前缀树
- 单个字符串中,字符从前到后的加到一颗多叉树上
- 字符放在路上,节点上有专属的数据项(常见的是pass和end值)
- 所有样本都这样添加,如果没有路就新建,如有路就复用
- 沿途节点的pass值增加1,每个字符串结束时来到的节点end值增加1
可以完成前缀相关的查询
package com.harrison.five;
import java.util.HashMap;
public class Code02_TrieTree {
// 字符串是所有的小写字母
public static class Node1 {
public int pass;
public int end;
public Node1[] nexts;
// 0 a
// 1 b
// ...
// 25 z
public Node1() {
pass = 0;
end = 0;
nexts = new Node1[26];
}
}
public static class Trie1 {
public Node1 root;
public Trie1() {
root = new Node1();
}
public void insert(String word) {
if (word == null) {
return;
}
char[] str = word.toCharArray();
Node1 node = root;
node.pass++;
int path = 0;
for (int i = 0; i < str.length; i++) {
// 从左往右遍历字符
path = str[i] - 'a';// 由字符,对应成走向哪条路
if (node.nexts[path] == null) {
// 无节点新建,有节点复用
node.nexts[path] = new Node1();
}
node = node.nexts[path];
node.pass++;
}
node.end++;
}
// word这个单词之前加入过几次
public int search(String word) {
if (word == null) {
return 0;
}
char[] str = word.toCharArray();
Node1 node = root;
int path = 0;
for (int i = 0; i < str.length; i++) {
path = str[i] - 'a';
if (node.nexts[path] == null) {
return 0;
}
node = node.nexts[path];
}
return node.end;
}
public void delete(String word) {
if (search(word) != 0) {
char[] str = word.toCharArray();
Node1 node = root;
int path = 0;
for (int i = 0; i < str.length; i++) {
path = str[i] - 'a';
if (--node.nexts[path].pass == 0) {
node.nexts[path] = null;
return;
}
node = node.nexts[path];
}
node.end--;
}
}
// 所有加入的字符串中,有几个是以pre这个字符串作为前缀的
public int prefixNumber(String word) {
if (word == null) {
return 0;
}
char[] str = word.toCharArray();
Node1 node = root;
int path = 0;
for (int i = 0; i < str.length; i++) {
path = str[i] - 'a';
if (node.nexts[path] == null) {
return 0;
}
node = node.nexts[path];
}
return node.pass;
}
}
public static class Node2 {
public int pass;
public int end;
public HashMap<Integer, Node2> nexts;
public Node2() {
pass = 0;
end = 0;
nexts = new HashMap<>();
}
}
public static class Trie2 {
public Node2 root;
public Trie2() {
root = new Node2();
}
public void insert(String word) {
if (word == null) {
return;
}
char[] str = word.toCharArray();
Node2 node = root;
node.pass++;
int path = 0;
for (int i = 0; i < str.length; i++) {
// 从左往右遍历字符
path = (int) str[i];
if (!node.nexts.containsKey(path)) {
// 无节点新建
node.nexts.put(path, new Node2());
}
// 有节点复用
node = node.nexts.get(path);
node.pass++;
}
node.end++;
}
// word这个单词之前加入过几次
public int search(String word) {
if (word == null) {
return 0;
}
char[] str = word.toCharArray();
Node2 node = root;
int path = 0;
for (int i = 0; i < str.length; i++) {
path = (int) str[i];
if (!node.nexts.containsKey(path)) {
return 0;
}
node = node.nexts.get(path);
}
return node.end;
}
public void delete(String word) {
if (search(word) != 0) {
char[] str = word.toCharArray();
Node2 node = root;
int path = 0;
for (int i = 0; i < str.length; i++) {
path = (int) str[i];
if (--node.nexts.get(path).pass == 0) {
node.nexts.remove(path);
return;
}
node = node.nexts.get(path);
}
node.end--;
}
}
// 所有加入的字符串中,有几个是以pre这个字符串作为前缀的
public int prefixNumber(String word) {
if (word == null) {
return 0;
}
char[] str = word.toCharArray();
Node2 node = root;
int path = 0;
for (int i = 0; i < str.length; i++) {
path = (int) str[i];
if (!node.nexts.containsKey(path)) {
return 0;
}
node = node.nexts.get(path);
}
return node.pass;
}
}
// 自己写的很暴力的方法,完全不用前缀树结构
public static class Right {
private HashMap<String, Integer> box;
public Right() {
box = new HashMap<>();
}
public void insert(String word) {
if (!box.containsKey(word)) {
box.put(word, 1);
} else {
box.put(word, box.get(word) + 1);
}
}
public void delete(String word) {
if (box.containsKey(word)) {
if (box.get(word) == 1) {
box.remove(word);
} else {
box.put(word, box.get(word) - 1);
}
}
}
public int search(String word) {
if (!box.containsKey(word)) {
return 0;
} else {
return box.get(word);
}
}
public int prefixNumber(String pre) {
int count = 0;
for (String cur : box.keySet()) {
if (cur.startsWith(pre)) {
count += box.get(cur);
}
}
return count;
}
}
public static String generateRandomString(int strLen) {
char[] ans = new char[(int) (Math.random() * (strLen) + 1)];
for (int i = 0; i < ans.length; i++) {
int value = (int) (Math.random() * 6);
ans[i] = (char) (97 + value);
}
return String.valueOf(ans);
}
public static String[] generateRandomStringArray(int arrLen, int strLen) {
String[] ans = new String[(int) (Math.random() * arrLen) + 1];
for (int i = 0; i < ans.length; i++) {
ans[i] = generateRandomString(strLen);
}
return ans;
}
public static void main(String[] args) {
int arrLen = 100;
int strLen = 20;
int testTimes = 100000;
for (int i = 0; i < testTimes; i++) {
String[] arr = generateRandomStringArray(arrLen, strLen);
Trie1 trie1 = new Trie1();
Trie2 trie2 = new Trie2();
Right right = new Right();
for (int j = 0; j < arr.length; j++) {
double decide = Math.random();
if (decide < 0.25) {
trie1.insert(arr[j]);
trie2.insert(arr[j]);
right.insert(arr[j]);
} else if (decide < 0.5) {
trie1.delete(arr[j]);
trie2.delete(arr[j]);
right.delete(arr[j]);
} else if (decide < 0.75) {
int ans1 = trie1.search(arr[j]);
int ans2 = trie2.search(arr[j]);
int ans3 = right.search(arr[j]);
if (ans1 != ans2 || ans2 != ans3) {
System.out.println("Oops!");
}
} else {
int ans1 = trie1.prefixNumber(arr[j]);
int ans2 = trie2.prefixNumber(arr[j]);
int ans3 = right.prefixNumber(arr[j]);
if (ans1 != ans2 || ans2 != ans3) {
System.out.println("Oops!");
}
}
}
}
System.out.println("finish!");
}
}
桶排序
桶排序是一个大思想,是不基于比较的排序,计数排序和基数排序是其中的体现。
不基于比较的排序
桶排序思想下的排序:计数排序&基数排序
1)桶排序思想下的排序都是不基于比较的排序
2)时间复杂度为O(N),额外空间复杂度为O(M)
3)应用范围有限,需要样本的数据状况满足桶的划分
弱点(极大):必须和样本的数据状况强相关,所以桶排序下的排序都对数据状况有要求
- 一般来讲,计数排序要求,样本是整数,且范围比较窄
- 一般来讲,基数排序要求,样本是10进制的正整数
- 一旦要求稍有升级,改写代价增加是显而易见的
计数排序
package com.harrison.five;
import java.util.Arrays;
public class Code03_CountSort {
public static void countSort(int[] arr) {
if (arr == null || arr.length < 2) {
return;
}
int max = Integer.MIN_VALUE;
for (int i = 0; i < arr.length; i++) {
max = Math.max(max, arr[i]);
}
int[] bucket = new int[max + 1];
for (int i = 0; i < arr.length; i++) {
bucket[arr[i]]++;
}
int i = 0;
for (int j = 0; j < bucket.length; j++) {
while (bucket[j]-- > 0) {
arr[i++] = j;
}
}
}
public static void comparator(int[] arr) {
Arrays.sort(arr);
}
// for test
public static int[] generateRandomArray(int maxSize, int maxValue) {
int[] arr = new int[(int) ((maxSize + 1) * Math.random())];
for (int i = 0; i < arr.length; i++) {
arr[i] = (int) ((maxValue + 1) * Math.random());
}
return arr;
}
// for test
public static int[] copyArray(int[] arr) {
if (arr == null) {
return null;
}
int[] res = new int[arr.length];
for (int i = 0; i < arr.length; i++) {
res[i] = arr[i];
}
return res;
}
// for test
public static boolean isEqual(int[] arr1, int[] arr2) {
if ((arr1 == null && arr2 != null) || (arr1 != null && arr2 == null)) {
return false;
}
if (arr1 == null && arr2 == null) {
return true;
}
if (arr1.length != arr2.length) {
return false;
}
for (int i = 0; i < arr1.length; i++) {
if (arr1[i] != arr2[i]) {
return false;
}
}
return true;
}
// for test
public static void printArray(int[] arr) {
if (arr == null) {
return;
}
for (int i = 0; i < arr.length; i++) {
System.out.print(arr[i] + " ");
}
System.out.println();
}
// for test
public static void main(String[] args) {
int testTime = 500000;
int maxSize = 100;
int maxValue = 150;
boolean succeed = true;
for (int i = 0; i < testTime; i++) {
int[] arr1 = generateRandomArray(maxSize, maxValue);
int[] arr2 = copyArray(arr1);
countSort(arr1);
comparator(arr2);
if (!isEqual(arr1, arr2)) {
succeed = false;
printArray(arr1);
printArray(arr2);
break;
}
}
System.out.println(succeed ? "Nice!" : "Fucking fucked!");
int[] arr = generateRandomArray(maxSize, maxValue);
printArray(arr);
countSort(arr);
printArray(arr);
}
}
基数排序
package com.harrison.five;
import java.util.Arrays;
public class Code04_RadixSort {
// only for no-negative value
public static void radixSort(int[] arr) {
if (arr == null || arr.length < 2) {
return;
}
radixSort(arr, 0, arr.length - 1, maxbits(arr));
}
public static int maxbits(int[] arr) {
int max = Integer.MIN_VALUE;
for (int i = 0; i < arr.length; i++) {
max = Math.max(max, arr[i]);
}
int res = 0;
while (max != 0) {
res++;
max /= 10;
}
return res;
}
// arr[L...R]排序,最大值的十进制位数digit
public static void radixSort(int[] arr, int L, int R, int digit) {
final int radix = 10;
int i = 0, j = 0;
// 有多少个数,准备多少个辅助空间
int[] help = new int[R - L + 1];
for (int d = 1; d <= digit; d++) {
// 有多少位就进出多少次
// 10个空间
// count[0] 当前位(d位)是0的数字有多少个
// count[1] 当前位(d位)是(0和1)的数字有多少个
// count[2] 当前位(d位)是(0、1和2)的数字有多少个
// count[i] 当前位(d位)是(0~i)的数字有多少个
int[] count = new int[radix];// count[0...9]
for (i = L; i <= R; i++) {
// 103 1 3
// 209 1 9
j = getDigit(arr[i], d);
count[j]++;
}
for (i = 1; i < radix; i++) {
count[i] = count[i] + count[i - 1];
}
for (i = R; i >= L; i--) {
j = getDigit(arr[i], d);
help[count[j] - 1] = arr[i];
count[j]--;
}
for (i = L, j = 0; i <= R; i++, j++) {
arr[i] = help[i];
}
}
}
public static int getDigit(int x, int d) {
return ((x / ((int) Math.pow(10, d - 1))) % 10);
}
public static void comparator(int[] arr) {
Arrays.sort(arr);
}
// for test
public static int[] generateRandomArray(int maxSize, int maxValue) {
int[] arr = new int[(int) ((maxSize + 1) * Math.random())];
for (int i = 0; i < arr.length; i++) {
arr[i] = (int) ((maxValue + 1) * Math.random());
}
return arr;
}
// for test
public static int[] copyArray(int[] arr) {
if (arr == null) {
return null;
}
int[] res = new int[arr.length];
for (int i = 0; i < arr.length; i++) {
res[i] = arr[i];
}
return res;
}
// for test
public static boolean isEqual(int[] arr1, int[] arr2) {
if ((arr1 == null && arr2 != null) || (arr1 != null && arr2 == null)) {
return false;
}
if (arr1 == null && arr2 == null) {
return true;
}
if (arr1.length != arr2.length) {
return false;
}
for (int i = 0; i < arr1.length; i++) {
if (arr1[i] != arr2[i]) {
return false;
}
}
return true;
}
// for test
public static void printArray(int[] arr) {
if (arr == null) {
return;
}
for (int i = 0; i < arr.length; i++) {
System.out.print(arr[i] + " ");
}
System.out.println();
}
// for test
public static void main(String[] args) {
int testTime = 500000;
int maxSize = 100;
int maxValue = 100000;
boolean succeed = true;
for (int i = 0; i < testTime; i++) {
int[] arr1 = generateRandomArray(maxSize, maxValue);
int[] arr2 = copyArray(arr1);
radixSort(arr1);
comparator(arr2);
if (!isEqual(arr1, arr2)) {
succeed = false;
printArray(arr1);
printArray(arr2);
break;
}
}
System.out.println(succeed ? "Nice!" : "Fucking fucked!");
int[] arr = generateRandomArray(maxSize, maxValue);
printArray(arr);
radixSort(arr);
printArray(arr);
}
}
排序总结
时间复杂度 | 额外空间复杂度 | 稳定性 | |
---|---|---|---|
选择排序 | O(N^2) | O(1) | 无 |
冒泡排序 | O(N^2) | O(1) | 有 |
插入排序 | O(N^2) | O(1) | 有 |
归并排序 | O(N*logN) | O(N) | 有 |
随机快排 | O(N*logN) | O(logN) | 无 |
堆排序 | O(N*logN) | O(1) | 无 |
计数排序 | O(N) | O(M) | 有 |
基数排序 | O(N) | O(N) | 有 |
- 不基于比较的排序,对样本数据有严格要求,不易改写
- 基于比较的排序,只要规定好两个样本怎么比大小就可以直接复用
- 基于比较的排序,时间复杂度的极限是O(N*logN)
- 时间复杂度O(N*logN)、额外空间复杂度低于O(N)、且稳定的基于比较的排序是不存在的
- 为了绝对的速度选快排,为了节省空间选堆排,为了稳定性选归并
常见的坑
- 归并排序的额外空间复杂度可以变成O(1),“归并排序 内部缓存法”,但是将变得不在稳定——直接用堆
- “原地归并排序”是垃圾贴,会让时间复杂度变成O(N^2)——插排
- 快速排序稳定性改进,“01 stable sort”,但是会对样本数据要求更多——桶排
- 在整型数组中,请把奇数放在数组左边,偶数放在数组右边,要求所有奇数之间原始的相对次序不变,所有偶数之间的相对次序不变。时间复杂度做到O(N),额外空间复杂度做到O(1)——快排不稳定