Eratosrhenes筛选法
1简介
埃拉托色尼选筛法(the Sieve of Eratosthenes)简称埃氏筛法,是古希腊数学家埃拉托色尼(Eratosthenes 274B.C.~194B.C.)提出的一种筛选法。 是针对自然数列中的自然数而实施的,用于求一定范围内的质数,它的容斥原理之完备性条件是p=H~。
2步骤
(1)先把1删除(现今数学界1既不是质数也不是合数)
(2)读取队列中当前最小的数2,然后把2的倍数删去
(3)读取队列中当前最小的数3,然后把3的倍数删去
(4)读取队列中当前最小的数5,然后把5的倍数删去
(5)如上所述直到需求的范围内所有的数均删除或读取
该算法是用空间换时间,代码实现如下
#include <stdio.h> #include <stdlib.h> #define SIZE 10000 #define TRUE 1 #define FALSE 0 int main() { char sieve[ SIZE ]; /* the sieve */ char *sp; /* pointer to access the sieve */ int number; /* number we’re computing */ /* ** Set the entire sieve to TRUE. */ for( sp = sieve; sp < &sieve[ SIZE ]; ) *sp++ = TRUE; /* Solution 6.4 continued . . . Pointers on C—Instructor′s Guide 33 ** Process each number from 3 to as many as the sieve holds. (Note: the ** loop is terminated from inside.) */ for( number = 3; ; number += 2 ){ /* ** Set the pointer to the proper element in the sieve, and stop ** the loop if we’ve gone too far. */ sp = &sieve[0]+(number-3) / 2; if( sp >= &sieve[ SIZE ] ) break; /* ** Now advance the pointer by multiples of the number and set ** each subsequent entry FALSE. */ while( sp += number, sp < &sieve[ SIZE ] ) *sp = FALSE; } /* ** Go through the entire sieve now and print the numbers corresponding ** to the locations that remain TRUE. */ number=2; printf( "%8d", number ); for( number = 3, sp = &sieve[ 0 ]; sp < &sieve[ SIZE ]; number += 2, sp++ ){ if( *sp ) printf( "%8d", number ); } return EXIT_SUCCESS; }
由于除了2之外,所有偶数都不是质数,所以令数组中的元素只对应奇数,可使程序的空间效率提高一倍。
n(数组下标) 0 1 2 3 4 5 6 7 8 9 10
s(奇数) 3 5 7 9 11 13 15 17 19 21 23
显然s=2*n+3 由于n增加1时,s增加2.
所以对s进行筛选时
读取3,然后将2*3=6的倍数删去(因为是对奇数进行筛选)——》对n将3k+0的数删去
所以对下标n进行筛选时,可删去3k,1+5k,2+7k。。。。。
for( number = 3; ; number += 2 ){ /* ** Set the pointer to the proper element in the sieve, and stop ** the loop if we’ve gone too far. */ sp = &sieve[0]+(number-3) / 2; if( sp >= &sieve[ SIZE ] ) break; /* ** Now advance the pointer by multiples of the number and set ** each subsequent entry FALSE. */ while( sp += number, sp < &sieve[ SIZE ] ) *sp = FALSE; }