简单算法学习笔记
[数据结构][算法][面向对象][Java]
1 设计接口
1.1 容器接口Container
package ds;
import java.util.NoSuchElementException;
public interface Container{
int size();
default boolean isEmpty(){
return size() == 0;
}
default void notEmptyCheck() {
if (isEmpty()) {
throw new NoSuchElementException("size == 0");
}
}
} 1.2 背包接口Bag
package ds;
public interface Bag<T> extends Container {
void add(T data);
}
1.3 栈接口Stack
package ds;
public interface Stack<T>extends Container {
void push(T data);
T pop();
T peek();
} 1.4 队列接口Queue
package ds;
public interface Queue<T> extends Container {
T peek();
void enqueue(T data);
T dequeue();
} 1.5 Union-Find算法接口UF
package ds;
public interface UF {
int size();
int find(int p);
boolean connected(int p, int q);
void union(int p, int q);
default void validate(int p, int n) {
if (p < 0 || p >= n) {
throw new IllegalArgumentException("p>=0 && p<n");
}
}
} 2 实现接口
为数组和链表定义了公共的结点和迭代器实现。
2.1 结点类Node
package ds.util;
public class Node<T> {
public T data;
public Node<T> next;
} 2.2 数组迭代器ArrayIterator
package ds.util;
import java.util.Iterator;
import java.util.NoSuchElementException;
public class ArrayIterator<T> implements Iterator<T> {
private int i;
private final int size;
private final T[] array;
public ArrayIterator(T[] array, int size) {
this.array = array;
this.size = size;
}
@Override
public boolean hasNext() {
return i < size;
}
@Override
public T next() {
if (!hasNext()) {
throw new NoSuchElementException();
}
return array[i++];
}
} 2.3 链表迭代器ListIterator
package ds.util;
import java.util.Iterator;
import java.util.NoSuchElementException;
public class ListIterator<T> implements Iterator<T> {
private Node<T> current;
public ListIterator(Node<T> first) {
current = first;
}
@Override
public boolean hasNext() {
return current != null;
}
@Override
public T next() {
if (!hasNext()) {
throw new NoSuchElementException();
}
T result = current.data;
current = current.next;
return result;
}
}
2.4 背包(Bag)的实现
背包(Bag)是一种不支持从中删除元素的集合数据类型。
2.4.1 能动态调整数组大小的Bag
package ds.impl;
import ds.Bag;
import java.util.Iterator;
import ds.util.ArrayIterator;
public class ArrayBag<T> implements Iterable<T>,Bag<T> {
private T[] array;
private int size;
public ArrayBag() {
array = (T[]) new Object[2];
size = 0;
}
@Override
public int size() {
return size;
}
@Override
public void add(T data) {
if (size == array.length) {
resize(2 * array.length);
}
array[size++] = data;
}
private void resize(int capacity) {
assert capacity >= 0;
T[] tmp = (T[]) new Object[capacity];
System.arraycopy(array, 0, tmp, 0, size);
array = tmp;
}
@Override
public Iterator<T> iterator() {
return new ArrayIterator(array,size);
}
} 2.4.2 链式Bag的实现
package ds.impl;
import ds.util.Node;
import ds.Bag;
import java.util.Iterator;
import ds.util.ListIterator;
public class LinkedBag<T> implements Iterable<T>, Bag<T> {
private Node<T> first;
private int size;
public LinkedBag() {
first = null;
size = 0;
}
@Override
public int size() {
return size;
}
@Override
public void add(T data) {
Node<T> old = first;
first = new Node<>();
first.data = data;
first.next = old;
size++;
}
@Override
public String toString() {
StringBuilder s = new StringBuilder();
for (T data : this) {
s.append(data).append(' ');
}
return s.toString();
}
@Override
public Iterator<T> iterator() {
return new ListIterator(first);
}
} 2.5 栈(Stack)的实现
栈(Stack)是一种先进后出的数据类型。
2.5.1 能动态调整数组大小的Stack
package ds.impl;
import ds.Stack;
import java.util.Iterator;
import ds.util.ArrayIterator;
public class ArrayStack<T> implements Iterable<T>,Stack<T> {
private T[] array;
private int size;
public ArrayStack(){
array = (T[]) new Object[2];
size = 0;
}
@Override
public Iterator<T> iterator() {
return new ArrayIterator(array,size);
}
@Override
public int size() {
return size;
}
@Override
public void push(T data) {
if(size == array.length){
resize(2* array.length);
}
array[size++] = data;
}
private void resize(int capacity) {
assert capacity >= size;
T[] tmp = (T[]) new Object[capacity];
System.arraycopy(array, 0, tmp, 0, size);
array = tmp;
}
@Override
public T pop() {
notEmptyCheck();
T result = array[size-1];
array[size-1] = null;
size--;
if(size>0 && size == array.length/4){
resize(array.length/2);
}
return result;
}
@Override
public T peek() {
notEmptyCheck();
return array[size-1];
}
} 2.5.2 链式Stack的实现
package ds.impl;
import ds.util.Node;
import ds.Stack;
import java.util.Iterator;
import ds.util.ListIterator;
public class LinkedStack<T> implements Iterable<T>,Stack<T> {
private Node<T> first;
private int size;
public LinkedStack() {
first = null;
size = 0;
}
@Override
public int size() {
return size;
}
/**
* 向栈中推入一个新元素
*
* @param data 新元素
*/
@Override
public void push(T data) {
Node<T> oldFirst = first;
first = new Node<>();
first.data = data;
first.next = oldFirst;
size++;
}
/**
* 从栈中弹出栈首元素
*
* @return 栈首元素
*/
@Override
public T pop() {
notEmptyCheck();
T result = first.data;
first = first.next;
size--;
return result;
}
/**
* 查看栈首元素
* @return 栈首元素
*/
@Override
public T peek() {
notEmptyCheck();
return first.data;
}
@Override
public String toString() {
StringBuilder s = new StringBuilder();
for (T data : this) {
s.append(data).append(' ');
}
return s.toString();
}
@Override
public Iterator<T> iterator() {
return new ListIterator(first);
}
} 2.6 队列(Queue)的实现
队列(Queue)是一种先进先出的数据类型。
2.6.1 能动态调整数组大小的Queue
package ds.impl;
import ds.Queue;
import java.util.Iterator;
public class ArrayQueue<T> implements Iterable<T>,Queue<T> {
private T[] array;
private int size;
private int first;
private int last;
public ArrayQueue() {
array = (T[]) new Object[2];
size = 0;
first = 0;
last = 0;
}
@Override
public int size() {
return size;
}
@Override
public void enqueue(T data) {
if (size == array.length) {
resize(2 * array.length);
}
array[last++] = data;
if (last == array.length) {
last = 0;
}
size++;
}
@Override
public T dequeue() {
notEmptyCheck();
T result = array[first];
array[first] = null;
size--;
first++;
if (first == array.length) {
first = 0;
}
if (size > 0 && size == array.length / 4) {
resize(array.length / 2);
}
return result;
}
@Override
public T peek(){
notEmptyCheck();
return array[first];
}
private void resize(int capacity) {
assert capacity >= size;
T[] tmp = (T[]) new Object[capacity];
for (int i = 0; i < size; i++) {
tmp[i] = array[(first + i) % array.length];
}
array = tmp;
first = 0;
last = size;
}
@Override
public Iterator<T> iterator() {
return new ds.util.ArrayIterator(array,size);
}
} 2.6.2 链式Queue的实现
package ds.impl;
import ds.util.Node;
import ds.Queue;
import java.util.Iterator;
import ds.util.ListIterator;
public class LinkedQueue<T> implements Iterable<T>, Queue<T> {
private Node<T> first;
private Node<T> last;
private int size;
public LinkedQueue() {
first = null;
last = null;
size = 0;
}
@Override
public int size() {
return size;
}
@Override
public T peek() {
notEmptyCheck();
return first.data;
}
@Override
public void enqueue(T data) {
Node<T> oldLast = last;
last = new Node<>();
last.data = data;
last.next = null;
if (isEmpty()) {
first = last;
} else {
oldLast.next = last;
}
size++;
}
@Override
public T dequeue() {
notEmptyCheck();
T result = first.data;
first = first.next;
size--;
if (isEmpty()) {
last = null;
}
return result;
}
@Override
public String toString() {
StringBuilder s = new StringBuilder();
for (T data : this) {
s.append(data).append(' ');
}
return s.toString();
}
@Override
public Iterator<T> iterator() {
return new ListIterator(first);
}
} 2.7 Union-Find算法的实现
并查集（Union-Find）是解决动态连通性问题的一类非常高效的数据结构。
2.7.1 DefaultUF
package ds.impl;
import ds.UF;
public class DefaultUF implements UF {
private int[] parent;
private byte[] rank;
private int size;
public DefaultUF(int n) {
if (n < 0) {
throw new IllegalArgumentException();
}
size = n;
parent = new int[n];
rank = new byte[n];
for (int i = 0; i < n; i++) {
parent[i] = i;
rank[i] = 0;
}
}
@Override
public int find(int p) {
validate(p, parent.length);
while (p != parent[p]) {
parent[p] = parent[parent[p]];
p = parent[p];
}
return p;
}
@Override
public int size() {
return size;
}
@Override
public boolean connected(int p, int q) {
return find(p) == find(q);
}
@Override
public void union(int p, int q) {
int rootP = find(p);
int rootQ = find(q);
if (rootP == rootQ) {
return;
}
if (rank[rootP] < rank[rootQ]) {
parent[rootP] = rootQ;
} else if (rank[rootQ] > rank[rootP]) {
parent[rootQ] = rootP;
} else {
parent[rootQ] = rootP;
rank[rootP]++;
}
size--;
}
} 2.7.2 QuickFindUF
package ds.impl;
import ds.UF;
public class QuickFindUF implements UF {
private final int[] id;
private int size;
public QuickFindUF(int n) {
size = n;
id = new int[n];
for (int i = 0; i < n; i++) {
id[i] = i;
}
}
@Override
public int size() {
return size;
}
@Override
public int find(int p) {
validate(p, id.length);
return id[p];
}
@Override
public boolean connected(int p, int q) {
validate(p, id.length);
validate(q, id.length);
return id[p] == id[q];
}
@Override
public void union(int p, int q) {
validate(p, id.length);
validate(q, id.length);
int pID = id[p];
int qID = id[q];
if (pID == qID) {
return;
}
for (int i = 0; i < id.length; i++) {
if (id[i] == pID) {
id[i] = qID;
}
}
size--;
}
} 2.7.3 QuickUnionUF
package ds.impl;
import ds.UF;
public class QuickUnionUF implements UF {
private final int[] parent;
private int size;
public QuickUnionUF(int n) {
parent = new int[n];
size = n;
for (int i = 0; i < n; i++) {
parent[i] = i;
}
}
@Override
public int size() {
return size;
}
@Override
public int find(int p) {
validate(p, parent.length);
while (p != parent[p]) {
p = parent[p];
}
return p;
}
@Override
public boolean connected(int p, int q) {
return find(p) == find(q);
}
@Override
public void union(int p, int q) {
int rootP = find(p);
int rootQ = find(q);
if (rootP == rootQ) {
return;
}
parent[rootP] = rootQ;
size--;
}
} 2.7.4 WeightedQuickUnionUF
加权 quick-union算法，是目前所有实现中最优的算法
package ds.impl;
import ds.UF;
public class WeightedQuickUnionUF implements UF {
private final int[] parent;
private final int[] num;
private int size;
public WeightedQuickUnionUF(int n) {
size = n;
parent = new int[n];
num = new int[n];
for (int i = 0; i < n; i++) {
parent[i] = i;
num[i] = 1;
}
}
@Override
public int size() {
return size;
}
@Override
public int find(int p) {
validate(p, parent.length);
while (p != parent[p]) {
p = parent[p];
}
return p;
}
@Override
public boolean connected(int p, int q) {
return find(p) == find(q);
}
@Override
public void union(int p, int q) {
int rootP = find(p);
int rootQ = find(q);
if (rootP == rootQ) {
return;
}
// make smaller root point to larger one
if (num[rootP] < num[rootQ]) {
parent[rootP] = rootQ;
num[rootQ] += num[rootP];
} else {
parent[rootQ] = rootP;
num[rootP] += num[rootQ];
}
size--;
}
} 2.8 测试
2.8.1 测试Stack
Dijkstra的双栈算数表达式求值算法
package test;
import java.util.Scanner;
import ds.impl.LinkedStack;
public class StackTest {
public static void main(String[] args) {
evaluate();
}
/**
* Dijkstra的双栈算数表达式求值算法
*/
public static void evaluate() {
String str = "( 1 + ( ( 2.3 + 3 ) / ( sqrt ( 16 ) * ( 5 % 9 ) ) ) )";
LinkedStack<String> opStack = new LinkedStack<>();
LinkedStack<Double> valStack = new LinkedStack<>();
Scanner scanner = new Scanner(str);
while (scanner.hasNext()) {
String s = scanner.next();
System.out.print(s + ' ');
switch (s) {
case "+":
case "-":
case "*":
case "/":
case "%":
case "sqrt":
opStack.push(s);
break;
case "(":
break;
case ")":
String op = opStack.pop();
double v = valStack.pop();
switch (op) {
case "+":
v = valStack.pop() + v;
break;
case "-":
v = valStack.pop() - v;
break;
case "*":
v = valStack.pop() * v;
break;
case "/":
v = valStack.pop() / v;
break;
case "%":
v = valStack.pop() % v;
break;
case "sqrt":
v = Math.sqrt(v);
break;
default:
break;
}
valStack.push(v);
break;
default:
valStack.push(Double.parseDouble(s));
break;
}
}
System.out.printf("= %f\n", valStack.pop());
}
} 2.8.2 测试Union-Find
package test;
import ds.UF;
import ds.impl.DefaultUF;
import ds.impl.QuickFindUF;
import ds.impl.QuickUnionUF;
public class UFTest {
public static void main(String[] args) {
UF uf;
int findResult;
uf = new QuickFindUF(10);
uf.union(4, 3);
uf.union(3, 8);
uf.union(6, 5);
uf.union(9, 4);
uf.union(2, 1);
uf.union(5, 0);
uf.union(7, 2);
uf.union(6, 1);
findResult = uf.find(3);
System.out.println(findResult);
System.out.println(uf.size());
System.out.println(uf.connected(4, 3));
System.out.println(uf.connected(9, 3));
uf = new DefaultUF(10);
uf.union(4, 3);
uf.union(3, 8);
uf.union(6, 5);
uf.union(9, 4);
uf.union(2, 1);
uf.union(5, 0);
uf.union(7, 2);
uf.union(6, 1);
findResult = uf.find(3);
System.out.println(findResult);
System.out.println(uf.size());
System.out.println(uf.connected(4, 3));
System.out.println(uf.connected(9, 3));
uf = new QuickUnionUF(10);
uf.union(4, 3);
uf.union(3, 8);
uf.union(6, 5);
uf.union(9, 4);
uf.union(2, 1);
uf.union(5, 0);
uf.union(7, 2);
uf.union(6, 1);
findResult = uf.find(3);
System.out.println(findResult);
System.out.println(uf.size());
System.out.println(uf.connected(4, 3));
System.out.println(uf.connected(9, 3));
}
} 3 排序算法
3.1 定义排序工具的类结构
需要实现排序器，以及通过静态方法调用。
package util;
import java.util.Arrays;
public class Sort {
public static void main(String[] args) {
String[] a = {
"1", "2", "3", "a", "a1", "b2", "a0", "_"
};
//Sort.heapSort(a);
//Sort.quickSort3way(a, 0, a.length - 1);
System.out.println(Arrays.toString(a));
}
/**
* 根据数组的两个下标交换数组中的元素
*
* @param <T>
* @param array
* @param i
* @param j
*/
private static <T extends Comparable> void swap(T[] array, int i, int j) {
T tmp = array[i];
array[i] = array[j];
array[j] = tmp;
}
/**
* 判断a是否小于b
*
* @param <T>
* @param a
* @param b
* @return
*/
private static <T extends Comparable> boolean less(T a, T b) {
return a.compareTo(b) < 0;
}
/**
* 判断a是否等于b
*
* @param <T>
* @param a
* @param b
* @return
*/
private static <T extends Comparable> boolean eq(T a, T b) {
return a.compareTo(b) == 0;
}
/**
* 判断数组是否排序
*
* @param <T>
* @param array
* @return
*/
public static <T extends Comparable> boolean isSorted(T[] array) {
for (int i = 1; i < array.length; i++) {
if (less(array[i], array[i - 1])) {
return false;
}
}
return true;
}
/**
* 打印数组
*
* @param <T>
* @param array
*/
public static <T extends Comparable> void print(T[] array) {
for (T t : array) {
System.out.print(t + " ");
}
System.out.println();
}
}
3.2 选择排序
时间复杂度O(n^2)，辅助空间O(1)，不稳定
public static <T extends Comparable> void selectSort(T[] a) {
int len = a.length;
for (int i = 0; i < len; i++) {
int min = i;
for (int j = i + 1; j < len; j++) {
if (less(a[j], a[min])) {
min = j;
}
}
swap(a, i, min);
}
} 3.3 插入排序
时间复杂度O(n^2)，辅助空间O(1)，稳定
public static <T extends Comparable> void insertSort(T[] a) {
insertSort(a, 0, a.length);
}
public static <T extends Comparable> void insertSort(T[] a, int low, int high) {
for (int i = low + 1; i < high; i++) {
for (int j = i; j > 0 && less(a[j], a[j - 1]); j--) {
swap(a, j, j - 1);
}
}
} 3.4 希尔排序
时间复杂度O(n^1.25)，辅助空间O(1)，不稳定
public static <T extends Comparable> void shellSort(T[] a) {
int len = a.length;
int h = 1;
while (h < len / 3) {
h = 3 * h + 1;
}
for (; h >= 1; h /= 3) {
for (int i = h; i < len; i++) {
for (int j = i; j >= h && less(a[j], a[j - h]); j -= h) {
swap(a, j, j - h);
}
}
}
} 3.5 归并排序
时间复杂度O(nlog(n))，辅助空间O(nlog(n))，稳定
3.5.1 归并排序的合并方法
/**
* 归并排序所需的辅助数组。不将其声明为方法内的局部变量，是为了避免重复创建数组
*/
private static Comparable[] mergeAux;
/**
* 归并排序的合并方法
* <pre>
* 该方法先将所有元素复制到辅助数组中，再归并回数组a中。
* 在归并时进行了4个条件判断：
* - 左半边用尽（取右半边的元素）
* - 右半边用尽（取左半边的元素）
* - 右半边当前元素小于左半边的当前元素（取右半边的元素）
* - 右半边当前元素大于等于左半边的当前元素（取左半边元素）
* </pre>
*
* @param <T>
* @param a
* @param low
* @param middle
* @param high
*/
private static <T extends Comparable>
void merge(T[] a, int low, int middle, int high) {
int i = low;
int j = middle + 1;
for (int k = low; k <= high; k++) {
mergeAux[k] = a[k];
}
for (int k = low; k <= high; k++) {
if (i > middle) {
a[k] = (T) mergeAux[j++];
} else if (j > high) {
a[k] = (T) mergeAux[i++];
} else if (less(mergeAux[j], mergeAux[i])) {
a[k] = (T) mergeAux[j++];
} else {
a[k] = (T) mergeAux[i++];
}
}
}
3.5.2 自顶向下的归并排序
public static <T extends Comparable> void mergeSort(T[] a) {
mergeAux = new Comparable[a.length];
mergeSort(a, 0, a.length - 1);
}
public static <T extends Comparable>
void mergeSort(T[] a, int low, int high) {
if (high <= low) {
return;
}
int middle = low + (high - low) / 2;
mergeSort(a, low, middle);
mergeSort(a, middle + 1, high);
merge(a, low, middle, high);
} 3.5.3 自底向上的归并排序
自底向上的归并排序适用于链表组织的数据
public static <T extends Comparable> void mergeSort2(T[] a) {
int N = a.length;
mergeAux = new Comparable[a.length];
for (int i = 1; i < N; i = i + i) {
for (int low = 0; low < N - i; low += i + i) {
merge(a, low, low + i - 1, Math.min(low + i + i - 1, N - 1));
}
}
} 3.6 快速排序
时间复杂度O(nlog(n))，辅助空间O(nlog(n))，不稳定。非特殊场景下，最快的排序算法
3.6.1 常规快速排序
/**
* 常规快速排序的划分方法
*
* @param <T>
* @param a
* @param low
* @param high
* @return
*/
private static <T extends Comparable>
int partition(T[] a, int low, int high) {
int i = low;
int j = high;
T p = a[low];
while (i < j) {
while (i < j && (less(p, a[j]) || eq(a[j], p))) {
j--;
}
if (i < j) {
swap(a, i++, j);
}
while (i < j && (less(a[i], p) || eq(a[i], p))) {
i++;
}
if (i < j) {
swap(a, i, j--);
}
}
return j;
}
/**
* 常规快速排序的排序方法
*
* @param <T>
* @param a
* @param low
* @param high
*/
public static <T extends Comparable>
void quickSort(T[] a, int low, int high) {
int p;
if (low < high) {
p = partition(a, low, high);
quickSort(a, low, p - 1);
quickSort(a, p + 1, high);
}
}
3.6.2 排序前先洗牌
/**
* 快速排序:在排序前先洗牌（可以参考随机化算法-舍伍德算法）
* <pre>
* 快速排序的算法改进：
* 【切换到插入排序】【三取样切分】【熵最优的排序，三向切分】
* <pre>
* 三取样切分
* (1)使用子数组的一小部分元素的中位数来切分数组，这样能切分得更好，但是需要计算中位数
* (2)人们发现将大小设为3并用大小居中的元素切分得效果最好
* </pre>
* </pre>
*
*
* @param <T>
* @param a
*/
public static <T extends Comparable> void quickSort(T[] a) {
Shuffle.shuffle(a);
quickSort(a, 0, a.length - 1);
// quickSortInsert(a, 0, a.length - 1);
// quickSort3way(a, 0, a.length - 1);
} 以下是该洗牌算法的实现
package util;
import java.util.Arrays;
/**
* 设计一种公平的洗牌算法（算法导论）
*
* @author zhaoxuyang
*/
public class Shuffle {
public static void main(String[] args) {
Integer[] a = {
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
};
shuffle(a);
System.out.println(Arrays.toString(a));
}
public static <T extends Comparable> void shuffle(T[] arr) {
for (int i = arr.length - 1; i >= 0; i--) {
swap(arr, i, rand(0, i));
}
}
private static int rand(int begin, int end) {
return (int) (begin + Math.random() * (end - begin + 1));
}
private static <T extends Comparable> void swap(T[] arr, int i, int j) {
T t = arr[i];
arr[i] = arr[j];
arr[j] = t;
}
}
3.6.3 快速排序的改进方法-小数据量转成插入排序
/**
* 快速排序的改进方法-小数据量转成插入排序
* <pre>
* (1)对于小数组，快速排序比插入排序慢；
* (2)因为递归，快速排序的sort方***调用自己。
* </pre>
*
* @param <T>
* @param a
* @param low
* @param high
*/
public static <T extends Comparable>
void quickSortInsert(T[] a, int low, int high) {
int m = 65535;
if (high <= low + m) {
insertSort(a, low, high);
return;
}
int lt = low;
int i = low + 1;
int gt = high;
T v = a[low];
while (i <= gt) {
int cmp = a[i].compareTo(v);
if (cmp < 0) {
swap(a, lt++, i++);
} else if (cmp > 0) {
swap(a, i, gt--);
} else {
i++;
}
}
quickSort(a, low, lt - 1);
quickSort(a, gt + 1, high);
} 3.6.4 快速排序的改进方法-三向切分
/**
* 快速排序的改进方法-三向切分（可以参考荷兰国旗问题）
* <pre>
* 对于存在大量重复元素的数组，三向切分比常规快排高效得多。
* </pre>
*
* @param <T>
* @param a
* @param low
* @param high
*/
public static <T extends Comparable>
void quickSort3way(T[] a, int low, int high) {
if (high <= low) {
return;
}
int lt = low;
int i = low + 1;
int gt = high;
T v = a[low];
while (i <= gt) {
int cmp = a[i].compareTo(v);
if (cmp < 0) {
swap(a, lt++, i++);
} else if (cmp > 0) {
swap(a, i, gt--);
} else {
i++;
}
}
quickSort(a, low, lt - 1);
quickSort(a, gt + 1, high);
} 3.7 堆排序
时间复杂度O(nlog(n))，辅助空间O(nlog(n))，不稳定
public static <T extends Comparable> void heapSort(T[] a) {
int n = a.length;
for (int k = n / 2; k >= 1; k--) {
sink(a, k, n);
}
while (n > 1) {
swap(a, 1 - 1, (n--) - 1);
sink(a, 1, n);
}
}
private static <T extends Comparable> void sink(T[] pq, int k, int n) {
while (2 * k <= n) {
int j = 2 * k;
if (j < n && less(pq[j - 1], pq[j])) {
j++;
}
if (!less(pq[k - 1], pq[j - 1])) {
break;
}
swap(pq, k - 1, j - 1);
k = j;
}
} 3.8 最终的排序工具
package util;
import java.util.Arrays;
/**
* 排序
*
* @author zhaoxuyang
*/
public class Sort {
public static void main(String[] args) {
String[] a = {
"1", "2", "3", "a", "a1", "b2", "a0", "_"
};
//Sort.heapSort(a);
//Sort.quickSort3way(a, 0, a.length - 1);
System.out.println(Arrays.toString(a));
}
/**
* 堆排序
*
* @param <T>
* @param a
*/
public static <T extends Comparable> void heapSort(T[] a) {
int n = a.length;
for (int k = n / 2; k >= 1; k--) {
sink(a, k, n);
}
while (n > 1) {
swap(a, 1 - 1, (n--) - 1);
sink(a, 1, n);
}
}
private static <T extends Comparable> void sink(T[] pq, int k, int n) {
while (2 * k <= n) {
int j = 2 * k;
if (j < n && less(pq[j - 1], pq[j])) {
j++;
}
if (!less(pq[k - 1], pq[j - 1])) {
break;
}
swap(pq, k - 1, j - 1);
k = j;
}
}
/**
* 快速排序:在排序前先洗牌（可以参考随机化算法-舍伍德算法）
* <pre>
* 快速排序的算法改进：
* 【切换到插入排序】【三取样切分】【熵最优的排序，三向切分】
* <pre>
* 三取样切分
* (1)使用子数组的一小部分元素的中位数来切分数组，这样能切分得更好，但是需要计算中位数
* (2)人们发现将大小设为3并用大小居中的元素切分得效果最好
* </pre>
* </pre>
*
*
* @param <T>
* @param a
*/
public static <T extends Comparable> void quickSort(T[] a) {
Shuffle.shuffle(a);
quickSort(a, 0, a.length - 1);
// quickSortInsert(a, 0, a.length - 1);
// quickSort3way(a, 0, a.length - 1);
}
/**
* 快速排序的改进方法-小数据量转成插入排序
* <pre>
* (1)对于小数组，快速排序比插入排序慢；
* (2)因为递归，快速排序的sort方***调用自己。
* </pre>
*
* @param <T>
* @param a
* @param low
* @param high
*/
public static <T extends Comparable>
void quickSortInsert(T[] a, int low, int high) {
int m = 65535;
if (high <= low + m) {
insertSort(a, low, high);
return;
}
int lt = low;
int i = low + 1;
int gt = high;
T v = a[low];
while (i <= gt) {
int cmp = a[i].compareTo(v);
if (cmp < 0) {
swap(a, lt++, i++);
} else if (cmp > 0) {
swap(a, i, gt--);
} else {
i++;
}
}
quickSort(a, low, lt - 1);
quickSort(a, gt + 1, high);
}
/**
* 快速排序的改进方法-三向切分（可以参考荷兰国旗问题）
* <pre>
* 对于存在大量重复元素的数组，三向切分比常规快排高效得多。
* </pre>
*
* @param <T>
* @param a
* @param low
* @param high
*/
public static <T extends Comparable>
void quickSort3way(T[] a, int low, int high) {
if (high <= low) {
return;
}
int lt = low;
int i = low + 1;
int gt = high;
T v = a[low];
while (i <= gt) {
int cmp = a[i].compareTo(v);
if (cmp < 0) {
swap(a, lt++, i++);
} else if (cmp > 0) {
swap(a, i, gt--);
} else {
i++;
}
}
quickSort(a, low, lt - 1);
quickSort(a, gt + 1, high);
}
/**
* 常规快速排序的排序方法
*
* @param <T>
* @param a
* @param low
* @param high
*/
public static <T extends Comparable>
void quickSort(T[] a, int low, int high) {
int p;
if (low < high) {
p = partition(a, low, high);
quickSort(a, low, p - 1);
quickSort(a, p + 1, high);
}
}
/**
* 常规快速排序的划分方法
*
* @param <T>
* @param a
* @param low
* @param high
* @return
*/
private static <T extends Comparable>
int partition(T[] a, int low, int high) {
int i = low;
int j = high;
T p = a[low];
while (i < j) {
while (i < j && (less(p, a[j]) || eq(a[j], p))) {
j--;
}
if (i < j) {
swap(a, i++, j);
}
while (i < j && (less(a[i], p) || eq(a[i], p))) {
i++;
}
if (i < j) {
swap(a, i, j--);
}
}
return j;
}
/**
* 归并排序所需的辅助数组。不将其声明为方法内的局部变量，是为了避免重复创建数组
*/
private static Comparable[] mergeAux;
/**
* 自底向上的归并排序（适用于链表组织的数据）
*
* @param <T>
* @param a
*/
public static <T extends Comparable> void mergeSort2(T[] a) {
int N = a.length;
mergeAux = new Comparable[a.length];
for (int i = 1; i < N; i = i + i) {
for (int low = 0; low < N - i; low += i + i) {
merge(a, low, low + i - 1, Math.min(low + i + i - 1, N - 1));
}
}
}
/**
* 自顶向下的归并排序
*
* @param <T>
* @param a
*/
public static <T extends Comparable> void mergeSort(T[] a) {
mergeAux = new Comparable[a.length];
mergeSort(a, 0, a.length - 1);
}
/**
* 自顶向下的归并排序
*
* @param <T>
* @param a
* @param low
* @param high
*/
public static <T extends Comparable>
void mergeSort(T[] a, int low, int high) {
if (high <= low) {
return;
}
int middle = low + (high - low) / 2;
mergeSort(a, low, middle);
mergeSort(a, middle + 1, high);
merge(a, low, middle, high);
}
/**
* 归并排序的合并方法
* <pre>
* 该方法先将所有元素复制到辅助数组中，再归并回数组a中。
* 在归并时进行了4个条件判断：
* - 左半边用尽（取右半边的元素）
* - 右半边用尽（取左半边的元素）
* - 右半边当前元素小于左半边的当前元素（取右半边的元素）
* - 右半边当前元素大于等于左半边的当前元素（取左半边元素）
* </pre>
*
* @param <T>
* @param a
* @param low
* @param middle
* @param high
*/
private static <T extends Comparable>
void merge(T[] a, int low, int middle, int high) {
int i = low;
int j = middle + 1;
for (int k = low; k <= high; k++) {
mergeAux[k] = a[k];
}
for (int k = low; k <= high; k++) {
if (i > middle) {
a[k] = (T) mergeAux[j++];
} else if (j > high) {
a[k] = (T) mergeAux[i++];
} else if (less(mergeAux[j], mergeAux[i])) {
a[k] = (T) mergeAux[j++];
} else {
a[k] = (T) mergeAux[i++];
}
}
}
/**
* 希尔排序
*
* @param <T>
* @param a
*/
public static <T extends Comparable> void shellSort(T[] a) {
int len = a.length;
int h = 1;
while (h < len / 3) {
h = 3 * h + 1;
}
for (; h >= 1; h /= 3) {
for (int i = h; i < len; i++) {
for (int j = i; j >= h && less(a[j], a[j - h]); j -= h) {
swap(a, j, j - h);
}
}
}
}
/**
* 插入排序
*
* @param <T>
* @param a
*/
public static <T extends Comparable> void insertSort(T[] a) {
insertSort(a, 0, a.length);
}
/**
* 插入排序
*
* @param <T>
* @param a
* @param low
* @param high
*/
public static <T extends Comparable> void insertSort(T[] a, int low, int high) {
for (int i = low + 1; i < high; i++) {
for (int j = i; j > 0 && less(a[j], a[j - 1]); j--) {
swap(a, j, j - 1);
}
}
}
/**
* 选择排序
*
* @param <T>
* @param a
*/
public static <T extends Comparable> void selectSort(T[] a) {
int len = a.length;
for (int i = 0; i < len; i++) {
int min = i;
for (int j = i + 1; j < len; j++) {
if (less(a[j], a[min])) {
min = j;
}
}
swap(a, i, min);
}
}
/**
* 根据数组的两个下标交换数组中的元素
*
* @param <T>
* @param array
* @param i
* @param j
*/
private static <T extends Comparable> void swap(T[] array, int i, int j) {
T tmp = array[i];
array[i] = array[j];
array[j] = tmp;
}
/**
* 判断a是否小于b
*
* @param <T>
* @param a
* @param b
* @return
*/
private static <T extends Comparable> boolean less(T a, T b) {
return a.compareTo(b) < 0;
}
/**
* 判断a是否等于b
*
* @param <T>
* @param a
* @param b
* @return
*/
private static <T extends Comparable> boolean eq(T a, T b) {
return a.compareTo(b) == 0;
}
/**
* 判断数组是否排序
*
* @param <T>
* @param array
* @return
*/
public static <T extends Comparable> boolean isSorted(T[] array) {
for (int i = 1; i < array.length; i++) {
if (less(array[i], array[i - 1])) {
return false;
}
}
return true;
}
/**
* 打印数组
*
* @param <T>
* @param array
*/
public static <T extends Comparable> void print(T[] array) {
for (T t : array) {
System.out.print(t + " ");
}
System.out.println();
}
}
洗牌算法(Shuffle)
package util;
import java.util.Arrays;
/**
* 设计一种公平的洗牌算法（算法导论）
*
* @author zhaoxuyang
*/
public class Shuffle {
public static void main(String[] args) {
Integer[] a = {
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
};
shuffle(a);
System.out.println(Arrays.toString(a));
}
public static <T extends Comparable> void shuffle(T[] arr) {
for (int i = arr.length - 1; i >= 0; i--) {
swap(arr, i, rand(0, i));
}
}
private static int rand(int begin, int end) {
return (int) (begin + Math.random() * (end - begin + 1));
}
private static <T extends Comparable> void swap(T[] arr, int i, int j) {
T t = arr[i];
arr[i] = arr[j];
arr[j] = t;
}
} 4 搜索
4.1 二分搜索(binarySearch)
public static int binarySearch(int[] a, int key) {
int low = 0;
int high = a.length - 1;
while (low <= high) {
int middle = low + (high - low) / 2;
if (key < a[middle]) {
high = middle - 1;
} else if (key > a[middle]) {
low = middle + 1;
} else {
return middle;
}
}
return -1;
} 4.2 优先队列(MaxPriorityQueue)
package ds.impl;
import java.util.Comparator;
import java.util.Iterator;
import java.util.NoSuchElementException;
public class MaxPriorityQueue<T> implements Iterable<T> {
private T[] pq;
private int size;
private Comparator<T> comparator;
public MaxPriorityQueue(int capacity, Comparator comparator) {
pq = (T[]) new Object[capacity + 1];
size = 0;
this.comparator = comparator;
}
public MaxPriorityQueue(int capacity) {
this(capacity, null);
}
public MaxPriorityQueue(Comparator comparator) {
this(1, comparator);
}
public MaxPriorityQueue() {
this(1);
}
public MaxPriorityQueue(T[] keys) {
size = keys.length;
pq = (T[]) new Object[size + 1];
System.arraycopy(keys, 0, pq, 1, size);
for (int k = size / 2; k >= 1; k--) {
sink(k);
}
}
public void insert(T key) {
if (size == pq.length - 1) {
resize(2 * pq.length);
}
pq[++size] = key;
swim(size);
}
public T max() {
notEmptyCheck();
return pq[1];
}
public T deleteMax() {
notEmptyCheck();
T result = pq[1];
swap(1, size--);
sink(1);
pq[size + 1] = null;
if ((size > 0) && (size == (pq.length - 1) / 4)) {
resize(pq.length / 2);
}
return result;
}
private void notEmptyCheck() {
if (isEmpty()) {
throw new NoSuchElementException();
}
}
@Override
public Iterator<T> iterator() {
return new HeapIterator();
}
private class HeapIterator implements Iterator<T> {
private MaxPriorityQueue<T> copy;
public HeapIterator() {
if (comparator == null) {
copy = new MaxPriorityQueue<>(size());
} else {
copy = new MaxPriorityQueue<>(size(), comparator);
}
for (int i = 1; i <= size; i++) {
copy.insert(pq[i]);
}
}
@Override
public boolean hasNext() {
return !copy.isEmpty();
}
@Override
public T next() {
if (!hasNext()) {
throw new NoSuchElementException();
}
return copy.deleteMax();
}
}
private void sink(int k) {
while (2 * k <= size) {
int j = 2 * k;
if (j < size && less(j, j + 1)) {
j++;
}
if (!less(k, j)) {
break;
}
swap(k, j);
k = j;
}
}
private boolean less(int a, int b) {
if (comparator == null) {
return ((Comparable<T>) pq[a]).compareTo(pq[b]) < 0;
} else {
return comparator.compare(pq[a], pq[b]) < 0;
}
}
private void swap(int a, int b) {
T tmp = pq[a];
pq[a] = pq[b];
pq[b] = tmp;
}
private void resize(int capacity) {
T[] tmp = (T[]) new Object[capacity];
for (int i = 1; i <= size; i++) {
tmp[i] = pq[i];
}
pq = tmp;
}
private void swim(int k) {
while (k > 1 && less(k / 2, k)) {
swap(k, k / 2);
k = k / 2;
}
}
private boolean isEmpty() {
return size() == 0;
}
private int size() {
return size;
}
}
4.3 二叉查找树(BST)
package ds.impl;
import ds.Queue;
public class BST<K extends Comparable<K>, V> {
private Node root;
private class Node {
private K key;
private V value;
private Node left, right;
private int size;
public Node(K key, V value, int size) {
this.key = key;
this.value = value;
this.size = size;
}
}
public int size() {
return size(root);
}
private int size(Node x) {
return x == null ? 0 : x.size;
}
public V get(K key) {
return get(root, key);
}
private V get(Node x, K key) {
if (x == null) {
return null;
}
int cmp = key.compareTo(x.key);
if (cmp < 0) {
return get(x.left, key);
} else if (cmp > 0) {
return get(x.right, key);
} else {
return x.value;
}
}
public void put(K key, V value) {
root = put(root, key, value);
}
private Node put(Node x, K key, V value) {
if (x == null) {
return new Node(key, value, 1);
}
int cmp = key.compareTo(x.key);
if (cmp < 0) {
x.left = put(x.left, key, value);
} else if (cmp > 0) {
x.right = put(x.right, key, value);
} else {
x.value = value;
}
x.size = size(x.left) + size(x.right) + 1;
return x;
}
public K minKey() {
return minKey(root).key;
}
private Node minKey(Node x) {
return x.left == null ? x : minKey(x.left);
}
public K maxKey() {
return maxKey(root).key;
}
private Node maxKey(Node x) {
return x.right == null ? x : maxKey(x.right);
}
public K floorKey(K key) {
Node x = floorKey(root, key);
return x == null ? null : x.key;
}
private Node floorKey(Node x, K key) {
if (x == null) {
return null;
}
int cmp = key.compareTo(x.key);
if (cmp == 0) {
return x;
}
if (cmp < 0) {
return floorKey(x, key);
}
Node t = floorKey(x.right, key);
return t != null ? t : x;
}
public K ceilKey(K key) {
Node x = ceilKey(root, key);
return x == null ? null : x.key;
}
private Node ceilKey(Node x, K key) {
if (x == null) {
return null;
}
int cmp = key.compareTo(x.key);
if (cmp == 0) {
return x;
}
if (cmp > 0) {
return ceilKey(x, key);
}
Node t = ceilKey(x.left, key);
return t != null ? t : x;
}
public K selectKey(int k) {
return selectKey(root, k).key;
}
private Node selectKey(Node x, int k) {
if (x == null) {
return null;
}
int t = size(x.left);
if (t > k) {
return selectKey(x.left, k);
} else if (t < k) {
return selectKey(x.right, k - t - 1);
} else {
return x;
}
}
public int rankKey(K key) {
return rankKey(root, key);
}
private int rankKey(Node x, K key) {
if (x == null) {
return 0;
}
int cmp = key.compareTo(x.key);
if (cmp < 0) {
return rankKey(x.left, key);
} else if (cmp > 0) {
return 1 + size(x.left) + rankKey(x.right, key);
} else {
return size(x.left);
}
}
public void deleteMin() {
root = deleteMin(root);
}
private Node deleteMin(Node x) {
if (x.left == null) {
return x.right;
}
x.left = deleteMin(x.left);
x.size = size(x.left) + size(x.right) + 1;
return x;
}
public void deleteMax() {
root = deleteMax(root);
}
private Node deleteMax(Node x) {
if (x.right == null) {
return x.left;
}
x.right = deleteMax(x.right);
x.size = size(x.left) + size(x.right) + 1;
return x;
}
public void delete(K key) {
root = delete(root, key);
}
private Node delete(Node x, K key) {
if (x == null) {
return null;
}
int cmp = key.compareTo(x.key);
if (cmp < 0) {
x.left = delete(x.left, key);
} else if (cmp > 0) {
x.right = delete(x.right, key);
} else {
if (x.right == null) {
return x.left;
}
if (x.left == null) {
return x.right;
}
Node t = x;
x = minKey(t.right);
x.right = deleteMin(t.right);
x.left = t.left;
}
x.size = size(x.left) + size(x.right) + 1;
return x;
}
public Iterable<K> keys() {
return keys(minKey(), maxKey());
}
private Iterable<K> keys(K lowKey, K highKey) {
LinkedQueue<K> queue = new LinkedQueue<>();
keys(root, queue, lowKey, highKey);
return queue;
}
private void keys(Node x, Queue<K> queue, K lowKey, K highKey) {
if (x == null) {
return;
}
int cmpLow = lowKey.compareTo(x.key);
int cmpHigh = highKey.compareTo(x.key);
if (cmpLow < 0) {
keys(x.left, queue, lowKey, highKey);
}
if (cmpLow <= 0 && cmpHigh >= 0) {
queue.enqueue(x.key);
}
if (cmpHigh > 0) {
keys(x.right, queue, lowKey, highKey);
}
}
}
4.4 红黑二叉查找树(RedBlackBST)
package ds.impl;
import java.util.NoSuchElementException;
public class RedBlackBST<K extends Comparable<K>, V> {
private Node root;
private class Node {
private K key;
private V value;
private Node left;
private Node right;
private boolean isRed;
private int size;
public Node(K key, V value, boolean isRed, int size) {
this.isRed = isRed;
this.key = key;
this.value = value;
this.size = size;
}
}
private boolean isRed(Node x) {
return x == null ? false : x.isRed;
}
private int size(Node x) {
return x == null ? 0 : x.size;
}
public int size() {
return size(root);
}
public boolean isEmpty() {
return root == null;
}
public V get(K key) {
if (key == null) {
throw new IllegalArgumentException();
}
return get(root, key);
}
private V get(Node x, K key) {
if (x == null) {
return null;
}
int cmp = key.compareTo(x.key);
if (cmp < 0) {
return get(x.left, key);
} else if (cmp > 0) {
return get(x.right, key);
} else {
return x.value;
}
// while (x != null) {
// int cmp = key.compareTo(x.key);
// if (cmp < 0) {
// x = x.left;
// } else if (cmp > 0) {
// x = x.right;
// } else {
// return x.value;
// }
// }
// return null;
}
public boolean containsKey(K key) {
return get(key) != null;
}
private void delete(K key) {
if (key == null) {
throw new IllegalArgumentException();
}
if (!containsKey(key)) {
return;
}
if (!isRed(root.left) && !isRed(root.right)) {
root.isRed = true;
}
root = delete(root, key);
if (!isEmpty()) {
root.isRed = false;
}
}
private Node delete(Node x, K key) {
int cmp = key.compareTo(x.key);
if (cmp < 0) {
if (!isRed(x.left) && !isRed(x.left.left)) {
x = moveRedLeft(x);
}
x.left = delete(x.left, key);
} else {
if (isRed(x.left)) {
x = rotateRight(x);
}
if (key.compareTo(x.key) == 0 && x.right == null) {
return null;
}
if (!isRed(x.right) && !isRed(x.right.left)) {
x = moveRedRight(x);
}
if (key.compareTo(x.key) == 0) {
Node rightMinNode = min();
x.key = rightMinNode.key;
x.value = rightMinNode.value;
x.right = deleteMin(x.right);
} else {
x.right = delete(x.right, key);
}
}
return balance(x);
}
public void put(K key, V value) {
if (key == null) {
throw new IllegalArgumentException();
}
if (value == null) {
delete(key);
return;
}
root = put(root, key, value);
root.isRed = false;
}
private Node put(Node x, K key, V value) {
if (x == null) {
return new Node(key, value, true, 1);
}
int cmp = key.compareTo(x.key);
if (cmp < 0) {
x.left = put(x.left, key, value);
} else if (cmp > 0) {
x.right = put(x.right, key, value);
} else {
x.value = value;
}
if (isRed(x.right) && !isRed(x.left)) {
x = rotateLeft(x);
} else if (isRed(x.left) && !isRed(x.right)) {
x = rotateRight(x);
} else if (isRed(x.left) && isRed(x.right)) {
flipColors(x);
}
x.size = size(x.left) + size(x.right) + 1;
return x;
}
private Node rotateLeft(Node x) {
Node result = x.right;
x.right = result.left;
result.left = x;
result.isRed = result.left.isRed;
result.left.isRed = true;
result.size = x.size;
x.size = size(x.left) + size(x.right) + 1;
return result;
}
private Node rotateRight(Node x) {
Node result = x.left;
x.left = result.right;
result.right = x;
result.isRed = result.right.isRed;
result.right.isRed = true;
result.size = x.size;
x.size = size(x.left) + size(x.right) + 1;
return result;
}
private void flipColors(Node x) {
x.isRed = !x.isRed;
x.left.isRed = !x.left.isRed;
x.right.isRed = !x.right.isRed;
}
private Node moveRedLeft(Node x) {
flipColors(x);
if (isRed(x.right.left)) {
x.right = rotateRight(x.right);
x = rotateLeft(x);
flipColors(x);
}
return x;
}
private Node moveRedRight(Node x) {
flipColors(x);
if (isRed(x.left.left)) {
x = rotateRight(x);
flipColors(x);
}
return x;
}
private Node min() {
throw new UnsupportedOperationException();
}
private Node deleteMin(Node right) {
throw new UnsupportedOperationException();
}
private Node balance(Node x) {
if (isRed(x.right)) {
x = rotateLeft(x);
}
if (isRed(x.left) && isRed(x.left.left)) {
x = rotateRight(x);
}
if (isRed(x.left) && isRed(x.right)) {
flipColors(x);
}
x.size = size(x.left) + size(x.right) + 1;
return x;
}
public int height() {
return height(root);
}
private int height(Node x) {
if (x == null) {
return -1;
}
return 1 + mathMax(height(x.left), height(x.right));
}
private int mathMax(int a, int b) {
return a > b ? a : b;
}
public K minKey() {
if (isEmpty()) {
throw new NoSuchElementException();
}
return minKey(root).key;
}
private Node minKey(Node x) {
return x.left == null ? x : minKey(x.left);
}
public K maxKey() {
if (isEmpty()) {
throw new NoSuchElementException();
}
return maxKey(root).key;
}
private Node maxKey(Node x) {
return x.right == null ? x : maxKey(x.right);
}
public Iterable<K> keys() {
if (isEmpty()) {
return new LinkedQueue<K>();
}
return keys(minKey(), maxKey());
}
public Iterable<K> keys(K lo, K hi) {
if (lo == null) {
throw new IllegalArgumentException("first argument to keys() is null");
}
if (hi == null) {
throw new IllegalArgumentException("second argument to keys() is null");
}
LinkedQueue<K> queue = new LinkedQueue<>();
// if (isEmpty() || lo.compareTo(hi) > 0) return queue;
keys(root, queue, lo, hi);
return queue;
}
private void keys(Node x, LinkedQueue<K> queue, K lo, K hi) {
if (x == null) {
return;
}
int cmplo = lo.compareTo(x.key);
int cmphi = hi.compareTo(x.key);
if (cmplo < 0) {
keys(x.left, queue, lo, hi);
}
if (cmplo <= 0 && cmphi >= 0) {
queue.enqueue(x.key);
}
if (cmphi > 0) {
keys(x.right, queue, lo, hi);
}
}
public static void main(String[] args) {
RedBlackBST<String, Integer> st = new RedBlackBST<>();
for (int i = 0; i < 100000; i++) {
String key = "key" + i;
Integer value = (int) (Math.random() * 10000);
st.put(key, value);
}
for (String s : st.keys()) {
System.out.println(s + " " + st.get(s));
}
System.out.println();
}
}
4.5 B-树(BTree)
package ds.impl;
import ds.Container;
public class BTree<K extends Comparable<K>, V> implements Container {
private static final int M = 4;
private Node root;
private int height;
private int size;
private static class Node {
private int m;
private final Entry[] children = new Entry[M];
public Node(int k) {
m = k;
}
}
private static class Entry {
private Comparable key;
private final Object value;
private Node next;
public Entry(Comparable key, Object value, Node next) {
this.key = key;
this.value = value;
this.next = next;
}
}
public BTree() {
root = new Node(0);
}
@Override
public int size() {
return size;
}
public int height() {
return height;
}
public V get(K key) {
if (key == null) {
throw new IllegalArgumentException();
}
return search(root, key, height);
}
private V search(Node node, K key, int height) {
Entry[] children = node.children;
if (height == 0) {
for (int i = 0; i < node.m; i++) {
if (eq(key, children[i].key)) {
return (V) children[i].value;
}
}
} else {
for (int i = 0; i < node.m; i++) {
if (i + 1 == node.m || less(key, children[i + 1].key)) {
return search(children[i].next, key, height - 1);
}
}
}
return null;
}
private boolean eq(Comparable k1, Comparable k2) {
return k1.compareTo(k2) == 0;
}
private boolean less(Comparable k1, Comparable k2) {
return k1.compareTo(k2) < 0;
}
public void put(K key, V value) {
if (key == null) {
throw new IllegalArgumentException();
}
Node u = insert(root, key, value, this.height);
size++;
if (u == null) {
return;
}
Node t = new Node(2);
t.children[0] = new Entry(root.children[0].key, null, root);
t.children[1] = new Entry(u.children[0].key, null, u);
root = t;
height++;
}
private Node insert(Node node, K key, V value, int height) {
int i;
Entry entry = new Entry(key, value, null);
if (height == 0) {
for (i = 0; i < node.m; i++) {
if (less(key, node.children[i].key)) {
break;
}
}
} else {
for (i = 0; i < node.m; i++) {
if ((i + 1 == node.m) || less(key, node.children[i + 1].key)) {
Node next = node.children[i++].next;
Node u = insert(next, key, value, height - 1);
if (u == null) {
return null;
}
entry.key = u.children[0].key;
entry.next = u;
break;
}
}
}
for (int j = node.m; j > i; j--) {
node.children[j] = node.children[j - 1];
}
node.children[i] = entry;
node.m++;
if (node.m < M) {
return null;
} else {
return split(node);
}
}
private Node split(Node node) {
int m = M / 2;
Node result = new Node(m);
node.m = m;
for (int j = 0; j < m; j++) {
result.children[j] = node.children[m + j];
}
return result;
}
@Override
public String toString() {
return toString(root, height, "") + "\n";
}
private String toString(Node node, int height, String indent) {
StringBuilder s = new StringBuilder();
Entry[] children = node.children;
if (height == 0) {
for (int j = 0; j < node.m; j++) {
s.append(indent)
.append(children[j].key)
.append('=')
.append(children[j].value)
.append('\n');
}
} else {
for (int j = 0; j < node.m; j++) {
if (j > 0) {
s.append(indent)
.append('(')
.append(children[j].key)
.append(')')
.append('\n');
}
s.append(toString(children[j].next, height - 1, indent + "#"));
}
}
return s.toString();
}
public static void main(String[] args) {
BTree<String, String> st = new BTree<>();
st.put("www.cs.princeton.edu", "128.112.136.12");
st.put("www.cs.princeton.edu", "128.112.136.11");
st.put("www.princeton.edu", "128.112.128.15");
st.put("www.yale.edu", "130.132.143.21");
st.put("www.simpsons.com", "209.052.165.60");
st.put("www.apple.com", "17.112.152.32");
st.put("www.amazon.com", "207.171.182.16");
st.put("www.ebay.com", "66.135.192.87");
st.put("www.cnn.com", "64.236.16.20");
st.put("www.google.com", "216.239.41.99");
st.put("www.nytimes.com", "199.239.136.200");
st.put("www.microsoft.com", "207.126.99.140");
st.put("www.dell.com", "143.166.224.230");
st.put("www.slashdot.org", "66.35.250.151");
st.put("www.espn.com", "199.181.135.201");
st.put("www.weather.com", "63.111.66.11");
st.put("www.yahoo.com", "216.109.118.65");
System.out.println("cs.princeton.edu: " + st.get("www.cs.princeton.edu"));
System.out.println("hardvardsucks.com: " + st.get("www.harvardsucks.com"));
System.out.println("simpsons.com: " + st.get("www.simpsons.com"));
System.out.println("apple.com: " + st.get("www.apple.com"));
System.out.println("ebay.com: " + st.get("www.ebay.com"));
System.out.println("dell.com: " + st.get("www.dell.com"));
System.out.println();
System.out.println("size: " + st.size());
System.out.println("height: " + st.height());
System.out.println(st);
System.out.println();
}
}
5 图
5.1 无向图(Graph)
package ds.impl;
public class Graph {
private static final String NEW_LINE = System.getProperty("line.separator");
private final int V;
private int E;
private LinkedBag<Integer>[] adj;
public Graph(int V) {
if (V < 0) {
throw new IllegalArgumentException();
}
this.V = V;
this.E = 0;
adj = (LinkedBag<Integer>[]) new LinkedBag[V];
for (int v = 0; v < V; v++) {
adj[v] = new LinkedBag<>();
}
}
public Graph(Graph G) {
this(G.V());
this.E = G.E();
for (int v = 0; v < V; v++) {
LinkedStack<Integer> reverse = new LinkedStack<>();
for (int w : G.adj[v]) {
reverse.push(w);
}
for (int w : reverse) {
adj[v].add(w);
}
}
}
public int V() {
return V;
}
public int E() {
return E;
}
private void validateVertex(int v) {
if (v < 0 || v >= V) {
throw new IllegalArgumentException();
}
}
public void addEdge(int v1, int v2) {
validateVertex(v1);
validateVertex(v2);
E++;
adj[v1].add(v2);
adj[v2].add(v1);
}
public Iterable<Integer> adj(int v) {
validateVertex(v);
return adj[v];
}
public int degree(int v) {
validateVertex(v);
return adj[v].size();
}
@Override
public String toString() {
StringBuilder s = new StringBuilder();
s.append("V=").append(V).append(", E=").append(E).append(NEW_LINE);
for (int v = 0; v < V; v++) {
s.append(v).append(": ");
for (int w : adj[v]) {
s.append(w).append(" ");
}
s.append(NEW_LINE);
}
return s.toString();
}
public static void main(String[] args) {
Graph g = new Graph(4);
g.addEdge(1, 2);
g.addEdge(2, 3);
System.out.println(g);
}
}
5.2 有向图(Digraph)
package ds.impl;
public class Digraph {
private static final String NEW_LINE = System.getProperty("line.separator");
private final int V;
private int E;
private LinkedBag<Integer>[] adj;
private int[] indegree;
public Digraph(int V) {
if (V < 0) {
throw new IllegalArgumentException();
}
this.V = V;
this.E = 0;
indegree = new int[V];
adj = (LinkedBag<Integer>[]) new LinkedBag[V];
for (int v = 0; v < V; v++) {
adj[v] = new LinkedBag<>();
}
}
public Digraph(Digraph G) {
this(G.V());
this.E = G.E();
for (int v = 0; v < V; v++) {
this.indegree[v] = G.indegree[v];
}
for (int v = 0; v < G.V(); v++) {
LinkedStack<Integer> reverse = new LinkedStack<>();
for (int w : G.adj[v]) {
reverse.push(w);
}
for (int w : reverse) {
adj[v].add(w);
}
}
}
public int V() {
return V;
}
public int E() {
return E;
}
private void validateVertex(int v) {
if (v < 0 || v >= V) {
throw new IllegalArgumentException();
}
}
public void addEdge(int v1, int v2) {
validateVertex(v1);
validateVertex(v2);
adj[v1].add(v2);
indegree[v2]++;
E++;
}
public Iterable<Integer> adj(int v) {
validateVertex(v);
return adj[v];
}
public int outdegree(int v) {
validateVertex(v);
return adj[v].size();
}
public int indegree(int v) {
validateVertex(v);
return indegree[v];
}
public Digraph reverse() {
Digraph result = new Digraph(V);
for (int v = 0; v < V; v++) {
for (int w : adj[v]) {
result.addEdge(w, v);
}
}
return result;
}
@Override
public String toString() {
StringBuilder s = new StringBuilder();
s.append("V=").append(V).append(", E=").append(E).append(NEW_LINE);
for (int v = 0; v < V; v++) {
s.append(v).append(": ");
for (int w : adj[v]) {
s.append(w).append(" ");
}
s.append(NEW_LINE);
}
return s.toString();
}
public static void main(String[] args) {
Digraph g = new Digraph(4);
g.addEdge(1, 2);
g.addEdge(2, 3);
System.out.println(g);
}
}
6 贪心
6.1 Dijkstra算法-单元最短路径
6.1.1 问题描述
给定一个有向带权图
G=(V，E)，其中每条边的权是一个非负实数。另外，给定V中的一个顶点，称为源点。现在要计算从源点到所有其它各个顶点的最短路径长度，这里的路径长度是指路径上经过的所有边上的权值之和。
6.1.2 贪心策略
Dijkstra算法思想: 按各个顶点与源点之间路径长度的递增次序，生成源点到各个顶点的最短路径的方法，即先求出长度最短的一条路径，再参照它求出长度次短的一条路径，依此类推，直到从源点到其它各个顶点的最短路径全部求出为止。
6.1.3 算法设计
- 源点u 。集合S中的顶点到源点的最短路径的长度已经确定，集合V-S中所包含的顶点到源点的最短路径的长度待定。
- 特殊路径：从源点出发只经过S中的点到达V-S中的点的路径。
- 贪心策略：选择特殊路径长度最短的路径，将其相连的V-S中的顶点加入到集合S中。
- 求解步骤:*
- 步骤1：设计合适的数据结构。带权邻接矩阵C，即如果< u, x >E，令
C[u][x]=<u, x >的权值，否则，C[u][x]=无穷；采用数组dist来记录从源点到其它顶点的最短路径长度；采用数组p来记录最短路径； - 步骤2：初始化。令集合S={u}，对于集合V-S中的所有顶点x，设置
dist[x]=C[u][x]；如果顶点i与源点相邻，设置p[i]=u，否则p[i]=-1； - 步骤3：在集合V-S中依照贪心策略来寻找使得dist[x]具有最小值的顶点t，t就是集合V-S中距离源点u最近的顶点。
- 步骤4：将顶点t加入集合S中，同时更新集合V-S；
- 步骤5：如果集合V-S为空，算法结束；否则，转步骤6；
- 步骤6：对集合V-S中的所有与顶点t相邻的顶点x，如果
dist[x]>dist[t]+C[t][x]，则dist[x]=dist[t]+C[t][x]并设置p[x]=t。转步骤3。6.1.4 示例
public class Dijkstra { private static final int N = Integer.MAX_VALUE; private static final int[][] Graph = { {0, 1, 5, N, N, N, N, N, N}, {1, 0, 3, 7, 5, N, N, N, N}, {5, 3, 0, N, 1, 7, N, N, N}, {N, 7, N, 0, 2, N, 3, N, N}, {N, 5, 1, 2, 0, 3, 6, 9, N}, {N, N, 7, N, 3, 0, N, 5, N}, {N, N, N, 3, 6, N, 0, 2, 7}, {N, N, N, N, 9, 5, 2, 0, 4}, {N, N, N, N, N, N, 7, 4, 0}}; public static void main(String[] args) { dijkstra(0, Graph); } /** * Dijkstra最短路径。 即图中"节点vs"到其它各个节点的最短路径。 * * @param vs 起始节点 * @param Graph 图 */ public static void dijkstra(int vs, int[][] Graph) { int NUM = Graph[0].length; // 前驱节点数组 int[] prenode = new int[NUM]; // 最短距离数组 int[] mindist = new int[NUM]; // 该节点是否已经找到最短路径 boolean[] find = new boolean[NUM]; int vnear = 0; for (int i = 0; i < mindist.length; i++) { prenode[i] = i; mindist[i] = Graph[vs][i]; find[i] = false; } find[vs] = true; for (int v = 1; v < Graph.length; v++) { // 每次循环求得距离vs最近的节点vnear和最短距离min int min = N; for (int j = 0; j < Graph.length; j++) { if (!find[j] && mindist[j] < min) { min = mindist[j]; vnear = j; } } find[vnear] = true; // 根据vnear修正vs到其他所有节点的前驱节点及距离 for (int k = 0; k < Graph.length; k++) { if (!find[k] && (min + Graph[vnear][k]) < mindist[k]) { prenode[k] = vnear; mindist[k] = min + Graph[vnear][k]; } } } for (int i = 0; i < NUM; i++) { System.out.println("v" + vs + "...v" + prenode[i] + "->v" + i + ", s=" + mindist[i]); } }
}
## 7 动态规划
### 7.1 最长公共子序列问题
```java
package net.zhaoxuyang.common.algorithm.dp;
/**
* 最长公共子序列
*
* @author zhaoxuyang
*/
public class LastestCommonSequence {
public static void main(String[] args) {
String x = "ABCBDAB";
String y = "BDCABA";
int m = x.length();
int n = y.length();
int[][] c = new int[m + 1][n + 1];
int[][] b = new int[m + 1][n + 1];
lscLength(x.toCharArray(), y.toCharArray(), c, b);
// for (int i = 0; i <= x.length(); i++) {
// for (int j = 0; j <= y.length(); j++) {
// System.out.print(b[i][j]+" ");
// }
// System.out.println();
// }
lcs(x.length(), y.length(), x.toCharArray(), b);
}
/**
* <pre>
* ------------------------------------------------------------
* = 0 i=0 || j=0
* c[i][j] = c[i-1][j-1]+1 i,j>0 && x[i]=y[j]
* = max(c[i][j-1],c[i-1][j]) i,j>0 && x[i]!=y[j]
*
* b[i][j]=1 表示c[i][j]由c[i-1][j-1]+1 得到
* b[i][j]=2 表示c[i][j]由c[i][j-1] 得到
* b[i][j]=3 表示c[i][j]由c[i-1][j] 得到
* ------------------------------------------------------------
* i行j列，初始时c[i][0]=0,c[0][j]=0
* B D C A B A
* 0 0 0 0 0 0 0
* A 0
* B 0
* C 0
* B 0
* D 0
* A 0
* B 0
* ------------------------------------------------------------
* </pre>
*
* @param x X序列
* @param y Y序列
* @param c 存入最优值
* @param b 存入最优值的来源
*/
private static void lscLength(char[] x, char[] y, int[][] c, int[][] b) {
for (int i = 1; i <= x.length; i++) {
for (int j = 1; j <= y.length; j++) {
if (x[i - 1] == y[j - 1]) {
c[i][j] = c[i - 1][j - 1] + 1;
b[i][j] = 1;
} else if (c[i][j - 1] > c[i - 1][j]) {
c[i][j] = c[i][j - 1];
b[i][j] = 2;
} else {
c[i][j] = c[i - 1][j];
b[i][j] = 3;
}
}
}
}
/**
* 根据记录下的信息构造最优解
*/
private static void lcs(int i, int j, char[] x, int[][] b) {
if (i == 0 || j == 0) {
return;
}
switch (b[i][j]) {
case 1:
lcs(i - 1, j - 1, x, b);
visit(x[i - 1]);
break;
case 2:
lcs(i, j - 1, x, b);
break;
default:
lcs(i - 1, j, x, b);
break;
}
}
/**
* 如何处理最优解，例如打印，或者添加到StringBuilder中
* @param c
*/
private static void visit(char c) {
System.out.print(c);
}
}
7.2 0-1背包问题
package net.zhaoxuyang.common.algorithm.dp;
import java.util.Arrays;
/**
* 0-1 背包问题
*
* <pre>
* n个物品和1个背包，对物品i，价值为v[i]，重量为w[i]，背包容量为W，如何装入使得总价值最大:
* - w[i]x[i]小于等于W
* - x[i]∈{0,1}
* - 目标函数为max(v[i]x[i])
* - 其中i∈[1,n]
*
* 假如(x[1], x[2], x[3], ..., x[n])是最优解，
* 那么(x[2], x[3], ..., x[n])则是以下问题的一个最优解：
* - w[i]x[i] 小于等于 W - w[1]x[1]
* - x[i]∈{0,1}
* - 目标函数为max(v[i]x[i])
* - 其中i∈[2,n]
*
* </pre>
*
* 时间复杂度为O(nW)
* 缺点：要求w数组中的元素为整数；当W>2^n时，时间复杂度为O(n2^n)
* @author zhaoxuyang
*/
public class KnapSack {
public static void main(String[] args) {
int[] w = {2, 2, 6, 5, 4};
int[] v = {6, 3, 5, 4, 6};
int W = 10;
byte[] result = fun(w, v, W);
System.out.println(Arrays.toString(result));
}
/**
*
* <pre>
* 数组c[w.length+1][W+1]存放每次迭代的执行结果
* 数组x[w.length]存放所装入的背包的物品状态
* @初始化 c[0][j] = c[i][0] = 0
*
* @递归式 c[i][j] = c[i-1][j] j 小于 w[i]
* = max{c[i-1][j],c[i-1][j-w[i]]+v[i]} j大于等于w[i]
* </pre>
* @param w 重量
* @param v 价值
* @param W 容量
* @return 最优解
*/
private static byte[] fun(int[] w, int[] v, int W) {
int row = w.length;
int col = W;
byte[] x = new byte[row];
int[][] c = new int[row + 1][col + 1];
//存放各个子问题的最优值
for (int i = 1; i <= row; i++) {
for (int j = 1; j <= col; j++) {
if (j < w[i - 1]) {
c[i][j] = c[i - 1][j];
} else {
int tmp = c[i - 1][j - w[i - 1]] + v[i - 1];
c[i][j] = Math.max(c[i - 1][j], tmp);
}
}
}
//构造最优解
for (int i = row, j = col; i > 0; i--) {
if (c[i][j] > c[i - 1][j]) {
x[i - 1] = 1;
j -= w[i - 1];
}
}
return x;
}
}
7.3 加工顺序问题
package net.zhaoxuyang.common.algorithm.dp;
import java.util.ArrayList;
import java.util.List;
/**
* 加工顺序
* 先a后b的加工顺序，给定a和b各工件的耗时，求加工耗时最短的序列。
* 解法：
* 1. 求a中小于b的位置组g1：1,4,6；a中大于等于a的位置组g2：2,3,5,7
* 2. 对g1非减序排序，对g2非增序排序
* 3. 将g1连接上g2
* @author zhaoxuyang
*/
public class ProcessingSquence {
public static void main(String[] args) {
int[] a = {3,8,10,12,6,9,15};
int[] b = {7,2,6,18,3,10,4};
List<Integer> result = flowShop(a,b);
System.out.println(result);
}
private static List<Integer> flowShop(int[] a, int[] b) {
int len = a.length;
List<Integer> g1 = new ArrayList<>();
List<Integer> g2 = new ArrayList<>();
for(int i=0;i<len;i++){
if(a[i]<b[i]){
g1.add(a[i]);
}else{
g2.add(b[i]);
}
}
g1.sort((left,right)->{
return left-right;
});
g2.sort((left,right)->{
return right-left;
});
g1.addAll(g2);
return g1;
}
}
8 搜索法
8.1 图的着色问题
package net.zhaoxuyang.common.algorithm.search;
/**
* 图的m着色问题
* @author zhaoxuyang
*/
public class MColoring {
public static void main(String[] args) {
int n = 5;
int m = 5;
int[][] a = {
{-1, -1, -1, -1, -1, -1},
{-1, 0, 1, 1, 1, 0},
{-1, 1, 0, 1, 1, 1},
{-1, 1, 1, 0, 1, 0},
{-1, 1, 1, 1, 0, 1},
{-1, 0, 1, 0, 1, 0}
};
MColoring c = new MColoring();
long sum = c.coloring(m, n, a);
System.out.println(sum);
}
int n; // 图的顶点数
int m; // 可用的颜色数
int[][] a;//图的邻接矩阵
int[] x;//当前解
long sum;//当前已找到的可m着色方案数
public long coloring(int m, int n, int[][] a) {
this.n = n;
this.a = a;
x = new int[n + 1];
this.m = m;
sum = 0;
backtrack(1);
return sum;
}
private void backtrack(int t) {
if (t > n) {
sum++;
for (int i = 1; i <= n; i++) {
visit(x[i]);
}
System.out.println();
} else {
for (int i = 1; i <= m; i++) {
x[t] = i;
if (ok(t)) {
backtrack(t + 1);
}
x[t] = 0;
}
}
}
private void visit(int item) {
System.out.printf("%d ", item);
}
private boolean ok(int k) {
for (int j = 1; j <= n; j++) {
if (a[k][j] == 1 && x[j] == x[k]) {
return false;
}
}
return true;
}
}
8.2 深度优先搜索
bool Visited[n + 1];
for(int i = 1; i <= n; i++){
Visited[i] = 0;
}
//从顶点k出发进行深度优先搜索
void Dfsk(int k){
Visited[k] = 1;
for(int j = 1; j <= n; j++){
if(c[k][j] == 1 && Visited[j] == 0){
Dfsk(j);
}
}
}
//深度优先搜索整个图
void Dfs(){
for(int i = 1; i <= n; i++){
if(Visited[i] == 0){
Dfsk(i);
}
}
} 8.3 回溯法
8.3.1 回溯法的算法框架
为定义搜索范围，应明确以下几个方面：
- 问题解的形式：表示成一个n元组的形式
(x[0],x[1],...,x[n-1]) - 显约束：对分量
x[i]的取值范围限定。 - 隐约束：为满足问题的解而对不同分量之间施加的约束。
- 解空间：对于问题的一个实例，解向量满足显约束的所有n元组构成了该实例的一个解空间。
在搜索的过程中，应了解几个名词：
- 扩展结点：一个正在生成孩子的结点。
- 活结点：一个自身已生成但其孩子还没有全部生成的结点。
- 死结点：一个所有孩子已生成的结点。
求解过程：
- 定义问题的解空间。
- 确定空间的组织结构。
- 搜索解空间：
- 确定约束条件
- 确定限界条件
- 搜索过程
算法描述：
(1)递归形式
// t为扩展节点在树中所处的层次
void Backtrack(int t){
if(t > n){
//搜索层次大于解空间的树的深度，说明找到了叶子结点，即找到了问题的一个解
output(x);
} else {
for (int i = s(n, t); i <= e(n, t); i++){
x[t] = d(i);
if(constraint(t) && bound(t)){
Backtrack(t + 1);
}
}
}
} (2)非递归形式
void NBacktrack(){
int t = 1;
while(t > 0){
if(s(n,t) <= e(n, t)){
for(int i = s(n, t); i <= e(n,t); i++){
x[t] = d(i);
if(constraint(t) && bound(t)){
if(t > n){
output(x);
} else {
t++;
}
}
}
} else {
t--;
}
}
} 8.3.2 子集树
当所给问题是从n个元素组成的集合S中找出满足某一性质的一个子集时，相应的解空间树称为子集树。
- 0-1 背包问题
- 子集和问题
- 装载问题
- 最大团问题
算法描述：//x用来存放当前解，constraint()为约束函数，bound()为限界函数 void Backtrack(int t){ if(t > n){ output(x); } //判断能否沿着扩展结点的左分支进行扩展 if(constraint(t)){ //做相关标识 Backtrack(t + 1); //做相关标识的反操作 } //判断能否沿着扩展结点的右分支进行扩展 if(bount(t)){ //做相关标识 Backstrack(t + 1); //做相关标识的反操作 } }8.3.3 排列树
当所给的问题是从n个元素的排列中找出满足某种性质的一个排列时，相应的解空间称为排列树。
- n皇后问题
- 旅行商问题
- 批处理作业调度问题
- 圆排列问题
- 电路板排序问题
算法描述：void Backtrack(int t){ if(t > n){ output(x); } else { for(int i = t; i <= n; i++){ swap(x[t], x[i]); if(constraint(t) && bound(t)){ Backtrack(t + 1); } swap(x[t], x[i]); } } }8.3.4 满m叉树(组合树)
当所给问题的n个元素中每一个元素均有m种选择，要求确定其中的一种选择，
使得对这n个元素的选择结果组成的向量满足某种性质，即寻找满足某种特性的n个元素取值的一个组合。
这类问题的解空间称为满m叉树。 - n皇后问题
- 图的m着色问题
- 最小机器设计问题
算法描述：void Backtrack(int t){ if(t > n) { output(x); } else { for (int i = 1; i <= m; i++) { if (constraint(t) && bound(t)) { x[t] = i; //做其他相关标识 Backtrack(t + 1); //做其他相关标识的反操作，退出相关标识 } } } }8.4 广度优先搜索
算法描述：
bool Visited[n + 1];
for(int i = 1; i <= n; i++){
Visited[i] = 0;
}
void BFSVO(int v0){
int w;
visit(v0);
Visited[v0] = 1;
InitQueue(&Q);
InsertQueue(&Q, v0);
while(!Empty(Q)){
DeleteQueue(&Q, &v);
for(int i = 1; i <= n; i++) {
if(g[v][i] != 0){
w = i;
}
if(!Visited(w)){
visit(w);
Visited[w] = 1;
InsertQueue(&Q, w);
}
}
}
}
BFS(){
for(int i = 1; i <= n; i++) {
if(Visited[i] = 0){
BFSVO(i);
}
}
}
8.5 分支限界法
分支限界发先将根结点加入活结点表，接着从活结点表中取出根结点，使其成为当前扩展结点，
一次性生成所有孩子结点，判断孩子结点是舍弃还是保留，舍弃那些导致可行解或者导致非最优解的孩子结点，
其余的被保留在活结点表中。
再从活结点表中取出一个活结点，重复上述过程。
解题步骤：
- 定义问题的解空间
- 确定问题的解空间组织结构(树或图)
- 搜索解空间。搜索要定义判断标准(约束函数或限界函数)，如果选用优先队列，需确定优先级。
9 随机化算法
随机化算法分类：
- 数值随机化算法：在原理上可能就不存在精确解，或者无法在可行时间内求得，因此用该算法得到相当满意的解。
- 蒙特卡罗算法：能求得问题的一个解，但这个解未必是正确的。
- 拉斯维加斯算法：绝不返回错误的解，但有时可能找不到解。
- 舍伍德算法：当一个确定性算法在最坏与平均情况时间复杂度相差较大时，引入随机性来降低最坏情况出现的概率，不会改变结果。
9.1 数值随机化算法
package net.zhaoxuyang.common.algorithm.random;
/**
* 数值随机化算法
*
* @author zhaoxuyang
*/
public class Numerical {
public static void main(String[] args) {
System.out.println(calculatePI(1));
System.out.println(calculatePI(10));
System.out.println(calculatePI(100));
System.out.println(calculatePI(1000));
System.out.println(calculatePI(100000));
System.out.println(calculatePI(1000000));
System.out.println(calculatePI(10000000));
System.out.println(calculatePI(100000000));
System.out.println(calculateDefiniteIntegral(100000000));
}
/**
* 计算π的值
* @param n
* @return
*/
static double calculatePI(int n) {
double x;
double y;
int k = 0;
for (int i = 1; i <= n; i++) {
x = Math.random();
y = Math.random();
if ((x * x + y * y) <= 1) {
k++;
}
}
return 4.0 * k / n;
}
/**
* 计算定积分(y=x^2)
* @param n
* @return
*/
static double calculateDefiniteIntegral(int n) {
int k = 0;
double x;
double y;
for (int i = 0; i <= n; i++) {
x = Math.random();
y = Math.random();
double fx = 2 * x;
if (y <= fx) {
k++;
}
}
return 1.0 * k / n;
}
}
9.2 蒙特卡罗算法
package net.zhaoxuyang.common.algorithm.random;
/**
*
* 蒙特卡罗算法
* <pre>
* 设p是一个实数，且 0.5 小于 p 小于 1，
* 如果蒙特卡罗算法对于问题的任一实例得到的正确解的概率不小于p，则称该算法是正确的。
* </pre>
* @author zhaoxuyang
*/
public class MonteCarlo {
public static void main(String[] args) {
//int[] array = {1, 2, 1};
// System.out.println(majority(array, array.length, 0.99));
// System.out.println(majority(array, array.length, 0.99));
System.out.println(checkPrimeByWilson(5));
System.out.println(checkPrimeByWilson(6));
while(true){
System.out.println(checkPrimeByMoteCarlo(97));
}
}
/**
* 判断一个数组是否存在主元素 一个含有n个元素的数组，当存在一个元素占比大于n/2时，称该元素是数组的主元素。
*/
static boolean majority(int[] array, double n, double p) {
int k = (int) Math.ceil(Math.log(n) / Math.log(1 - 0.9));
for (int i = 1; i <= k; i++) {
if (checkMajority(array, n)) {
System.out.println(array[i]);
return true;
}
}
return false;
}
static boolean checkMajority(int[] array, double n) {
int randomIndex = (int) (Math.random() * n);
int item = array[randomIndex];
int k = 0;
for (int i = 0; i < n; i++) {
if (item == array[i]) {
k++;
}
}
return (k > 1.0 * n / 2);
}
/**
* 常规的判断一个数是否为素数
*
* @param n
* @return
*/
boolean checkPrime(long n) {
int m = (int) Math.floor(Math.sqrt(n));
for (int i = 2; i <= m; i++) {
if (n % i == 0) {
return false;
}
}
return true;
}
/**
*
* <pre>
* Wilson定理有很高的理论价值，定义为：对于给定的正整数n，判定n是素数的充要条件是：
* (n-1)! === -1(mod n)
* 例如n = 5,6,7
* (5-1)!=24, 24 mod 5 = -1(mod 5)，故5是素数
* (6-1)!=120, 120 mod 6 = 0(mod 6)，故6不是素数
* (7-1)!=720, 720 mod 7 = -1(mod 7)，故6不是素数
*
* </pre>
* @param n
* @return
*/
static boolean checkPrimeByWilson(long n) {
return fan(n - 1) % n == n - 1;
}
static long fan(long n) {
return n == 0 ? 1 : n * fan(n - 1);
}
static boolean checkPrimeByMoteCarlo(int n){
int m = (int) Math.floor(Math.sqrt(n));
int min = 2;
int max = m - 1;
int i = (int)(Math.random()*(max-min+1)+min);
System.out.println(i);
return n % i != 0;
}
} 9.3 拉斯维加斯算法
package net.zhaoxuyang.common.algorithm.random;
/**
*
* 拉斯维加斯算法
*
* <pre>
* void RLV(Type x, Type &y){
* bool success = fasle;
* while(!success){
* success = RLV(x, y);
* }
* }
* </pre>
*
* @author zhaoxuyang
*/
public class LasVegas {
public static void main(String[] args) {
split(182);
}
static void pollard(int n) {
int i = 1;
int x = (int) (Math.random() * n +1);
int y = x;
int k = 2;
while (true) {
i++;
x = (x * x - 1) % n;
int d = gcd(y - x, n);
if ((d > 1) && (d < n)) {
System.out.print(d + " ");
}
if (i == k) {
y = x;
k *= 2;
}
}
}
/**
* 整数因子分解
*/
static void split(int n) {
int k = (int) Math.floor(Math.sqrt(n));
for (int i = 2; i <= k; i++) {
if (n % i == 0) {
System.out.print(i + " ");
}
}
}
static int gcd(int a, int b) {
if (b == 0) {
return a;
}
return gcd(b, a % b);
}
} 9.4 舍伍德算法
package net.zhaoxuyang.common.algorithm.random;
import java.util.Arrays;
import net.zhaoxuyang.common.algorithm.other.QuickSort;
import net.zhaoxuyang.common.algorithm.other.Shuffle;
/**
*
* 舍伍德算法
* @author zhaoxuyang
*/
public class Sherwood {
public static void main(String[] args){
int[] array = {1,2,3,4,5,6,7,8,7,6};
randomQuickSort(array);
System.out.println(Arrays.toString(array));
}
/**
* 打乱顺序，再排序
* @param array
*/
static void randomQuickSort(int[] array){
Shuffle.shuffle(array);
QuickSort.quickSort(array);
}
}
10 数论算法
10.1 Stein求最大公约数
package net.zhaoxuyang.common.algorithm.math;
/**
*
* @author zhaoxuyang
*/
public class Stein {
public static void main(String[] args) {
System.out.println(gcd(2412122241212121212L, 2131424432543544656L));
}
static long gcd(long a, long b) {
if (a == 0) {
return b;
}
if (b == 0) {
return a;
}
if (a % 2 == 0 && b % 2 == 0) {
return 2 * gcd(a >> 1, b >> 1);
} else if (a % 2 == 0) {
return gcd(a >> 1, b);
} else if (b % 2 == 0) {
return gcd(a, b >> 1);
} else {
return gcd(Math.abs(a - b), Math.min(a, b));
}
}
}
10.2 矩阵求斐波那切数列
package net.zhaoxuyang.common.algorithm.other;
/**
* 暴力递归，记忆搜索，矩阵求法，流处理求斐波那契数列
* 结论：
* （1）流处理并没有想象中的快
* （2）暴力递归效率及其低下，N稍微大点（40左右）就已经算不出结果
* （3）矩阵求法并没有记忆搜索快，但是在计算量较大的【别的程序】中却是最快的
*/
import java.util.stream.Stream;
/**
*
* @author zhaoxuyang
*/
public class Fibonacci {
public static void main(String[] args) throws InterruptedException {
int n = 50;
int res;
long start;
// start = System.nanoTime();
// res = f1(n);
// System.out.println(res);
// System.out.println(System.nanoTime() - start);
start = System.nanoTime();
res = f2(n);
System.out.println(res);
System.out.println(System.nanoTime() - start);
start = System.nanoTime();
res = f3(n);
System.out.println(res);
System.out.println(System.nanoTime() - start);
start = System.nanoTime();
res = Stream.iterate(new int[]{0, 1}, t -> new int[]{t[1], t[0] + t[1]})
.skip(n)
.map(t -> t[0])
.findFirst()
.get();
System.out.println(res);
System.out.println(System.nanoTime() - start);
/*
n=20时，输出
6765
705588
6765
21391
6765
38888
6765
72026229
n=100时，注释掉O(2^N)的第一段因为要很久时间，所以后三个的输出
-980107325
166519
-980107325
43165
-980107325
72650035
*/
}
/**
* O(2^N)
*
* @param n
* @return
*/
public static int f1(int n) {
if (n < 1) {
return 0;
} else if (n == 1 || n == 2) {
return 1;
} else {
return f1(n - 1) + f1(n - 2);
}
}
/**
* 记忆搜索 O(N)
*
* @param n
* @return
*/
public static int f2(int n) {
if (n < 1) {
return 0;
}
if (n == 1 || n == 2) {
return 1;
}
int res = 1;
int pre = 1;
for (int i = 3; i <= n; i++) {
int tmp = res;
res = res + pre;
pre = tmp;
}
return res;
}
/**
* 矩阵求法 O(logN)
* <pre>
* 如果递归式严格遵循F(N)=F(N-1)+F(N-2)，
* 对于求第N项值，有矩阵乘法的方式可以将时间复杂度降至O(ogN)
* 二阶递推数列，状态矩阵为2*2的矩阵：
*
* (F(n),F(n-1)) = (F(n-1),F(n-2)) * | a b |
* | c d |
* 斐波那契数列的前4项代入求出状态矩阵：
* | a b | | 1 1 |
* | c d | = | 1 0 |
*
* (F(n),F(n-1)) = (F(n-1), F(n-2)) * | 1 1 | = (1,1) * | 1 1 |^(n-2)
* | 1 0 | | 1 0 |
* 问题变成求矩阵N次方
* 以整数10的75次方为例：
* 75的二进制为1001011，则10的75次方=10^64 * 10^8 * 10^2 * 10^1
* 把累乘的值相乘即可
*
* 对于矩阵，求矩阵m的p次方请参看matrixPower方法，其中muliMatrix是两个矩阵相乘的具体实现
* </pre>
*
* @param n
* @return
*/
public static int f3(int n) {
if (n < 1) {
return 0;
}
if (n == 1 || n == 2) {
return 1;
}
int[][] base = {{1, 1}, {1, 0}};
int[][] res = matrixPower(base, n - 2);
return res[0][0] + res[1][0];
}
/**
* @param m
* @param p
* @return
*/
private static int[][] matrixPower(int[][] m, int p) {
int[][] res = new int[m.length][m[0].length];
for (int i = 0; i < res.length; i++) {
res[i][i] = 1;
}
int[][] tmp = m;
for (; p != 0; p >>= 1) {
if ((p & 1) != 0) {
res = muliMatrix(res, tmp);
}
tmp = muliMatrix(tmp, tmp);
}
return res;
}
private static int[][] muliMatrix(int[][] m1, int[][] m2) {
int[][] res = new int[m1.length][m2[0].length];
for (int i = 0; i < m2[0].length; i++) {
for (int j = 0; j < m1.length; j++) {
for (int k = 0; k < m2.length; k++) {
res[i][j] += m1[i][k] * m2[k][j];
}
}
}
return res;
}
}
看哭了，恋爱那一段你女朋友说你不够关心她，可你毕竟也愿意遇到矛盾写几千字来和她慢慢分析；说不愿意给她花钱，我感觉可能只是消费观不一样；如果她想留在长沙，也应该提前跟你说开。不过她也许会心疼你放弃大厂offer转向数字马力？我也因为同样的原因有过一段幸福而充满遗憾的感情，不过跟爱情相比确实前途更重要一点。至于offer的选择，换我我也会这么选。把这些旧事记录下来以后，接下来就好好向前看吧，加油兄弟
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