随机化选择标定点
为了解决对有序数组排序，快速排序法复杂度变成O(n)的问题，在partition()方法中随机选择一个元素作为标定点
import java.util.Arrays;
import java.util.Random;
public class Algorithm {
public static void main(String[] args) {
Integer[] testScale = {10000, 100000};
for (Integer n : testScale) {
Integer[] randomArr = ArrayGenerator.generatorRandomArray(n, n);
Integer[] sortedArr = ArrayGenerator.generatorSortedArray(n, n);
Integer[] arr1 = Arrays.copyOf(randomArr, randomArr.length);
Integer[] arr3 = Arrays.copyOf(randomArr, randomArr.length);
Integer[] arr2 = Arrays.copyOf(sortedArr, sortedArr.length);
Integer[] arr4 = Arrays.copyOf(sortedArr, sortedArr.length);
System.out.println("测试随机数组排序性能");
System.out.println();
Verify.testTime("MergeSort", arr1);
Verify.testTime("QuickSortOptimized", arr3);
System.out.println();
System.out.println("测试有序数组排序性能");
System.out.println();
Verify.testTime("MergeSort", arr2);
Verify.testTime("QuickSortOptimized", arr4);
System.out.println();
}
}
}
class QuickSort {
private QuickSort(){}
public static<E extends Comparable<E>> void sortOptimized(E[] arr){
/**
* 将重复使用的temp和random变量作为参数传递，进行内存优化
*/
Random random = new Random();
E temp = null;
sortOptimized(arr, 0, arr.length - 1, random, temp);
}
public static<E extends Comparable<E>> void sortOptimized(E[] arr, int left, int right, Random random, E temp){
if (left >= right){
return;
}
int p = partition(arr, left, right, random, temp);
sortOptimized(arr, left, p - 1, random, temp);
sortOptimized(arr, p + 1, right, random, temp);
}
public static <E extends Comparable<E>> int partition(E[] arr, int left, int right, Random random, E temp){
/**
* 生成[left, right]之间的随机索引
* random.nextInt()方法只能生成[0, bound)的整数，因此还要加上偏移left
*/
int p = random.nextInt(right - left + 1) + left;
temp = arr[p];
arr[p] = arr[left];
arr[left] = temp;
int j = left;
for (int i = left + 1; i <= right; i++) {
if (arr[i].compareTo(arr[left]) < 0){
j++;
temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
temp = arr[j];
arr[j] = arr[left];
arr[left] = temp;
return j;
}
}
class MergeSort {
private MergeSort(){}
public static<E extends Comparable<E>> void sort(E[] arr){
E[] temp = Arrays.copyOf(arr, arr.length);
sort(arr, 0, arr.length - 1, temp);
}
private static<E extends Comparable<E>> void sort(E[] arr, int left, int right, E[] temp){
if (left >= right){
return;
}
int mid = left + (right - left) / 2;
sort(arr, left, mid, temp);
sort(arr, mid + 1, right, temp);
if (arr[mid].compareTo(arr[mid + 1]) > 0) {
merge(arr, left, mid, right, temp);
}
}
public static<E extends Comparable<E>> void merge(E[] arr, int left, int mid, int right, E[] temp) {
int i = left;
int j = mid + 1;
System.arraycopy(arr, left, temp, left, right - left + 1);
for (int n = left; n < right + 1; n++) {
if (i == mid + 1){
arr[n] = temp[j];
j++;
}
else if (j == right + 1) {
arr[n] = temp[i];
i++;
}
else if (temp[i].compareTo(temp[j]) <= 0) {
arr[n] = temp[i];
i++;
}
else{
arr[n] = temp[j];
j++;
}
}
}
}
class ArrayGenerator {
private ArrayGenerator (){}
public static Integer[] generatorRandomArray (Integer n, Integer maxBound){
Integer[] arr = new Integer[n];
Random random = new Random();
for (int i = 0; i < n; i++) {
arr[i] = random.nextInt(maxBound);
}
return arr;
}
public static Integer[] generatorSortedArray (Integer n, Integer maxBound){
Integer[] arr = new Integer[n];
for (int i = 0; i < n; i++) {
arr[i] = i;
}
return arr;
}
}
class Verify {
private Verify (){}
public static<E extends Comparable<E>> boolean isSorted(E[] arr){
for (int i = 0; i < arr.length - 1; i++) {
if (arr[i].compareTo(arr[i + 1]) > 0) {
return false;
}
}
return true;
}
public static<E extends Comparable<E>> void testTime(String AlgorithmName, E[] arr) {
long startTime = System.nanoTime();
if (AlgorithmName.equals("MergeSort")) {
MergeSort.sort(arr);
}
if (AlgorithmName.equals("QuickSortOptimized")) {
QuickSort.sortOptimized(arr);
}
long endTime = System.nanoTime();
if (!Verify.isSorted(arr)){
throw new RuntimeException(AlgorithmName + "算法排序失败！");
}
System.out.println(String.format("%s算法，测试用例为%d，执行时间：%f秒", AlgorithmName, arr.length, (endTime - startTime) / 1000000000.0));
}
}
练习：
对于总是将中间的元素作为标定点的快速排序，编写一个算例，生成一个中间元素总是最小的数组，即让这个快速排序法“失效”
import java.util.Arrays;
import java.util.Random;
public class Algorithm {
public static void main(String[] args) {
Integer[] testScale = {10000, 100000};
for (Integer n : testScale) {
Integer[] randomArr = ArrayGenerator.generatorRandomArray(n, n);
Integer[] sortedArr = ArrayGenerator.generatorSortedArray(n, n);
Integer[] specialArr = ArrayGenerator.generateSpecialArray(n);
Integer[] arr1 = Arrays.copyOf(randomArr, randomArr.length);
Integer[] arr3 = Arrays.copyOf(randomArr, randomArr.length);
Integer[] arr2 = Arrays.copyOf(sortedArr, sortedArr.length);
Integer[] arr4 = Arrays.copyOf(sortedArr, sortedArr.length);
Integer[] arr5 = Arrays.copyOf(specialArr, specialArr.length);
Integer[] arr6 = Arrays.copyOf(specialArr, specialArr.length);
System.out.println("测试随机数组排序性能");
System.out.println();
Verify.testTime("MergeSort", arr1);
Verify.testTime("QuickSortOptimized", arr3);
System.out.println();
System.out.println("测试有序数组排序性能");
System.out.println();
Verify.testTime("MergeSort", arr2);
Verify.testTime("QuickSortOptimized", arr4);
System.out.println();
System.out.println("测试特殊数组排序性能");
System.out.println();
Verify.testTime("MergeSort", arr5);
Verify.testTime("QuickSortOptimized", arr6);
}
}
}
class QuickSort {
private QuickSort(){}
public static<E extends Comparable<E>> void sortOptimized(E[] arr){
E temp = null;
sortOptimized(arr, 0, arr.length - 1, temp);
}
public static<E extends Comparable<E>> void sortOptimized(E[] arr, int left, int right, E temp){
if (left >= right){
return;
}
int p = partition(arr, left, right, temp);
sortOptimized(arr, left, p - 1, temp);
sortOptimized(arr, p + 1, right, temp);
}
public static <E extends Comparable<E>> int partition(E[] arr, int left, int right, E temp){
int mid = left + (right - left) / 2;
temp = arr[mid];
arr[mid] = arr[left];
arr[left] = temp;
int j = left;
for (int i = left + 1; i <= right; i++) {
if (arr[i].compareTo(arr[left]) < 0){
j++;
temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
temp = arr[j];
arr[j] = arr[left];
arr[left] = temp;
return j;
}
}
class MergeSort {
private MergeSort(){}
public static<E extends Comparable<E>> void sort(E[] arr){
E[] temp = Arrays.copyOf(arr, arr.length);
sort(arr, 0, arr.length - 1, temp);
}
private static<E extends Comparable<E>> void sort(E[] arr, int left, int right, E[] temp){
if (left >= right){
return;
}
int mid = left + (right - left) / 2;
sort(arr, left, mid, temp);
sort(arr, mid + 1, right, temp);
if (arr[mid].compareTo(arr[mid + 1]) > 0) {
merge(arr, left, mid, right, temp);
}
}
public static<E extends Comparable<E>> void merge(E[] arr, int left, int mid, int right, E[] temp) {
int i = left;
int j = mid + 1;
System.arraycopy(arr, left, temp, left, right - left + 1);
for (int n = left; n < right + 1; n++) {
if (i == mid + 1){
arr[n] = temp[j];
j++;
}
else if (j == right + 1) {
arr[n] = temp[i];
i++;
}
else if (temp[i].compareTo(temp[j]) <= 0) {
arr[n] = temp[i];
i++;
}
else{
arr[n] = temp[j];
j++;
}
}
}
}
class ArrayGenerator {
private ArrayGenerator() {
}
public static Integer[] generatorRandomArray(Integer n, Integer maxBound) {
Integer[] arr = new Integer[n];
Random random = new Random();
for (int i = 0; i < n; i++) {
arr[i] = random.nextInt(maxBound);
}
return arr;
}
public static Integer[] generatorSortedArray(Integer n, Integer maxBound) {
Integer[] arr = new Integer[n];
for (int i = 0; i < n; i++) {
arr[i] = i;
}
return arr;
}
public static Integer[] generateSpecialArray(int n) {
Integer[] arr = new Integer[n];
generateSpecialArray(arr, 0, n - 1, 0);
return arr;
}
public static void generateSpecialArray(Integer[] arr, int left, int right, int value) {
if (left >= right) {
return;
}
int mid = left + (right - left) / 2;
/**
* 总是将最小值赋值给mid，这样在每个子区间，mid都是最小的元素
*/
arr[mid] = value;
generateSpecialArray(arr, left, mid, value + 1);
generateSpecialArray(arr, mid + 1, right, value + 1);
/**
* 因为mid是左区间最后一个元素，因此左区间的元素都会被赋值，可是右区间最后一个元素可能不会被递归到
* 为了防止最后一个元素没有被赋值，所以要额外判断一下
*/
if (arr[right] == null){
arr[right] = value + 2;
}
}
}
class Verify {
private Verify (){}
public static<E extends Comparable<E>> boolean isSorted(E[] arr){
for (int i = 0; i < arr.length - 1; i++) {
if (arr[i].compareTo(arr[i + 1]) > 0) {
return false;
}
}
return true;
}
public static<E extends Comparable<E>> void testTime(String AlgorithmName, E[] arr) {
long startTime = System.nanoTime();
if (AlgorithmName.equals("MergeSort")) {
MergeSort.sort(arr);
}
if (AlgorithmName.equals("QuickSortOptimized")) {
QuickSort.sortOptimized(arr);
}
long endTime = System.nanoTime();
if (!Verify.isSorted(arr)){
throw new RuntimeException(AlgorithmName + "算法排序失败！");
}
System.out.println(String.format("%s算法，测试用例为%d，执行时间：%f秒", AlgorithmName, arr.length, (endTime - startTime) / 1000000000.0));
}
}
普通快速排序法还有问题
在目前的partition()方法，数组的元素越无序，快速排序法的性能越好。虽然随机化选择标定点对有序的数组也有了性能的提升
但还有一个问题：如果数组的所有元素都一样，那就等价于是完全有序数组，而且随机化标定点的策略也会失效，此时快速排序法的复杂度仍会降到O(n^2)
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