/null +++ b/代码模板/基础算法.md @@ -0,0 +1,313 @@ +##### 快速排序算法模板 —— 模板题 AcWing 785. 快速排序 + +```cpp +void quick_sort(int q[], int l, int r) +{ + if (l >= r) return; + + int i = l - 1, j = r + 1, x = q[l + r >> 1]; + while (i < j) + { + do i ++ ; while (q[i] < x); + do j -- ; while (q[j] > x); + if (i < j) swap(q[i], q[j]); + } + quick_sort(q, l, j), quick_sort(q, j + 1, r); +} +``` + +------------------------ + +##### 归并排序算法模板 —— 模板题 AcWing 787. 归并排序 + +```cpp +void merge_sort(int q[], int l, int r) +{ + if (l >= r) return; + + int mid = l + r >> 1; + merge_sort(q, l, mid); + merge_sort(q, mid + 1, r); + + int k = 0, i = l, j = mid + 1; + while (i <= mid && j <= r) + if (q[i] <= q[j]) tmp[k ++ ] = q[i ++ ]; + else tmp[k ++ ] = q[j ++ ]; + + while (i <= mid) tmp[k ++ ] = q[i ++ ]; + while (j <= r) tmp[k ++ ] = q[j ++ ]; + + for (i = l, j = 0; i <= r; i ++, j ++ ) q[i] = tmp[j]; +} +``` + +------------ + +##### 整数二分算法模板 —— 模板题 AcWing 789. 数的范围 + +```cpp +bool check(int x) {/* ... */} // 检查x是否满足某种性质 + +// 区间[l, r]被划分成[l, mid]和[mid + 1, r]时使用: +int bsearch_1(int l, int r) +{ + while (l < r) + { + int mid = l + r >> 1; + if (check(mid)) r = mid; // check()判断mid是否满足性质 + else l = mid + 1; + } + return l; +} +// 区间[l, r]被划分成[l, mid - 1]和[mid, r]时使用: +int bsearch_2(int l, int r) +{ + while (l < r) + { + int mid = l + r + 1 >> 1; + if (check(mid)) l = mid; + else r = mid - 1; + } + return l; +} +``` + +------------------- + +##### 浮点数二分算法模板 —— 模板题 AcWing 790. 数的三次方根 + +```cpp +bool check(double x) {/* ... */} // 检查x是否满足某种性质 + +double bsearch_3(double l, double r) +{ + const double eps = 1e-6; // eps 表示精度,取决于题目对精度的要求 + while (r - l > eps) + { + double mid = (l + r) / 2; + if (check(mid)) r = mid; + else l = mid; + } + return l; +} +``` + +-------------------- + +##### 高精度加法 —— 模板题 AcWing 791. 高精度加法 + +```cpp +// C = A + B, A >= 0, B >= 0 +vector add(vector &A, vector &B) +{ + if (A.size() < B.size()) return add(B, A); + + vector C; + int t = 0; + for (int i = 0; i < A.size(); i ++ ) + { + t += A[i]; + if (i < B.size()) t += B[i]; + C.push_back(t % 10); + t /= 10; + } + + if (t) C.push_back(t); + return C; +} +``` + +-------------------- + +##### 高精度减法 —— 模板题 AcWing 792. 高精度减法 + +```cpp +// C = A - B, 满足A >= B, A >= 0, B >= 0 +vector sub(vector &A, vector &B) +{ + vector C; + for (int i = 0, t = 0; i < A.size(); i ++ ) + { + t = A[i] - t; + if (i < B.size()) t -= B[i]; + C.push_back((t + 10) % 10); + if (t < 0) t = 1; + else t = 0; + } + + while (C.size() > 1 && C.back() == 0) C.pop_back(); + return C; +} +``` + +------------------ + +##### 高精度乘低精度 —— 模板题 AcWing 793. 高精度乘法 + +```cpp +// C = A * b, A >= 0, b >= 0 +vector mul(vector &A, int b) +{ + vector C; + + int t = 0; + for (int i = 0; i < A.size() || t; i ++ ) + { + if (i < A.size()) t += A[i] * b; + C.push_back(t % 10); + t /= 10; + } + + while (C.size() > 1 && C.back() == 0) C.pop_back(); + return C; +} +``` + +----------------- + +##### 高精度除以低精度 —— 模板题 AcWing 794. 高精度除法 + +```cpp +// A / b = C ... r, A >= 0, b > 0 +vector div(vector &A, int b, int &r) +{ + vector C; + r = 0; + for (int i = A.size() - 1; i >= 0; i -- ) + { + r = r * 10 + A[i]; + C.push_back(r / b); + r %= b; + } + reverse(C.begin(), C.end()); + while (C.size() > 1 && C.back() == 0) C.pop_back(); + return C; +} +``` + +---------------------- + +##### 一维前缀和 —— 模板题 AcWing 795. 前缀和 + +$S[i] = a[1] + a[2] + ... a[i]$ + +$a[l] + ... + a[r] = S[r] - S[l - 1]$ + +--------------------- + +##### 二维前缀和 —— 模板题 AcWing 796. 子矩阵的和 + +$S[i, j] = 第i行j列格子左上部分所有元素的和$ + +$以(x_1, y_1)为左上角,(x_2, y_2)为右下角的子矩阵的和为:$ + +```cpp +S[x2, y2] - S[x1 - 1, y2] - S[x2, y1 - 1] + S[x1 - 1, y1 - 1] +``` + +---------------- + +##### 一维差分 —— 模板题 AcWing 797. 差分 + +$给区间[l, r]中的每个数加上c:$ + +```cpp +B[l] += c, B[r + 1] -= c +``` + +---------------- + +##### 二维差分 —— 模板题 AcWing 798. 差分矩阵 + +$给以(x_1, y_1)为左上角,(x_2, y_2)为右下角的子矩阵中的所有元素加上c:$ + +```cpp +S[x1, y1] += c, S[x2 + 1, y1] -= c, S[x1, y2 + 1] -= c, S[x2 + 1, y2 + 1] += c +``` + +----------------------- + +##### 位运算 —— 模板题 AcWing 801. 二进制中1的个数 + +```cpp +求n的第k位数字: n >> k & 1 +返回n的最后一位1:lowbit(n) = n & -n +``` + +---------------------- + +##### 双指针算法 —— 模板题 AcWIng 799. 最长连续不重复子序列, AcWing 800. 数组元素的目标和 + +```cpp +for (int i = 0, j = 0; i < n; i ++ ) +{ + while (j < i && check(i, j)) j ++ ; + + // 具体问题的逻辑 +} +常见问题分类: + (1) 对于一个序列,用两个指针维护一段区间 + (2) 对于两个序列,维护某种次序,比如归并排序中合并两个有序序列的操作 +``` + +----------------------- + +##### 离散化 —— 模板题 AcWing 802. 区间和 + +```cpp +vector alls; // 存储所有待离散化的值 +sort(alls.begin(), alls.end()); // 将所有值排序 +alls.erase(unique(alls.begin(), alls.end()), alls.end()); // 去掉重复元素 + +// 二分求出x对应的离散化的值 +int find(int x) // 找到第一个大于等于x的位置 +{ + int l = 0, r = alls.size() - 1; + while (l < r) + { + int mid = l + r >> 1; + if (alls[mid] >= x) r = mid; + else l = mid + 1; + } + return r + 1; // 映射到1, 2, ...n +} +``` + +-------------------- + +##### 区间合并 —— 模板题 AcWing 803. 区间合并 + +```cpp +// 将所有存在交集的区间合并 +void merge(vector &segs) +{ + vector res; + + sort(segs.begin(), segs.end()); + + int st = -2e9, ed = -2e9; + for (auto seg : segs) + if (ed < seg.first) + { + if (st != -2e9) res.push_back({st, ed}); + st = seg.first, ed = seg.second; + } + else ed = max(ed, seg.second); + + if (st != -2e9) res.push_back({st, ed}); + + segs = res; + +} +``` + +-------------- + +> 作者:yxc +> +> 链接:https://www.acwing.com/blog/content/277/ +> +> 来源:AcWing +> +> 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。 + diff --git a/代码模板/基础算法.pdf b/代码模板/基础算法.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f1bb405ee0a51f59bc6eb993ad917addde13592e GIT binary patch literal 155843 zcma%>V{j)y+wM2EZQHhOI~&`!Z9Cc6wz08|jcwx}>*T3(zPzX256?{1H9dWI&s0s# zTs{4}n@mwuoQ{c}1DY)GXM=`jVkTrHv^TPX=H-QE5Vy2(F?Awj5VtXOF%>m6wl^__ zW{@?tGk38dWMgOLfad3ic6Mh%;5Ht&n?ME^(GsLbWfXv{K|g)JH(_|2`1(0|@bbeOyfH z>leEO%k)hJ^HSFgFtI^eC^!{+~M&_9!c!|4vQAR8;sf-lX&}N 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zZbh5>l^Zu77HmkAf82ZX^u`mF>woGW-&ECIzbQ2O+~?hs_aFJ1cB$?5E^%q$+gR DZATgr literal 0 HcmV?d00001 diff --git a/代码模板/搜索与图论.md b/代码模板/搜索与图论.md new file mode 100644 index 0000000..8911eda --- /dev/null +++ b/代码模板/搜索与图论.md @@ -0,0 +1,538 @@ +##### 树与图的存储 + +树是一种特殊的图,与图的存储方式相同。 + +$对于无向图中的边ab,存储两条有向边a->b, b->a。$ + +因此我们可以只考虑有向图的存储。 + +(1) 邻接矩阵:$g[a][b] 存储边a->b$ + +(2) 邻接表: + +```cpp +// 对于每个点k,开一个单链表,存储k所有可以走到的点。h[k]存储这个单链表的头结点 +int h[N], e[N], ne[N], idx; + +// 添加一条边a->b +void add(int a, int b) +{ + e[idx] = b, ne[idx] = h[a], h[a] = idx ++ ; +} + +// 初始化 +idx = 0; +memset(h, -1, sizeof h); +``` + +------------- + +##### 树与图的遍历 + +###### $时间复杂度 O(n+m), n 表示点数,m表示边数$ + +###### (1) 深度优先遍历 —— 模板题 AcWing 846. 树的重心 + +```cpp +int dfs(int u) +{ + st[u] = true; // st[u] 表示点u已经被遍历过 + + for (int i = h[u]; i != -1; i = ne[i]) + { + int j = e[i]; + if (!st[j]) dfs(j); + } +} +``` + +###### (2) 宽度优先遍历 —— 模板题 AcWing 847. 图中点的层次 + +```cpp +queue q; +st[1] = true; // 表示1号点已经被遍历过 +q.push(1); + +while (q.size()) +{ + int t = q.front(); + q.pop(); + + for (int i = h[t]; i != -1; i = ne[i]) + { + int j = e[i]; + if (!st[j]) + { + st[j] = true; // 表示点j已经被遍历过 + q.push(j); + } + } +} +``` + +---------------- + +##### 拓扑排序 —— 模板题 AcWing 848. 有向图的拓扑序列 + +###### $时间复杂度 O(n+m), n 表示点数,m表示边数$ + +```cpp +bool topsort() +{ + int hh = 0, tt = -1; + + // d[i] 存储点i的入度 + for (int i = 1; i <= n; i ++ ) + if (!d[i]) + q[ ++ tt] = i; + + while (hh <= tt) + { + int t = q[hh ++ ]; + + for (int i = h[t]; i != -1; i = ne[i]) + { + int j = e[i]; + if (-- d[j] == 0) + q[ ++ tt] = j; + } + } + + // 如果所有点都入队了,说明存在拓扑序列;否则不存在拓扑序列。 + return tt == n - 1; + +} +``` + +----------------- + +##### 朴素dijkstra算法 —— 模板题 AcWing 849. Dijkstra求最短路 I + +###### $时间复杂是 O(n^2+m), n 表示点数,m表示边数$ + +```cpp +int g[N][N]; // 存储每条边 +int dist[N]; // 存储1号点到每个点的最短距离 +bool st[N]; // 存储每个点的最短路是否已经确定 + +// 求1号点到n号点的最短路,如果不存在则返回-1 +int dijkstra() +{ + memset(dist, 0x3f, sizeof dist); + dist[1] = 0; + + for (int i = 0; i < n - 1; i ++ ) + { + int t = -1; // 在还未确定最短路的点中,寻找距离最小的点 + for (int j = 1; j <= n; j ++ ) + if (!st[j] && (t == -1 || dist[t] > dist[j])) + t = j; + + // 用t更新其他点的距离 + for (int j = 1; j <= n; j ++ ) + dist[j] = min(dist[j], dist[t] + g[t][j]); + + st[t] = true; + } + + if (dist[n] == 0x3f3f3f3f) return -1; + return dist[n]; +} +``` + +---------------- + +##### 堆优化版dijkstra —— 模板题 AcWing 850. Dijkstra求最短路 II + +###### $时间复杂度 O(mlogn), n 表示点数,m表示边数$ + +```cpp +typedef pair PII; + +int n; // 点的数量 +int h[N], w[N], e[N], ne[N], idx; // 邻接表存储所有边 +int dist[N]; // 存储所有点到1号点的距离 +bool st[N]; // 存储每个点的最短距离是否已确定 + +// 求1号点到n号点的最短距离,如果不存在,则返回-1 +int dijkstra() +{ + memset(dist, 0x3f, sizeof dist); + dist[1] = 0; + priority_queue, greater> heap; + heap.push({0, 1}); // first存储距离,second存储节点编号 + + while (heap.size()) + { + auto t = heap.top(); + heap.pop(); + + int ver = t.second, distance = t.first; + + if (st[ver]) continue; + st[ver] = true; + + for (int i = h[ver]; i != -1; i = ne[i]) + { + int j = e[i]; + if (dist[j] > distance + w[i]) + { + dist[j] = distance + w[i]; + heap.push({dist[j], j}); + } + } + } + + if (dist[n] == 0x3f3f3f3f) return -1; + return dist[n]; + +} +``` + +------------------ + +##### Bellman-Ford算法 —— 模板题 AcWing 853. 有边数限制的最短路 + +###### $时间复杂度 O(nm), n 表示点数,m表示边数$ + +注意在模板题中需要对下面的模板稍作修改,加上备份数组,详情见模板题。 + +```cpp +int n, m; // n表示点数,m表示边数 +int dist[N]; // dist[x]存储1到x的最短路距离 + +struct Edge // 边,a表示出点,b表示入点,w表示边的权重 +{ + int a, b, w; +}edges[M]; + +// 求1到n的最短路距离,如果无法从1走到n,则返回-1。 +int bellman_ford() +{ + memset(dist, 0x3f, sizeof dist); + dist[1] = 0; + + // 如果第n次迭代仍然会松弛三角不等式,就说明存在一条长度是n+1的最短路径,由抽屉原理,路径中至少存在两个相同的点,说明图中存在负权回路。 + for (int i = 0; i < n; i ++ ) + { + for (int j = 0; j < m; j ++ ) + { + int a = edges[j].a, b = edges[j].b, w = edges[j].w; + if (dist[b] > dist[a] + w) + dist[b] = dist[a] + w; + } + } + + if (dist[n] > 0x3f3f3f3f / 2) return -1; + return dist[n]; +} +``` + +------------------- + +##### spfa 算法(队列优化的Bellman-Ford算法) —— 模板题 AcWing 851. spfa求最短路 + +###### $时间复杂度 平均情况下 O(m),最坏情况下 O(nm), n 表示点数,m表示边数$ + +```cpp +int n; // 总点数 +int h[N], w[N], e[N], ne[N], idx; // 邻接表存储所有边 +int dist[N]; // 存储每个点到1号点的最短距离 +bool st[N]; // 存储每个点是否在队列中 + +// 求1号点到n号点的最短路距离,如果从1号点无法走到n号点则返回-1 +int spfa() +{ + memset(dist, 0x3f, sizeof dist); + dist[1] = 0; + + queue q; + q.push(1); + st[1] = true; + + while (q.size()) + { + auto t = q.front(); + q.pop(); + + st[t] = false; + + for (int i = h[t]; i != -1; i = ne[i]) + { + int j = e[i]; + if (dist[j] > dist[t] + w[i]) + { + dist[j] = dist[t] + w[i]; + if (!st[j]) // 如果队列中已存在j,则不需要将j重复插入 + { + q.push(j); + st[j] = true; + } + } + } + } + + if (dist[n] == 0x3f3f3f3f) return -1; + return dist[n]; +} +``` + +----------------- + +##### spfa判断图中是否存在负环 —— 模板题 AcWing 852. spfa判断负环 + +###### $时间复杂度是 O(nm), n 表示点数,m表示边数$ + +```cpp +int n; // 总点数 +int h[N], w[N], e[N], ne[N], idx; // 邻接表存储所有边 +int dist[N], cnt[N]; // dist[x]存储1号点到x的最短距离,cnt[x]存储1到x的最短路中经过的点数 +bool st[N]; // 存储每个点是否在队列中 + +// 如果存在负环,则返回true,否则返回false。 +bool spfa() +{ + // 不需要初始化dist数组 + // 原理:如果某条最短路径上有n个点(除了自己),那么加上自己之后一共有n+1个点,由抽屉原理一定有两个点相同,所以存在环。 + + queue q; + for (int i = 1; i <= n; i ++ ) + { + q.push(i); + st[i] = true; + } + + while (q.size()) + { + auto t = q.front(); + q.pop(); + + st[t] = false; + + for (int i = h[t]; i != -1; i = ne[i]) + { + int j = e[i]; + if (dist[j] > dist[t] + w[i]) + { + dist[j] = dist[t] + w[i]; + cnt[j] = cnt[t] + 1; + if (cnt[j] >= n) return true; // 如果从1号点到x的最短路中包含至少n个点(不包括自己),则说明存在环 + if (!st[j]) + { + q.push(j); + st[j] = true; + } + } + } + } + + return false; +} +``` + +------------------ + +##### floyd算法 —— 模板题 AcWing 854. Floyd求最短路 + +###### $时间复杂度是 O(n^3), n表示点数$ + +```cpp +初始化: + for (int i = 1; i <= n; i ++ ) + for (int j = 1; j <= n; j ++ ) + if (i == j) d[i][j] = 0; + else d[i][j] = INF; + +// 算法结束后,d[a][b]表示a到b的最短距离 +void floyd() +{ + for (int k = 1; k <= n; k ++ ) + for (int i = 1; i <= n; i ++ ) + for (int j = 1; j <= n; j ++ ) + d[i][j] = min(d[i][j], d[i][k] + d[k][j]); +} +``` + +------------------- + +##### 朴素版prim算法 —— 模板题 AcWing 858. Prim算法求最小生成树 + +###### $时间复杂度是 O(n^2+m), n 表示点数,m表示边数$ + +```cpp +int n; // n表示点数 +int g[N][N]; // 邻接矩阵,存储所有边 +int dist[N]; // 存储其他点到当前最小生成树的距离 +bool st[N]; // 存储每个点是否已经在生成树中 + + +// 如果图不连通,则返回INF(值是0x3f3f3f3f), 否则返回最小生成树的树边权重之和 +int prim() +{ + memset(dist, 0x3f, sizeof dist); + + int res = 0; + for (int i = 0; i < n; i ++ ) + { + int t = -1; + for (int j = 1; j <= n; j ++ ) + if (!st[j] && (t == -1 || dist[t] > dist[j])) + t = j; + + if (i && dist[t] == INF) return INF; + + if (i) res += dist[t]; + st[t] = true; + + for (int j = 1; j <= n; j ++ ) dist[j] = min(dist[j], g[t][j]); + } + + return res; +} +``` + +------------------ + +##### Kruskal算法 —— 模板题 AcWing 859. Kruskal算法求最小生成树 + +###### $时间复杂度是 O(mlogm), n 表示点数,m表示边数$ + +```cpp +int n, m; // n是点数,m是边数 +int p[N]; // 并查集的父节点数组 + +struct Edge // 存储边 +{ + int a, b, w; + + bool operator< (const Edge &W)const + { + return w < W.w; + } + +}edges[M]; + +int find(int x) // 并查集核心操作 +{ + if (p[x] != x) p[x] = find(p[x]); + return p[x]; +} + +int kruskal() +{ + sort(edges, edges + m); + + for (int i = 1; i <= n; i ++ ) p[i] = i; // 初始化并查集 + + int res = 0, cnt = 0; + for (int i = 0; i < m; i ++ ) + { + int a = edges[i].a, b = edges[i].b, w = edges[i].w; + + a = find(a), b = find(b); + if (a != b) // 如果两个连通块不连通,则将这两个连通块合并 + { + p[a] = b; + res += w; + cnt ++ ; + } + } + + if (cnt < n - 1) return INF; + return res; +} +``` + +------------------ + +##### 染色法判别二分图 —— 模板题 AcWing 860. 染色法判定二分图 + +###### $时间复杂度是 O(n+m), n 表示点数,m表示边数$ + +```cpp +int n; // n表示点数 +int h[N], e[M], ne[M], idx; // 邻接表存储图 +int color[N]; // 表示每个点的颜色,-1表示未染色,0表示白色,1表示黑色 + +// 参数:u表示当前节点,c表示当前点的颜色 +bool dfs(int u, int c) +{ + color[u] = c; + for (int i = h[u]; i != -1; i = ne[i]) + { + int j = e[i]; + if (color[j] == -1) + { + if (!dfs(j, !c)) return false; + } + else if (color[j] == c) return false; + } + + return true; + +} + +bool check() +{ + memset(color, -1, sizeof color); + bool flag = true; + for (int i = 1; i <= n; i ++ ) + if (color[i] == -1) + if (!dfs(i, 0)) + { + flag = false; + break; + } + return flag; +} +``` + +------------------- + +##### 匈牙利算法 —— 模板题 AcWing 861. 二分图的最大匹配 + +###### $时间复杂度是 O(nm), n 表示点数,m表示边数$ + +```cpp +int n1, n2; // n1表示第一个集合中的点数,n2表示第二个集合中的点数 +int h[N], e[M], ne[M], idx; // 邻接表存储所有边,匈牙利算法中只会用到从第一个集合指向第二个集合的边,所以这里只用存一个方向的边 +int match[N]; // 存储第二个集合中的每个点当前匹配的第一个集合中的点是哪个 +bool st[N]; // 表示第二个集合中的每个点是否已经被遍历过 + +bool find(int x) +{ + for (int i = h[x]; i != -1; i = ne[i]) + { + int j = e[i]; + if (!st[j]) + { + st[j] = true; + if (match[j] == 0 || find(match[j])) + { + match[j] = x; + return true; + } + } + } + + return false; +} + +// 求最大匹配数,依次枚举第一个集合中的每个点能否匹配第二个集合中的点 +int res = 0; +for (int i = 1; i <= n1; i ++ ) +{ + memset(st, false, sizeof st); + if (find(i)) res ++ ; +} +``` + +------------------- + +> 作者:yxc +> +> 链接:https://www.acwing.com/blog/content/405/ +> +> 来源:AcWing +> +> 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。 \ No newline at end of file diff --git a/代码模板/搜索与图论.pdf b/代码模板/搜索与图论.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c50fe088c367127f94527177e0c584871ba1835d GIT binary patch literal 261503 zcma%>Q*bU!@aALNwr$(CZQHhO+iz?qC$??d$%%2Y-~Ma&V()gUYpVO{>8hEUnv1Fa zkt>Nw&@(e|!jK35H(+2`xQUpE98GLs`1oKLC9Lh-%w32WCG3pd%*D)29nH*P80E|z zEZwY#n3>pFnP3D2U|ii?%#H0~yz*}Sw-fNVolboN#rbioSne?gm=@^UzHfGczkyq9 z88K3TeSZlzxP&E=)OWe{H?wz_X38W=O;}jHp~v_`W@7)Q|K`E{c=pR>c6Be`3ljXh z9^7)1en(YMUG%nO81)T&qDYUOXV3fh6)5<A@TP}@%QnU;2{9vPEA$r;$Z({5pM=?wnf9RI^vV_Tkc_?p^`h9FY;7r zHa+>`xeNFirB$h*EDZ2{A9rMv3jBoUXK}lU-t&BU-cF1RWcj!*g^s`paI4_cOpRkd zWd8np+41}J@QvGs3+j$nH(x2MK;DMnKwDn5qCU zVNI;!+{*iGTEigPuT}LZTJPg!bj9BZ^A+-AE1ty@8@XyW<+st1ljV+2I@6E!{yApI 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zQ-S@byF8T<;XkQy{MGl}lG$k2a)#C70frNhO_`3aLoEW$h636?7zPe%hUSf}-E$eU znUV69D{$?()oG%_p_A}2;lecg=9w(m9br$!RQ>iLAq7d76Du**Nh+mZ;tC(dh#}w} zwlX|o!s()Oc8KN?!^?NI?s5y^6@{`5x<`fPAP{ZSGuCR|#D^244FKqSL(O#cP;@i8!g& zHjSSPPDp!BU)FZs_*jDp7F;jbNZ}KRjEF;+=}2bK0DErv>Q87&xrSeRl2Sr-z%c=oBb_B7fs= zQY71itrqp)rAYQ!_#yKqpU|=wim49cP-FJf&5BR@f`R}I$L&Nno=>9p&b5VFQ^p>k zbgk5u?HWihn^q4E^%6|4Ov6pi4B~81a=^lwQ7sI)6E^o!sjrO+~(e<`+&%2FFQkjT;M76M*s;iOFJF2Rh zCNSjP(}2=yd4_o3IlnIvwh7A(CTkX-Wh|EM_w1YQpbUI?I)9Cli_ukhQJ~wQ3?y|o zVeqhfxxRESQmKq;h+%QU8z9}iOtf-Q{+W^HfitlsDfs(_6Vc&i@Tm0Q-Q!o`+t?>@ zpxsm3xu0HD3oEN%iFccLYyb+UoIgITI>+6#AgW**jjR`tn9{m|U#+Is3;0;y{M`4V zR~t>(B`xt+JUI-hu;2OOnj^JyDyWGy{m22JbmcJ+O1HOGKb2}ueAYXd3l6wv&n zNqJmF?D6K^<(}sd-CT74=A|Zh|My=g8E?n_w!!@Sc#5TWxCa=fslf}y`upR-FpZx^ z2$mO$^7Hinf&a9SenkI6a@%4(Jpa&xV7G@eFh=oM@0K z670-CZ`!_cfqB(C>ck)w=OFi8PAa%oo}Jp4zdwH`(>T}hWw}f8s`$ZGH(j4Du}xfV zF9)boRw#ISfevb~Hko_gD>D{TF(z9RHHw~m#!(0o(S)&z13v-x8eEf1*k!#UnZW*o z_itKX9m@!FF>TQv%I(kWr+Leiru>%bhnwc3TlM>_WVH#AQvqF<q-B_+(h^&78e$dG*LWvc7ZjBi=TYkPy r;&Al0?S2loLk`vE7J 1) cout << x << ' ' << 1 << endl; + cout << endl; +} +``` + +--------------- + +##### 朴素筛法求素数 —— 模板题 AcWing 868. 筛质数 + +```cpp +int primes[N], cnt; // primes[]存储所有素数 +bool st[N]; // st[x]存储x是否被筛掉 + +void get_primes(int n) +{ + for (int i = 2; i <= n; i ++ ) + { + if (st[i]) continue; + primes[cnt ++ ] = i; + for (int j = i + i; j <= n; j += i) + st[j] = true; + } +} +``` + +--------------- + +##### 线性筛法求素数 —— 模板题 AcWing 868. 筛质数 + +```cpp +int primes[N], cnt; // primes[]存储所有素数 +bool st[N]; // st[x]存储x是否被筛掉 + +void get_primes(int n) +{ + for (int i = 2; i <= n; i ++ ) + { + if (!st[i]) primes[cnt ++ ] = i; + for (int j = 0; primes[j] <= n / i; j ++ ) + { + st[primes[j] * i] = true; + if (i % primes[j] == 0) break; + } + } +} +``` + +--------------- + +##### 试除法求所有约数 —— 模板题 AcWing 869. 试除法求约数 + +```cpp +vector get_divisors(int x) +{ + vector res; + for (int i = 1; i <= x / i; i ++ ) + if (x % i == 0) + { + res.push_back(i); + if (i != x / i) res.push_back(x / i); + } + sort(res.begin(), res.end()); + return res; +} +``` + +----- + +##### 约数个数和约数之和 —— 模板题 AcWing 870. 约数个数, AcWing 871. 约数之和 + +如果 N = $p_1^{c_1}$ * $p_2^{c_2}$ * ... *$p_k^{c_k}$ + +约数个数:$(c_1 + 1) * (c_2 + 1) * ... * (c_k + 1)$ + +约数之和: ($p_1^0$ + $p_1^1$ + ... +$ p_1^{c_1}$) * ... * ($p_k^0$ + $p_k^1$ + ... + $p_k^{c_k}$) + +-------- + +##### 欧几里得算法 —— 模板题 AcWing 872. 最大公约数 + +```cpp +int gcd(int a, int b) +{ + return b ? gcd(b, a % b) : a; +} +``` + +---------- + +##### 求欧拉函数 —— 模板题 AcWing 873. 欧拉函数 + +```cpp +int phi(int x) +{ + int res = x; + for (int i = 2; i <= x / i; i ++ ) + if (x % i == 0) + { + res = res / i * (i - 1); + while (x % i == 0) x /= i; + } + if (x > 1) res = res / x * (x - 1); + + return res; +} +``` + +---------- + +##### 筛法求欧拉函数 —— 模板题 AcWing 874. 筛法求欧拉函数 + +```cpp +int primes[N], cnt; // primes[]存储所有素数 +int euler[N]; // 存储每个数的欧拉函数 +bool st[N]; // st[x]存储x是否被筛掉 + +void get_eulers(int n) +{ + euler[1] = 1; + for (int i = 2; i <= n; i ++ ) + { + if (!st[i]) + { + primes[cnt ++ ] = i; + euler[i] = i - 1; + } + for (int j = 0; primes[j] <= n / i; j ++ ) + { + int t = primes[j] * i; + st[t] = true; + if (i % primes[j] == 0) + { + euler[t] = euler[i] * primes[j]; + break; + } + euler[t] = euler[i] * (primes[j] - 1); + } + } +} +``` + +---- + +##### 快速幂 —— 模板题 AcWing 875. 快速幂 + +求 $m^k$ mod p,时间复杂度 O($logk$)。 + +```cpp +int qmi(int m, int k, int p) +{ + int res = 1 % p, t = m; + while (k) + { + if (k&1) res = res * t % p; + t = t * t % p; + k >>= 1; + } + return res; +} +``` + +---- + +##### 扩展欧几里得算法 —— 模板题 AcWing 877. 扩展欧几里得算法 + +```cpp +// 求x, y,使得ax + by = gcd(a, b) +int exgcd(int a, int b, int &x, int &y) +{ + if (!b) + { + x = 1; y = 0; + return a; + } + int d = exgcd(b, a % b, y, x); + y -= (a/b) * x; + return d; +} +``` + +--- + +##### 高斯消元 —— 模板题 AcWing 883. 高斯消元解线性方程组 + +```cpp +// a[N][N]是增广矩阵 +int gauss() +{ + int c, r; + for (c = 0, r = 0; c < n; c ++ ) + { + int t = r; + for (int i = r; i < n; i ++ ) // 找到绝对值最大的行 + if (fabs(a[i][c]) > fabs(a[t][c])) + t = i; + + if (fabs(a[t][c]) < eps) continue; + + for (int i = c; i <= n; i ++ ) swap(a[t][i], a[r][i]); // 将绝对值最大的行换到最顶端 + for (int i = n; i >= c; i -- ) a[r][i] /= a[r][c]; // 将当前行的首位变成1 + for (int i = r + 1; i < n; i ++ ) // 用当前行将下面所有的列消成0 + if (fabs(a[i][c]) > eps) + for (int j = n; j >= c; j -- ) + a[i][j] -= a[r][j] * a[i][c]; + + r ++ ; + } + + if (r < n) + { + for (int i = r; i < n; i ++ ) + if (fabs(a[i][n]) > eps) + return 2; // 无解 + return 1; // 有无穷多组解 + } + + for (int i = n - 1; i >= 0; i -- ) + for (int j = i + 1; j < n; j ++ ) + a[i][n] -= a[i][j] * a[j][n]; + + return 0; // 有唯一解 +} +``` + +--- + +##### 递推法求组合数 —— 模板题 AcWing 885. 求组合数 I + +```cpp +// c[a][b] 表示从a个苹果中选b个的方案数 +for (int i = 0; i < N; i ++ ) + for (int j = 0; j <= i; j ++ ) + if (!j) c[i][j] = 1; + else c[i][j] = (c[i - 1][j] + c[i - 1][j - 1]) % mod; +``` + +--- + +##### 通过预处理逆元的方式求组合数 —— 模板题 AcWing 886. 求组合数 II + +首先预处理出所有阶乘取模的余数fact[N],以及所有阶乘取模的逆元infact[N] + +如果取模的数是质数,可以用费马小定理求逆元 + +```cpp +int qmi(int a, int k, int p) // 快速幂模板 +{ + int res = 1; + while (k) + { + if (k & 1) res = (LL)res * a % p; + a = (LL)a * a % p; + k >>= 1; + } + return res; +} + +// 预处理阶乘的余数和阶乘逆元的余数 +fact[0] = infact[0] = 1; +for (int i = 1; i < N; i ++ ) +{ + fact[i] = (LL)fact[i - 1] * i % mod; + infact[i] = (LL)infact[i - 1] * qmi(i, mod - 2, mod) % mod; +} +``` + +--- + +##### Lucas定理 —— 模板题 AcWing 887. 求组合数 III + +$若p是质数,则对于任意整数 1 <= m <= n,有:$ + +​ C(n, m) = C(n % p, m % p) * C(n / p, m / p) (mod p) + +```cpp +int qmi(int a, int k, int p) // 快速幂模板 +{ + int res = 1 % p; + while (k) + { + if (k & 1) res = (LL)res * a % p; + a = (LL)a * a % p; + k >>= 1; + } + return res; +} + +int C(int a, int b, int p) // 通过定理求组合数C(a, b) +{ + if (a < b) return 0; + + LL x = 1, y = 1; // x是分子,y是分母 + for (int i = a, j = 1; j <= b; i --, j ++ ) + { + x = (LL)x * i % p; + y = (LL) y * j % p; + } + + return x * (LL)qmi(y, p - 2, p) % p; +} + +int lucas(LL a, LL b, int p) +{ + if (a < p && b < p) return C(a, b, p); + return (LL)C(a % p, b % p, p) * lucas(a / p, b / p, p) % p; +} +``` + +--- + +##### 分解质因数法求组合数 —— 模板题 AcWing 888. 求组合数 IV + +当我们需要求出组合数的真实值,而非对某个数的余数时,分解质因数的方式比较好用: + + 1. 筛法求出范围内的所有质数 + + 2. $通过 C(a, b) = a! / b! / (a - b)! 这个公式求出每个质因子的次数。$ + + $n! 中p的次数是 n / p + n / p^2+ n / p^3 + ...$ + + 3. 用高精度乘法将所有质因子相乘 + +```cpp +int primes[N], cnt; // 存储所有质数 +int sum[N]; // 存储每个质数的次数 +bool st[N]; // 存储每个数是否已被筛掉 + +void get_primes(int n) // 线性筛法求素数 +{ + for (int i = 2; i <= n; i ++ ) + { + if (!st[i]) primes[cnt ++ ] = i; + for (int j = 0; primes[j] <= n / i; j ++ ) + { + st[primes[j] * i] = true; + if (i % primes[j] == 0) break; + } + } +} + +int get(int n, int p) // 求n!中的次数 +{ + int res = 0; + while (n) + { + res += n / p; + n /= p; + } + return res; +} + +vector mul(vector a, int b) // 高精度乘低精度模板 +{ + vector c; + int t = 0; + for (int i = 0; i < a.size(); i ++ ) + { + t += a[i] * b; + c.push_back(t % 10); + t /= 10; + } + + while (t) + { + c.push_back(t % 10); + t /= 10; + } + + return c; + +} + +get_primes(a); // 预处理范围内的所有质数 + +for (int i = 0; i < cnt; i ++ ) // 求每个质因数的次数 +{ + int p = primes[i]; + sum[i] = get(a, p) - get(b, p) - get(a - b, p); +} + +vector res; +res.push_back(1); + +for (int i = 0; i < cnt; i ++ ) // 用高精度乘法将所有质因子相乘 + for (int j = 0; j < sum[i]; j ++ ) + res = mul(res, primes[i]); +``` + +--- + +##### 卡特兰数 —— 模板题 AcWing 889. 满足条件的01序列 + +$给定n个0和n个1,它们按照某种顺序排成长度为2n的序列,满足任意前缀中0的个数都不少于1的$ + +$个数的序列的数量为: Cat(n) = C(2n, n) / (n + 1)$ + +--- + +##### NIM游戏 —— 模板题 AcWing 891. Nim游戏 + +给定N堆物品,第i堆物品有Ai个。两名玩家轮流行动,每次可以任选一堆,取走任意多个物品,可把一堆取光,但不能不取。取走最后一件物品者获胜。两人都采取最优策略,问先手是否必胜。 + +我们把这种游戏称为NIM博弈。把游戏过程中面临的状态称为局面。整局游戏第一个行动的称为先手,第二个行动的称为后手。若在某一局面下无论采取何种行动,都会输掉游戏,则称该局面必败。 + +所谓采取最优策略是指,若在某一局面下存在某种行动,使得行动后对面面临必败局面,则优先采取该行动。同时,这样的局面被称为必胜。我们讨论的博弈问题一般都只考虑理想情况,即两人均无失误,都采取最优策略行动时游戏的结果。 + +NIM博弈不存在平局,只有先手必胜和先手必败两种情况。 + +定理: NIM博弈先手必胜,当且仅当 A1 ^ A2 ^ … ^ An != 0 + +--- + +##### 公平组合游戏ICG + +若一个游戏满足: + +1. 由两名玩家交替行动; + +2. 在游戏进程的任意时刻,可以执行的合法行动与轮到哪名玩家无关; + +3. 不能行动的玩家判负; + +则称该游戏为一个公平组合游戏。 + +NIM博弈属于公平组合游戏,但城建的棋类游戏,比如围棋,就不是公平组合游戏。因为围棋交战双方分别只能落黑子和白子,胜负判定也比较复杂,不满足条件2和条件3。 + +--- + +##### 有向图游戏 + +给定一个有向无环图,图中有一个唯一的起点,在起点上放有一枚棋子。两名玩家交替地把这枚棋子沿有向边进行移动,每次可以移动一步,无法移动者判负。该游戏被称为有向图游戏。 + +任何一个公平组合游戏都可以转化为有向图游戏。具体方法是,把每个局面看成图中的一个节点,并且从每个局面向沿着合法行动能够到达的下一个局面连有向边。 + +--- + +##### Mex运算 + +$设S表示一个非负整数集合。定义mex(S)为求出不属于集合S的最小非负整数的运算,即:$ + +$mex(S) = min{x}, x属于自然数,且x不属于S$ + +--- + +##### SG函数 + +$在有向图游戏中,对于每个节点x,设从x出发共有k条有向边,分别到达节点y_1, y_2, …, y_k,定$ + +$义SG(x)为x的后继节点y_1, y_2, …, y_k 的SG函数值构成的集合再执行mex(S)运算的结果,即:$ + +$SG(x) = mex({SG(y_1), SG(y_2), …, SG(y_k)})$ + +特别地,整个有向图游戏G的SG函数值被定义为有向图游戏起点s的SG函数值,即SG(G) = SG(s)。 + +--- + +##### 有向图游戏的和 —— 模板题 AcWing 893. 集合-Nim游戏 + +$设G_1, G_2, …, G_m 是m个有向图游戏。定义有向图游戏G,它的行动规则是任选某个有向图游戏$ + +$G_i,并在G_i上行动一步。G被称为有向图游戏G_1, G_2, …, G_m的和。$ + +$有向图游戏的和的SG函数值等于它包含的各个子游戏SG函数值的异或和,即:$ + +SG(G) = SG($G_1$) ^ SG($G_2$) ^ … ^ SG($G_m$) + +--- + +##### 定理 + +$有向图游戏的某个局面必胜,当且仅当该局面对应节点的SG函数值大于0。$ + +$有向图游戏的某个局面必败,当且仅当该局面对应节点的SG函数值等于0。$ + +--- + +> 作者:yxc +> +> 链接:https://www.acwing.com/blog/content/406/ +> +> 来源:AcWing +> +> 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。 \ No newline at end of file diff --git a/代码模板/数学知识.pdf b/代码模板/数学知识.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6ce04e28d5665f81c4b271aa292cc2bf7348195c GIT binary patch literal 331838 zcma&MQ;;UnwuM`^ZQFJi|FXK!W!pBpY}>YN+qP}nxA#6LZk+peGa^Thxe)o1v0~;o zC;4xH7(FusCk(mYzY7M2nT3dn$kxCDhK~=1QOw-R$=HF2QOru;$rxa4XlrB)!zg2H 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+```cpp +常见模型:找出滑动窗口中的最大值/最小值 +int hh = 0, tt = -1; +for (int i = 0; i < n; i ++ ) +{ + while (hh <= tt && check_out(q[hh])) hh ++ ; // 判断队头是否滑出窗口 + while (hh <= tt && check(q[tt], i)) tt -- ; + q[ ++ tt] = i; +} +``` + +----------------------- + +##### KMP —— 模板题 AcWing 831. KMP字符串 + +```cpp +// s[]是长文本,p[]是模式串,n是s的长度,m是p的长度 +求模式串的Next数组: +for (int i = 2, j = 0; i <= m; i ++ ) +{ + while (j && p[i] != p[j + 1]) j = ne[j]; + if (p[i] == p[j + 1]) j ++ ; + ne[i] = j; +} + +// 匹配 +for (int i = 1, j = 0; i <= n; i ++ ) +{ + while (j && s[i] != p[j + 1]) j = ne[j]; + if (s[i] == p[j + 1]) j ++ ; + if (j == m) + { + j = ne[j]; + // 匹配成功后的逻辑 + } +} +``` + +----------------- + +##### Trie树 —— 模板题 AcWing 835. Trie字符串统计 + +```cpp +int son[N][26], cnt[N], idx; +// 0号点既是根节点,又是空节点 +// son[][]存储树中每个节点的子节点 +// cnt[]存储以每个节点结尾的单词数量 + +// 插入一个字符串 +void insert(char *str) +{ + int p = 0; + for (int i = 0; str[i]; i ++ ) + { + int u = str[i] - 'a'; + if (!son[p][u]) son[p][u] = ++ idx; + p = son[p][u]; + } + cnt[p] ++ ; +} + +// 查询字符串出现的次数 +int query(char *str) +{ + int p = 0; + for (int i = 0; str[i]; i ++ ) + { + int u = str[i] - 'a'; + if (!son[p][u]) return 0; + p = son[p][u]; + } + return cnt[p]; +} +``` + +--------------------- + +##### 并查集 —— 模板题 AcWing 836. 合并集合, AcWing 837. 连通块中点的数量 + +###### (1) 朴素并查集: + +```cpp +int p[N]; //存储每个点的祖宗节点 + +// 返回x的祖宗节点 +int find(int x) +{ + if (p[x] != x) p[x] = find(p[x]); + return p[x]; +} + +// 初始化,假定节点编号是1~n +for (int i = 1; i <= n; i ++ ) p[i] = i; + +// 合并a和b所在的两个集合: +p[find(a)] = find(b); +``` + +###### (2) 维护size的并查集: + +```cpp +int p[N], size[N]; +//p[]存储每个点的祖宗节点, size[]只有祖宗节点的有意义,表示祖宗节点所在集合中的点的数量 + +// 返回x的祖宗节点 +int find(int x) +{ + if (p[x] != x) p[x] = find(p[x]); + return p[x]; +} + +// 初始化,假定节点编号是1~n +for (int i = 1; i <= n; i ++ ) +{ + p[i] = i; + size[i] = 1; +} + +// 合并a和b所在的两个集合: +size[find(b)] += size[find(a)]; +p[find(a)] = find(b); +``` + +###### (3) 维护到祖宗节点距离的并查集: + +```cpp +int p[N], d[N]; +//p[]存储每个点的祖宗节点, d[x]存储x到p[x]的距离 + +// 返回x的祖宗节点 +int find(int x) +{ + if (p[x] != x) + { + int u = find(p[x]); + d[x] += d[p[x]]; + p[x] = u; + } + return p[x]; +} + +// 初始化,假定节点编号是1~n +for (int i = 1; i <= n; i ++ ) +{ + p[i] = i; + d[i] = 0; +} + +// 合并a和b所在的两个集合: +p[find(a)] = find(b); +d[find(a)] = distance; // 根据具体问题,初始化find(a)的偏移量 +``` + +---------------- + +##### 堆 —— 模板题 AcWing 838. 堆排序, AcWing 839. 模拟堆 + +```cpp +// h[N]存储堆中的值, h[1]是堆顶,x的左儿子是2x, 右儿子是2x + 1 +// ph[k]存储第k个插入的点在堆中的位置 +// hp[k]存储堆中下标是k的点是第几个插入的 +int h[N], ph[N], hp[N], size; + +// 交换两个点,及其映射关系 +void heap_swap(int a, int b) +{ + swap(ph[hp[a]],ph[hp[b]]); + swap(hp[a], hp[b]); + swap(h[a], h[b]); +} + +void down(int u) +{ + int t = u; + if (u * 2 <= size && h[u * 2] < h[t]) t = u * 2; + if (u * 2 + 1 <= size && h[u * 2 + 1] < h[t]) t = u * 2 + 1; + if (u != t) + { + heap_swap(u, t); + down(t); + } +} + +void up(int u) +{ + while (u / 2 && h[u] < h[u / 2]) + { + heap_swap(u, u / 2); + u >>= 1; + } +} + +// O(n)建堆 +for (int i = n / 2; i; i -- ) down(i); +``` + +----------------------- + +##### 一般哈希 —— 模板题 AcWing 840. 模拟散列表 + +###### (1) 拉链法 + +```cpp +int h[N], e[N], ne[N], idx; + +// 向哈希表中插入一个数 +void insert(int x) +{ + int k = (x % N + N) % N; + e[idx] = x; + ne[idx] = h[k]; + h[k] = idx ++ ; +} + +// 在哈希表中查询某个数是否存在 +bool find(int x) +{ + int k = (x % N + N) % N; + for (int i = h[k]; i != -1; i = ne[i]) + if (e[i] == x) + return true; + + return false; +} +``` + +###### (2) 开放寻址法 + +```cpp +int h[N]; + +// 如果x在哈希表中,返回x的下标;如果x不在哈希表中,返回x应该插入的位置 +int find(int x) +{ + int t = (x % N + N) % N; + while (h[t] != null && h[t] != x) + { + t ++ ; + if (t == N) t = 0; + } + return t; +} +``` + +----------------- + +##### 字符串哈希 —— 模板题 AcWing 841. 字符串哈希 + +$核心思想:将字符串看成P进制数,P的经验值是131或13331,取这两个值的冲突概率低$ + +$小技巧:取模的数用2^{64},这样直接用unsigned\ long\ long存储,溢出的结果就是取模的结果$ + +```cpp +typedef unsigned long long ULL; +ULL h[N], p[N]; // h[k]存储字符串前k个字母的哈希值, p[k]存储 P^k mod 2^64 + +// 初始化 +p[0] = 1; +for (int i = 1; i <= n; i ++ ) +{ + h[i] = h[i - 1] * P + str[i]; + p[i] = p[i - 1] * P; +} + +// 计算子串 str[l ~ r] 的哈希值 +ULL get(int l, int r) +{ + return h[r] - h[l - 1] * p[r - l + 1]; +} +``` + +---------------------- + +##### C++ STL简介 + +```cpp +vector, 变长数组,倍增的思想 + size() 返回元素个数 + empty() 返回是否为空 + clear() 清空 + front()/back() + push_back()/pop_back() + begin()/end() + [] + 支持比较运算,按字典序 + +pair + first, 第一个元素 + second, 第二个元素 + 支持比较运算,以first为第一关键字,以second为第二关键字(字典序) + +string,字符串 + size()/length() 返回字符串长度 + empty() + clear() + substr(起始下标,(子串长度)) 返回子串 + c_str() 返回字符串所在字符数组的起始地址 + +queue, 队列 + size() + empty() + push() 向队尾插入一个元素 + front() 返回队头元素 + back() 返回队尾元素 + pop() 弹出队头元素 + +priority_queue, 优先队列,默认是大根堆 + size() + empty() + push() 插入一个元素 + top() 返回堆顶元素 + pop() 弹出堆顶元素 + 定义成小根堆的方式:priority_queue, greater> q; + +stack, 栈 + size() + empty() + push() 向栈顶插入一个元素 + top() 返回栈顶元素 + pop() 弹出栈顶元素 + +deque, 双端队列 + size() + empty() + clear() + front()/back() + push_back()/pop_back() + push_front()/pop_front() + begin()/end() + [] + +set, map, multiset, multimap, 基于平衡二叉树(红黑树),动态维护有序序列 + size() + empty() + clear() + begin()/end() + ++, -- 返回前驱和后继,时间复杂度 O(logn) + + set/multiset + insert() 插入一个数 + find() 查找一个数 + count() 返回某一个数的个数 + erase() + (1) 输入是一个数x,删除所有x O(k + logn) + (2) 输入一个迭代器,删除这个迭代器 + lower_bound()/upper_bound() + lower_bound(x) 返回大于等于x的最小的数的迭代器 + upper_bound(x) 返回大于x的最小的数的迭代器 + map/multimap + insert() 插入的数是一个pair + erase() 输入的参数是pair或者迭代器 + find() + [] 注意multimap不支持此操作。 时间复杂度是 O(logn) + lower_bound()/upper_bound() + +unordered_set, unordered_map, unordered_multiset, unordered_multimap, 哈希表 + 和上面类似,增删改查的时间复杂度是 O(1) + 不支持 lower_bound()/upper_bound(), 迭代器的++,-- + +bitset, 圧位 + bitset<10000> s; + ~, &, |, ^ + >>, << + ==, != + [] + + count() 返回有多少个1 + + any() 判断是否至少有一个1 + none() 判断是否全为0 + + set() 把所有位置成1 + set(k, v) 将第k位变成v + reset() 把所有位变成0 + flip() 等价于~ + flip(k) 把第k位取反 +``` + +------------------- + +> 作者:yxc +> +> 链接:https://www.acwing.com/blog/content/404/ +> +> 来源:AcWing +> +> 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。 \ No newline at end of file diff --git a/代码模板/数据结构.pdf b/代码模板/数据结构.pdf new file mode 100644 index 0000000000000000000000000000000000000000..18e922f16343d8b15ad37701652ecab92fdc9971 GIT binary patch literal 191999 zcma&NV~{98mo3`1ZQHhO+qUh~wtd>RZQHhOyH9t&{@%QJH}1szn2Jd4UCF4bti4ul zQUwt)T1Gl{DAIu63kr&nk${1~-pC4yhX;yY%+kii)QNyz%*N2gRK(QS-ozA&UdGhU z+{J=`nS+G^ijNP<*~Q7!&=$%g>l)WO?x^MZ^Bd|9or3r+I0DMr`sVa$3-}Xwk!|Eb zjljYWd~&jF!b`MURRRyU#Oq7LX@WS4Xu^Ro5i=F;Vye_!XT~p~ z#tA{t7O9@|d+0sW$DJRaRmTp{8U#8F^inuRI+8;vMZ&^}i1w)j*WY36?Z~0481(^71~N>IoTcEt z-3}pajtn6b;zqL$zMjud>|^s`q5!dK0McN=Fwph5ZcxP7ZirmcwzN~Igs+iBkCf)t zyT|KoFr;&Mr314JL0r|=_Y(|;^;;sa7AKFg`79uHUYvJh%A|jI{3}_d6AQD2z<{`# 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