素?cái)?shù)是除了1和它本身之外再不能被其他數(shù)整除的自然數(shù)。由于找不到一個(gè)通項(xiàng)公式來表示所有的素?cái)?shù)，所以對于數(shù)學(xué)家來說，素?cái)?shù)一直是一個(gè)未解之謎。像著名的 哥德巴赫猜想、孿生素?cái)?shù)猜想，幾百年來不知吸引了世界上多少優(yōu)秀的數(shù)學(xué)家。盡管他們苦心鉆研，嘔心瀝血，但至今仍然未見分曉。
求素?cái)?shù)的方法有很多種，最簡單的方法是根據(jù)素?cái)?shù)的定義來求。對于一個(gè)自然數(shù)N，用大于1小于N的各個(gè)自然數(shù)都去除一下N，如果都除不盡，則N為素?cái)?shù)，否則N為合數(shù)。
第三，對于N來說，不必用從2到N一1的所有素?cái)?shù)去除，只需用小于等于√N(yùn)(根號N)的所有素?cái)?shù)去除就可以了。這一點(diǎn)可以用反證法來證明：
首先，將2，3，5，7分別存放在a[1]、a[2]、a[3]、a[4]中，以后每求出一個(gè)素?cái)?shù)，只要不大于100，就依次存放在A數(shù)組中的一個(gè)單元 中。當(dāng)我們求100—10000之間的素?cái)?shù)時(shí)，可依次用a[1]－a[2]的素?cái)?shù)去試除N，這個(gè)范圍內(nèi)的素?cái)?shù)可以不保存，直接打印。
簡單介紹一下厄拉多塞篩法。厄拉多塞是一位古希臘數(shù)學(xué)家，他在尋找素?cái)?shù)時(shí)，采用了一種與眾不同的方法：先將2－N的各數(shù)寫在紙上：
在2的上面畫一個(gè)圓圈，然后劃去2的其他倍數(shù)；第一個(gè)既未畫圈又沒有被劃去的數(shù)是3，將它畫圈，再劃去3的其他倍數(shù)；現(xiàn)在既未畫圈又沒有被劃去的第一個(gè)數(shù) 是5，將它畫圈，并劃去5的其他倍數(shù)……依次類推，一直到所有小于或等于N的各數(shù)都畫了圈或劃去為止。這時(shí)，表中畫了圈的以及未劃去的那些數(shù)正好就是小于 N的素?cái)?shù)。
這很像一面篩子，把滿足條件的數(shù)留下來，把不滿足條件的數(shù)篩掉。由于這種方法是厄拉多塞首先發(fā)明的，所以，后人就把這種方法稱作厄拉多塞篩法。
在計(jì)算機(jī)中，篩法可以用給數(shù)組單元置零的方法來實(shí)現(xiàn)。具體來說就是：首先開一個(gè)數(shù)組：a[i]，i=1，2，3，…，同時(shí)，令所有的數(shù)組元素都等于下標(biāo) 值，即a[i]=i，當(dāng)i不是素?cái)?shù)時(shí)，令a[i]=0 。當(dāng)輸出結(jié)果時(shí)，只要判斷a[i]是否等于零即可，如果a[i]=0，則令i=i+1，檢查下一個(gè)a[i]。
顯然，當(dāng)N≥1時(shí)，6N，6N+2，6N+3，6N+4都不是素?cái)?shù)，只有形如6N+1和6N+5的自然數(shù)有可能是素?cái)?shù)。所以，除了2和3之外，所有的素?cái)?shù)都可以表示成6N±1的形式(N為自然數(shù))。
根據(jù)上述分析，我們可以構(gòu)造另一面篩子，只對形如6 N±1的自然數(shù)進(jìn)行篩選，這樣就可以大大減少篩選的次數(shù)，從而進(jìn)一步提高程序的運(yùn)行效率和速度。
在程序上，我們可以用一個(gè)二重循環(huán)實(shí)現(xiàn)這一點(diǎn)，外循環(huán)i按3的倍數(shù)遞增，內(nèi)循環(huán)j為0－1的循環(huán)，則2(i+j)-1恰好就是形如6N±1的自然數(shù)。
插入排序:
package org.rut.util.algorithm.support;
import org.rut.util.algorithm.SortUtil;
/**
?* @author treeroot
?* @since 2006-2-2
?* @version 1.0
?*/
public class InsertSort implements SortUtil.Sort{
??? /* (non-Javadoc)
???? * @see org.rut.util.algorithm.SortUtil.Sort#sort(int[])
???? */
??? public void sort(int[] data) {
??????? int temp;
??????? for(int i=1;i<data.length;i++){
??????????? for(int j=i;(j>0)&&(data[j]<data[j-1]);j--){
??????????????? SortUtil.swap(data,j,j-1);
??????????? }
??????? }???????
??? }
}
冒泡排序:
package org.rut.util.algorithm.support;
import org.rut.util.algorithm.SortUtil;
/**
?* @author treeroot
?* @since 2006-2-2
?* @version 1.0
?*/
public class BubbleSort implements SortUtil.Sort{
??? /* (non-Javadoc)
???? * @see org.rut.util.algorithm.SortUtil.Sort#sort(int[])
???? */
??? public void sort(int[] data) {
??????? int temp;
??????? for(int i=0;i<data.length;i++){
??????????? for(int j=data.length-1;j>i;j--){
??????????????? if(data[j]<data[j-1]){
??????????????????? SortUtil.swap(data,j,j-1);
??????????????? }
??????????? }
??????? }
??? }
}
選擇排序:
package org.rut.util.algorithm.support;
import org.rut.util.algorithm.SortUtil;
/**
?* @author treeroot
?* @since 2006-2-2
?* @version 1.0
?*/
public class SelectionSort implements SortUtil.Sort {
??? /*
???? * (non-Javadoc)
???? *
???? * @see org.rut.util.algorithm.SortUtil.Sort#sort(int[])
???? */
??? public void sort(int[] data) {
??????? int temp;
??????? for (int i = 0; i < data.length; i++) {
??????????? int lowIndex = i;
??????????? for (int j = data.length - 1; j > i; j--) {
??????????????? if (data[j] < data[lowIndex]) {
??????????????????? lowIndex = j;
??????????????? }
??????????? }
??????????? SortUtil.swap(data,i,lowIndex);
??????? }
??? }
}
Shell排序:
package org.rut.util.algorithm.support;
import org.rut.util.algorithm.SortUtil;
/**
?* @author treeroot
?* @since 2006-2-2
?* @version 1.0
?*/
public class ShellSort implements SortUtil.Sort{
??? /* (non-Javadoc)
???? * @see org.rut.util.algorithm.SortUtil.Sort#sort(int[])
???? */
??? public void sort(int[] data) {
??????? for(int i=data.length/2;i>2;i/=2){
??????????? for(int j=0;j<i;j++){
??????????????? insertSort(data,j,i);
??????????? }
??????? }
??????? insertSort(data,0,1);
??? }
??? /**
???? * @param data
???? * @param j
???? * @param i
???? */
??? private void insertSort(int[] data, int start, int inc) {
??????? int temp;
??????? for(int i=start+inc;i<data.length;i+=inc){
??????????? for(int j=i;(j>=inc)&&(data[j]<data[j-inc]);j-=inc){
??????????????? SortUtil.swap(data,j,j-inc);
??????????? }
??????? }
??? }
}
快速排序:
package org.rut.util.algorithm.support;
import org.rut.util.algorithm.SortUtil;
/**
?* @author treeroot
?* @since 2006-2-2
?* @version 1.0
?*/
public class QuickSort implements SortUtil.Sort{
??? /* (non-Javadoc)
???? * @see org.rut.util.algorithm.SortUtil.Sort#sort(int[])
???? */
??? public void sort(int[] data) {
??????? quickSort(data,0,data.length-1);???????
??? }
??? private void quickSort(int[] data,int i,int j){
??????? int pivotIndex=(i+j)/2;
??????? //swap
??????? SortUtil.swap(data,pivotIndex,j);
???????
??????? int k=partition(data,i-1,j,data[j]);
??????? SortUtil.swap(data,k,j);
??????? if((k-i)>1) quickSort(data,i,k-1);
??????? if((j-k)>1) quickSort(data,k+1,j);
???????
??? }
??? /**
???? * @param data
???? * @param i
???? * @param j
???? * @return
???? */
??? private int partition(int[] data, int l, int r,int pivot) {
??????? do{
?????????? while(data[++l]<pivot);
?????????? while((r!=0)&&data[--r]>pivot);
?????????? SortUtil.swap(data,l,r);
??????? }
??????? while(l<r);
??????? SortUtil.swap(data,l,r);???????
??????? return l;
??? }
}
改進(jìn)后的快速排序:
package org.rut.util.algorithm.support;
import org.rut.util.algorithm.SortUtil;
/**
?* @author treeroot
?* @since 2006-2-2
?* @version 1.0
?*/
public class ImprovedQuickSort implements SortUtil.Sort {
??? private static int MAX_STACK_SIZE=4096;
??? private static int THRESHOLD=10;
??? /* (non-Javadoc)
???? * @see org.rut.util.algorithm.SortUtil.Sort#sort(int[])
???? */
??? public void sort(int[] data) {
??????? int[] stack=new int[MAX_STACK_SIZE];
???????
??????? int top=-1;
??????? int pivot;
??????? int pivotIndex,l,r;
???????
??????? stack[++top]=0;
??????? stack[++top]=data.length-1;
???????
??????? while(top>0){
??????????? int j=stack[top--];
??????????? int i=stack[top--];
???????????
??????????? pivotIndex=(i+j)/2;
??????????? pivot=data[pivotIndex];
???????????
??????????? SortUtil.swap(data,pivotIndex,j);
???????????
??????????? //partition
??????????? l=i-1;
??????????? r=j;
??????????? do{
??????????????? while(data[++l]<pivot);
??????????????? while((r!=0)&&(data[--r]>pivot));
??????????????? SortUtil.swap(data,l,r);
??????????? }
??????????? while(l<r);
??????????? SortUtil.swap(data,l,r);
??????????? SortUtil.swap(data,l,j);
???????????
??????????? if((l-i)>THRESHOLD){
??????????????? stack[++top]=i;
??????????????? stack[++top]=l-1;
??????????? }
??????????? if((j-l)>THRESHOLD){
??????????????? stack[++top]=l+1;
??????????????? stack[++top]=j;
??????????? }
???????????
??????? }
??????? //new InsertSort().sort(data);
??????? insertSort(data);
??? }
??? /**
???? * @param data
???? */
??? private void insertSort(int[] data) {
??????? int temp;
??????? for(int i=1;i<data.length;i++){
??????????? for(int j=i;(j>0)&&(data[j]<data[j-1]);j--){
??????????????? SortUtil.swap(data,j,j-1);
??????????? }
??????? }??????
??? }
}
歸并排序:
package org.rut.util.algorithm.support;
import org.rut.util.algorithm.SortUtil;
/**
?* @author treeroot
?* @since 2006-2-2
?* @version 1.0
?*/
public class MergeSort implements SortUtil.Sort{
??? /* (non-Javadoc)
???? * @see org.rut.util.algorithm.SortUtil.Sort#sort(int[])
???? */
??? public void sort(int[] data) {
??????? int[] temp=new int[data.length];
??????? mergeSort(data,temp,0,data.length-1);
??? }
???
??? private void mergeSort(int[] data,int[] temp,int l,int r){
??????? int mid=(l+r)/2;
??????? if(l==r) return ;
??????? mergeSort(data,temp,l,mid);
??????? mergeSort(data,temp,mid+1,r);
??????? for(int i=l;i<=r;i++){
??????????? temp[i]=data[i];
??????? }
??????? int i1=l;
??????? int i2=mid+1;
??????? for(int cur=l;cur<=r;cur++){
??????????? if(i1==mid+1)
??????????????? data[cur]=temp[i2++];
??????????? else if(i2>r)
??????????????? data[cur]=temp[i1++];
??????????? else if(temp[i1]<temp[i2])
??????????????? data[cur]=temp[i1++];
??????????? else
??????????????? data[cur]=temp[i2++];???????????
??????? }
??? }
}
改進(jìn)后的歸并排序:
package org.rut.util.algorithm.support;
import org.rut.util.algorithm.SortUtil;
/**
?* @author treeroot
?* @since 2006-2-2
?* @version 1.0
?*/
public class ImprovedMergeSort implements SortUtil.Sort {
??? private static final int THRESHOLD = 10;
??? /*
???? * (non-Javadoc)
???? *
???? * @see org.rut.util.algorithm.SortUtil.Sort#sort(int[])
???? */
??? public void sort(int[] data) {
??????? int[] temp=new int[data.length];
??????? mergeSort(data,temp,0,data.length-1);
??? }
??? private void mergeSort(int[] data, int[] temp, int l, int r) {
??????? int i, j, k;
??????? int mid = (l + r) / 2;
??????? if (l == r)
??????????? return;
??????? if ((mid - l) >= THRESHOLD)
??????????? mergeSort(data, temp, l, mid);
??????? else
??????????? insertSort(data, l, mid - l + 1);
??????? if ((r - mid) > THRESHOLD)
??????????? mergeSort(data, temp, mid + 1, r);
??????? else
??????????? insertSort(data, mid + 1, r - mid);
??????? for (i = l; i <= mid; i++) {
??????????? temp[i] = data[i];
??????? }
??????? for (j = 1; j <= r - mid; j++) {
??????????? temp[r - j + 1] = data[j + mid];
??????? }
??????? int a = temp[l];
??????? int b = temp[r];
??????? for (i = l, j = r, k = l; k <= r; k++) {
??????????? if (a < b) {
??????????????? data[k] = temp[i++];
??????????????? a = temp[i];
??????????? } else {
??????????????? data[k] = temp[j--];
??????????????? b = temp[j];
??????????? }
??????? }
??? }
??? /**
???? * @param data
???? * @param l
???? * @param i
???? */
??? private void insertSort(int[] data, int start, int len) {
??????? for(int i=start+1;i<start+len;i++){
??????????? for(int j=i;(j>start) && data[j]<data[j-1];j--){
??????????????? SortUtil.swap(data,j,j-1);
??????????? }
??????? }
??? }
}
堆排序:
package org.rut.util.algorithm.support;
import org.rut.util.algorithm.SortUtil;
/**
?* @author treeroot
?* @since 2006-2-2
?* @version 1.0
?*/
public class HeapSort implements SortUtil.Sort{
??? /* (non-Javadoc)
???? * @see org.rut.util.algorithm.SortUtil.Sort#sort(int[])
???? */
??? public void sort(int[] data) {
??????? MaxHeap h=new MaxHeap();
??????? h.init(data);
??????? for(int i=0;i<data.length;i++)
??????????? h.remove();
??????? System.arraycopy(h.queue,1,data,0,data.length);
??? }
???? private static class MaxHeap{
????????
???????
??????? void init(int[] data){
??????????? this.queue=new int[data.length+1];
??????????? for(int i=0;i<data.length;i++){
??????????????? queue[++size]=data[i];
??????????????? fixUp(size);
??????????? }
??????? }
????????
??????? private int size=0;
??????? private int[] queue;
???????????????
??????? public int get() {
??????????? return queue[1];
??????? }
??????? public void remove() {
??????????? SortUtil.swap(queue,1,size--);
??????????? fixDown(1);
??????? }
??????? //fixdown
??????? private void fixDown(int k) {
??????????? int j;
??????????? while ((j = k << 1) <= size) {
??????????????? if (j < size && queue[j]<queue[j+1])
??????????????????? j++;
??????????????? if (queue[k]>queue[j]) //不用交換
??????????????????? break;
??????????????? SortUtil.swap(queue,j,k);
??????????????? k = j;
??????????? }
??????? }
??????? private void fixUp(int k) {
??????????? while (k > 1) {
??????????????? int j = k >> 1;
??????????????? if (queue[j]>queue[k])
??????????????????? break;
??????????????? SortUtil.swap(queue,j,k);
??????????????? k = j;
??????????? }
??????? }
??? }
}
?
SortUtil:
package org.rut.util.algorithm;
import org.rut.util.algorithm.support.BubbleSort;
import org.rut.util.algorithm.support.HeapSort;
import org.rut.util.algorithm.support.ImprovedMergeSort;
import org.rut.util.algorithm.support.ImprovedQuickSort;
import org.rut.util.algorithm.support.InsertSort;
import org.rut.util.algorithm.support.MergeSort;
import org.rut.util.algorithm.support.QuickSort;
import org.rut.util.algorithm.support.SelectionSort;
import org.rut.util.algorithm.support.ShellSort;
/**
?* @author treeroot
?* @since 2006-2-2
?* @version 1.0
?*/
public class SortUtil {
??? public final static int INSERT = 1;
??? public final static int BUBBLE = 2;
??? public final static int SELECTION = 3;
??? public final static int SHELL = 4;
??? public final static int QUICK = 5;
??? public final static int IMPROVED_QUICK = 6;
??? public final static int MERGE = 7;
??? public final static int IMPROVED_MERGE = 8;
??? public final static int HEAP = 9;
??? public static void sort(int[] data) {
??????? sort(data, IMPROVED_QUICK);
??? }
??? private static String[] name={
??????????? "insert","bubble","selection","shell","quick","improved_quick","merge","improved_merge","heap"
??? };
???
??? private static Sort[] impl=new Sort[]{
??????????? new InsertSort(),
??????????? new BubbleSort(),
??????????? new SelectionSort(),
??????????? new ShellSort(),
??????????? new QuickSort(),
??????????? new ImprovedQuickSort(),
??????????? new MergeSort(),
??????????? new ImprovedMergeSort(),
??????????? new HeapSort()
??? };
??? public static String toString(int algorithm){
??????? return name[algorithm-1];
??? }
???
??? public static void sort(int[] data, int algorithm) {
??????? impl[algorithm-1].sort(data);
??? }
??? public static interface Sort {
??????? public void sort(int[] data);
??? }
??? public static void swap(int[] data, int i, int j) {
??????? int temp = data[i];
??????? data[i] = data[j];
??????? data[j] = temp;
??? }
}