为什么这些筛子优化破坏了我的代码？

Question

我怎样才能纠正他们的工作呢？我试图根据以前的建议优化我的筛子，但在这两种情况下，代码都会中断：

递增j = j + ( i * 2)会破坏代码。

显然，我和其他人一样，也错过了一些关于优化的概念。但是一般说来，你只需要把一个素数的所有倍数标记为非素数。优化是下一步。

​

// prime-3
// sieve implementation
function prime3(n) {
  const sieve = (new Array(n)).fill(true);
  // Only iterate up to the square root n for marking
  for (let i = 2; i * i <= n; i += 1) {
    if (sieve[i]) {
      // recommended optimization breaks the code
      // j = j + ( i * 2 )
      for (let j = i * i; j <= n; j = j + i) {
        sieve[j] = false;
      }
    }
  }
  return makePrimes(sieve, n);
};

function makePrimes(sieve, n) {
  let primes = [];
  for (let i = 2; i < n; i++) {
    if (sieve[i]) {
      primes.push(i);
    }
  }
  return primes;
}
console.log(prime3(100));

​

Answer 1

你犯了一个简单的错误，没有准备你的筛子。首先，您应该消除2的所有倍数：

function makeSieve(n){
  const sieve = new Array(n).fill(true);
  for(let i = 2; i < sieve.length; i += 2){
    sieve[i] = false;
  }
}

现在，当您进入标记非素数时，可以增加i * 2。

例如

3, 6, 9, 12, 15, 18, 21

会变成

3, 9, 15, 21

Answer 2

您的优化需要应用于先去掉偶数之后(2除外)。因为当使用i==2时，可以通过i*2增量有效地跳过所有偶数。

下面是一个工作代码：

// prime-3
// sieve implementation
function prime3(n) {
  let sieve = (new Array(n)).fill(true);
  for (let i = 4; i < n; i+=2) {
    sieve[i] = false;
  }
  
  // Only iterate up to the square root n for marking
  for (let i = 2; i * i <= n; i += 1) {
    if (sieve[i]) {
      // now it works
      // j = j + ( i * 2 )
      for (let j = i * i; j <= n; j = j + i*2) {
        sieve[j] = false;
      }
    }
  }
  return makePrimes(sieve, n);
};

function makePrimes(sieve, n) {
  let primes = [];
  for (let i = 2; i < n; i++) {
    if (sieve[i]) {
      primes.push(i);
    }
  }
  return primes;
}
console.log(prime3(100));

编辑

擦掉这个。进一步的测试表明，prime3比简单的筛子快3倍。

的代码虽然有效，但似乎花费了太多的技巧，并引入了额外的计算和混淆。在我的比较中，有一个简单的筛子代码执行答案中的一个。再说一次，亲吻是原则。

用原语答案317 1M来筛选1M的数字，而简单的筛子只需要241 1M。

function simpleSieve(n) {
  let a = new Array(n)
  let answer = []
  let p
  for (let i = 2; i < n; i ++) {
    a[i] = i
  }
  for (let i = 2; i < n; i++) {
    if (a[i]) {
      answer.push(a[i])
      p = a[i]
      for(let j = p; j < n;  j += p) {
        a[j] = 0
      }
    }
  }
  return answer
}

编辑2

用cpp和prime3进行重新测试确实比简单的筛子提高了大约3倍：

p3:
n = 100000000, t = 866717 microseconds.
n = 200000000, t = 2354425 microseconds.
n = 300000000, t = 3689165 microseconds.
n = 400000000, t = 4950224 microseconds.
n = 500000000, t = 6119779 microseconds.
n = 600000000, t = 7375925 microseconds.
n = 700000000, t = 8647293 microseconds.
n = 800000000, t = 10477116 microseconds.
n = 900000000, t = 11589894 microseconds.
n = 1000000000, t = 12806997 microseconds.
simple:
n = 100000000, t = 2316019 microseconds.
n = 200000000, t = 6063749 microseconds.
n = 300000000, t = 9783295 microseconds.
n = 400000000, t = 13315450 microseconds.
n = 500000000, t = 16640474 microseconds.
n = 600000000, t = 20282461 microseconds.
n = 700000000, t = 24219469 microseconds.
n = 800000000, t = 29203786 microseconds.
n = 900000000, t = 32965856 microseconds.
n = 1000000000, t = 37694084 microseconds.

这里的代码是完整的：

void simpleSieve(int n) {
  bool *a = (bool *)calloc(n, sizeof(bool));
  int p;
  memset(a, true, sizeof(bool) * n);
  for (int i = 2; i < n; i++) {
    if (a[i]) {
      p = i;
      for (int j = p; j < n; j += p) {
        a[j] = 0;
      }
    }
  }
  free(a);
}

void prime3(int n) {
  bool *sieve = (bool*)calloc(n, sizeof(bool));
  sieve[2] = true;
  for (int i = 3; i < n; i+=2) {
    sieve[i] = true;
  }
  int step;
  for (int i = 2; i * i <= n; i += 1) {
    if (sieve[i]) {
      step = i*2;
      for (int j = i * i; j <= n; j = j + step) {
        sieve[j] = false;
      }
    }
  }
  free(sieve);
}

Answer 3

实际上，优化对我来说是有效的：

​

 // prime-3
    // sieve implementation
    function prime3 (n) {
      const sieve = (new Array(n)).fill(true);
    
      // Only iterate up to the square root n for marking
      for (let i = 2; i * i <= n; i += 1) {
        if (sieve[i]) {
    
          // recomended optimizations break the code
          // j = i * i 
          // j = j + ( i * 2 )
          for (let j = i * i; j <= n; j = j+i) {
            sieve[j] = false;
          }
        }
      }
      return makePrimes(sieve, n);
    };
    
    function makePrimes(sieve, n){
      let primes = [];
      for (let i = 2; i < n; i++) {
        if(sieve[i]) {
          primes.push(i);
        }
      }
      return primes;
    }
    
    console.log(prime3(100));

​

但是你的j = j + ( i * 2 )‘优化’实际上破坏了算法。结果将包含非素数。这里有一个要检查的片段：

​

function prime3 (n) {
      const sieve = (new Array(n)).fill(true);
    
      // Only iterate up to the square root n for marking
      for (let i = 2; i * i <= n; i += 1) {
        if (sieve[i]) {
    
          // recomended optimizations break the code
          // j = i * i 
          // j = j + ( i * 2 )
          for (let j = i * i; j <= n; j = j+(i*2)) {
            sieve[j] = false;
          }
        }
      }
      return makePrimes(sieve, n);
    };
    
    function makePrimes(sieve, n){
      let primes = [];
      for (let i = 2; i < n; i++) {
        if(sieve[i]) {
          primes.push(i);
        }
      }
      return primes;
    }
    
    console.log(prime3(100));

​