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字符串

KMP字符串

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致敬三位前辈：D.E.Knuth、J.H.Morris 和 V.R.Pratt！

字符串模式匹配算法，大大避免重复遍历的情况，克努特-莫里斯-普拉特算法。

• 例题：AcWing 831. KMP字符串

import java.io.*;
import java.util.*;public class Main {static final int N = 100010;static final int M = 1000010;static char[] p = new char[N];static char[] s = new char[M];static int[] ne = new int[N];public static void main(String[] args) throws IOException {BufferedReader br = new BufferedReader(new InputStreamReader(System.in));BufferedWriter bw = new BufferedWriter(new OutputStreamWriter(System.out));int n = Integer.parseInt(br.readLine());String pStr = " " + br.readLine();int m = Integer.parseInt(br.readLine());String sStr = " " + br.readLine();p = pStr.toCharArray();s = sStr.toCharArray();// 计算 next 数组for (int i = 2, j = 0; i <= n; i++) {while (j != 0 && p[i] != p[j + 1])j = ne[j];if (p[i] == p[j + 1])j++;ne[i] = j;}// KMP 匹配for (int i = 1, j = 0; i <= m; i++) {while (j != 0 && s[i] != p[j + 1])j = ne[j];if (s[i] == p[j + 1])j++;if (j == n) {bw.write((i - n) + " ");j = ne[j];}}br.close();bw.close();}
}

Trie树

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用于高效地存储和查找字符串集合的数据结构。

• 例题：AcWing 835. Trie字符串统计

import java.io.*;
import java.util.*;public class Main {static final int N = 100010;static int[][] son = new int[N][26]; // 存储所有节点的儿子是什么static int[] cnt = new int[N]; // 字符串结束标记且进行计数static int idx; // 下标为 0 的点，既是跟节点，又是空节点public static void main(String[] args) throws IOException {BufferedReader br = new BufferedReader(new InputStreamReader(System.in));BufferedWriter bw = new BufferedWriter(new OutputStreamWriter(System.out));int n = Integer.parseInt(br.readLine());while (n-- != 0) {String[] lines = br.readLine().split(" ");char op = lines[0].charAt(0);String str = lines[1];switch (op) {case 'I': {insert(str.toCharArray());break;}case 'Q': {bw.write(query(str.toCharArray()) + "");bw.newLine();}}}br.close();bw.close();}// 插入字符串public static void insert(char[] str) {int p = 0;for (int i = 0; i < str.length; i++) {int u = str[i] - 'a';if (son[p][u] == 0)son[p][u] = ++idx;p = son[p][u];}cnt[p]++;}// 查询字符串出现的次数public static int query(char[] str) {int p = 0;for (int i = 0; i < str.length; i++) {int u = str[i] - 'a';if (son[p][u] == 0)return 0;p = son[p][u];}return cnt[p];}
}

并查集

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两个操作：

1. 将两个集合合并。
2. 询问两个元素是否在一个集合当中。

基本原理：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。每个节点存储
它的父节点，P[x] 表示 x 的父节点。

问题 1：如何判断树根 if (p[x] == ×) {return true;}
问题 2：如何求 x 的集合编号 while (p[x] != x) x = p[x];
问题 3：如何合并两个集合，p[x] 是 x 的集合编号，p[y] 是 y 的集合编号 p[x] = y;

• 例题：AcWing 836. 合并集合

import java.util.Scanner;public class Main {static int[] p;public static void main(String[] args) {Scanner scanner = new Scanner(System.in);int n = scanner.nextInt();p = new int[n + 10];for (int i = 1; i <= n; i++)p[i] = i;int m = scanner.nextInt();while (m-- != 0) {char op = scanner.next().charAt(0);int a = scanner.nextInt();int b = scanner.nextInt();switch (op) {case 'M': {p[find(a)] = find(b);break;}case 'Q': {System.out.println((find(a) == find(b)) ? "Yes" : "No");break;}}}}// 返回祖宗节点 + 路径压缩public static int find(int x) {if (x != p[x])p[x] = find(p[x]);return p[x];}
}

堆

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手写堆，功能：

1. 插入一个数：heap[++size] = x; up(size);
2. 求集合中的最小值：heap[1];
3. 删除最小值：heap[1] = heap[size]; size--; down(1);
4. 删除任意一个元素：heap[1] = heap[k]; size--; down(1); up(1);
5. 修改任意一个元素：heap[k] = x; down(k); up(k);

使用一维数组存储：

• x 左儿子：2x
• x 右儿子：2x + 1

• 例题：AcWing 839. 模拟堆

import java.util.Scanner;public class Main {static final int N = 100010;static int[] h;static int[] ph;static int[] hp;static int size;public static void main(String[] args) {Scanner scanner = new Scanner(System.in);int n = scanner.nextInt();h = new int[n + 10];ph = new int[n + 10];hp = new int[n + 10];int m = 0;while (n-- != 0) {String op = scanner.next();switch (op) {case "I": {int x = scanner.nextInt();size++;m++;ph[m] = size;hp[size] = m;h[size] = x;up(size);break;}case "PM": {System.out.println(h[1]);break;}case "DM": {heapSwap(1, size);size--;down(1);break;}case "D": {int k = scanner.nextInt();k = ph[k];heapSwap(k, size);size--;down(k);up(k);break;}case "C": {int k = scanner.nextInt();int x = scanner.nextInt();k = ph[k];h[k] = x;down(k);up(k);break;}}}}public static void heapSwap(int a, int b) {int temp = ph[hp[a]];ph[hp[a]] = ph[hp[b]];ph[hp[b]] = temp;temp = hp[a];hp[a] = hp[b];hp[b] = temp;temp = h[a];h[a] = h[b];h[b] = temp;}public static void up(int u) {while (u / 2 != 0 && h[u / 2] > h[u]) {heapSwap(u / 2, u);u /= 2;}}public static void down(int u) {int t = u; // t 表示三个节点中最小节点的编号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) {heapSwap(u, t);down(t);}}
}