Eratosthenes的素数比同时更快？

Question

I am currently writing a program which first generates prime numbers by the Sieve of Eratosthenes sequentially, then concurrently. The concurrent version of the algorithm is supposed to be quicker than the sequential one, but in my case the concurrent version is approx. 10 times slower. I am wondering where I am putting the extra work on my threads, compared to the main thread in the sequential solution. Here's my program (prepare to read a bit!):

Primes.java :

public abstract class Primes {

    byte[] bitArr;
    int maxNum;
    final int[] BITMASK = { 1, 2, 4, 8, 16, 32, 64 };
    final int[] BITMASK2 = { 255 - 1, 255 - 2, 255 - 4, 255 - 8, 
                             255 - 16, 255 - 32, 255 - 64 };

    void setAllPrime() {
        for (int i = 0; i < bitArr.length; i++) {
            bitArr[i] = (byte) 127;
        }
    }

    void crossOut(int i) {
        bitArr[i/14] = (byte) (bitArr[i/14] - BITMASK[((i/2)%7)]);
    }

    boolean isPrime(int i) {
        if(i == 2){
            return true;
        }
        if((i%2) == 0){
            return false;
        }

        return (bitArr[i/14] & BITMASK[(i%14)>>1]) != 0;

    }

    int nextPrime(int i) {
        int k;
        if ((i%2) == 0){
            k =i+1;
        }
        else {
            k = i+2;
        }
        while (!isPrime(k) && k < maxNum){
            k+=2;
        }
        return k;
    }

    void printAllPrimes() {
        for (int i = 2; i <= maxNum; i++){
            if (isPrime(i)){
                System.out.println("Prime: " + i);
            }
        }
    }
}

PrimesSeq.java :

import java.util.ArrayList;

public class PrimesSeq extends Primes{

    PrimesSeq(int maxNum) {
        this.maxNum = maxNum;
        bitArr = new byte[(maxNum / 14) + 1];
        setAllPrime();
        generatePrimesByEratosthenes();
    }

    void generatePrimesByEratosthenes() {
        crossOut(1); // 1 is not a prime

        int curr = 3;
        while(curr < Math.sqrt(maxNum)){
            for(int i = curr*curr; i < maxNum; i+=2*curr){ 
                if(isPrime(i)){                // 2*curr because odd*2 = even!
                    crossOut(i);
                }
            }
            curr = nextPrime(curr);
        }
    }
}

PrimesPara.java :

import java.util.ArrayList;


public class PrimesPara extends Primes {

    PrimeThread[] threads;
    int processors;
    int currentState = 0;
    //0 = Init
    //1 = Generate primes after thread #0 finish
    //2 = Factorize

    public PrimesPara(int maxNum){
        this.maxNum = maxNum;
        this.processors = Runtime.getRuntime().availableProcessors();
        bitArr = new byte[(maxNum / 14) + 1];
        setAllPrime();
        this.threads = new PrimeThread[processors*2];
        generateErastothenesConcurrently();
        //printAllPrimes();
    }

    public void generateErastothenesConcurrently(){
        int[] starts = generateThreadIndexes();

        for(int i = 0; i < threads.length; i++){
            if(i != threads.length-1){
                threads[i] = new PrimeThread(starts[i], starts[i+1]-1, i);
            } else {
                threads[i] = new PrimeThread(starts[i], maxNum, i);
            }
        }

        //Start generating the first primes
        crossOut(1);
        Thread th = new Thread(threads[0]);
        th.start();
        try {
            th.join();
        } catch (InterruptedException e) {
            // TODO Auto-generated catch block
            e.printStackTrace();
        }
        currentState = 1;

        //Start generating the rest of the primes
        Thread[] thrs = new Thread[threads.length];
        for(int i = 0; i < thrs.length; i++){
            thrs[i] = new Thread(threads[i]);
            thrs[i].start();
        }
        for(int i = 0; i < thrs.length; i++){
            try {
                thrs[i].join();
            } catch (InterruptedException e) {
                e.printStackTrace();
            }
        }
        currentState = 2;
    }

    private int[] generateThreadIndexes(){
        int[] indexes = new int[processors*2];

        for(int i = 0; i < indexes.length; i++){
            indexes[i] = (i*((maxNum/(processors*2))));
        }

        indexes[indexes.length-1]++;

        return indexes;
    }

    public class PrimeThread implements Runnable {

        int start;
        int end;
        int thridx;

        public PrimeThread(int start, int end, int thridx){
            this.start = start;
            this.end = end;
            this.thridx = thridx;
        }

        public void run() {
            switch(currentState){
            case 0:
                generateSqrtPrimes();
                break;
            case 1:
                generateMyPrimes();
                break;
            case 2:
                break;
            }           
        }

        private void generateSqrtPrimes(){
            int curr = 3;           
            while(curr < Math.sqrt(maxNum)+1){
                for(int i = curr*curr; i < Math.sqrt(maxNum)+1; i+=2*curr){ 
                    if(isPrime(i)){                  // 2*curr because odd*2 = even!
                        crossOut(i);
                    }
                }
                curr = nextPrime(curr);
            }
        }

        private void generateMyPrimes(){
            int curr = start>(int)Math.sqrt(maxNum)?start:(int)Math.sqrt(maxNum);

            while(curr < end){
                for(int i = 3; i < Math.sqrt(maxNum)+1; i = nextPrime(i)){
                    if((curr%i) == 0){
                        if(isPrime(curr)){
                            crossOut(curr);
                        }
                    }
                }
                curr = nextPrime(curr);
            }
        }
    }
}

If someone could tell me where the bottleneck on the concurrent program is, I'd be very happy. Thanks in advance!

Answer 1

I am no JAVA coder so I stick with C++. Also this is not an direct answer to your question (sorry for that but I can not debug JAVA) take this as some pointers which way to go or check...

1. Sieves of Eratosthenes

   Parallelization is possible but not with big enough speed gain. Instead I use more sieve-tabs where each one have its own sub-divisions and each table size is an common multiply of all its sub-divisors. This way you need initiate tables just once and then just check them in O(1)

2. Parallelization

   After checking all of the sieves then I would use threads to do the obvious division testing for all of the unused divisors

3. Memoize

   If you have active table of all found primes then divide just by primes and add all new primes found

I am using non parallel prime search which is fast enough for me ...

• You can adapt this to your parallel code ...

[Edit1] updated code

//---------------------------------------------------------------------------
int bits(DWORD p)
    {
    DWORD m=0x80000000; int b=32;
    for (;m;m>>=1,b--)
     if (p>=m) break;
    return b;
    }
//---------------------------------------------------------------------------
DWORD sqrt(const DWORD &x)
    {
    DWORD m,a;
    m=(bits(x)>>1);
    if (m) m=1<<m; else m=1;
    for (a=0;m;m>>=1) { a|=m; if (a*a>x) a^=m; }
    return a;
    }
//---------------------------------------------------------------------------
List<int> primes_i32;                   // list of precomputed primes
const int primes_map_sz=4106301;        // max size of map for speedup search for primes max(LCM(used primes per bit)) (not >>1 because SOE is periodic at double LCM size and only odd values are stored 2/2=1)
const int primes_map_N[8]={ 4106301,3905765,3585337,4026077,3386981,3460469,3340219,3974653 };
const int primes_map_i0=33;             // first index of prime not used in mask
const int primes_map_p0=137;            // biggest prime used in mask
BYTE primes_map[primes_map_sz];         // factors map for first i0-1 primes
bool primes_i32_alloc=false;
int isprime(int p)
    {
    int i,j,a,b,an,im[8]; BYTE u;
    an=0;
    if (!primes_i32.num)                // init primes vars
        {
        primes_i32.allocate(1024*1024);
        primes_i32.add(  2); for (i=1;i<primes_map_sz;i++) primes_map[i]=255; primes_map[0]=254;
        primes_i32.add(  3); for (u=255-  1,j=  3,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add(  5); for (u=255-  2,j=  5,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add(  7); for (u=255-  4,j=  7,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 11); for (u=255-  8,j= 11,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 13); for (u=255- 16,j= 13,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 17); for (u=255- 32,j= 17,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 19); for (u=255- 64,j= 19,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 23); for (u=255-128,j= 23,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 29); for (u=255-  1,j=137,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 31); for (u=255-  2,j=131,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 37); for (u=255-  4,j=127,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 41); for (u=255-  8,j=113,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 43); for (u=255- 16,j= 83,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 47); for (u=255- 32,j= 61,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 53); for (u=255- 64,j=107,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 59); for (u=255-128,j=101,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 61); for (u=255-  1,j=103,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 67); for (u=255-  2,j= 67,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 71); for (u=255-  4,j= 37,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 73); for (u=255-  8,j= 41,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 79); for (u=255- 16,j= 43,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 83); for (u=255- 32,j= 47,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 89); for (u=255- 64,j= 53,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add( 97); for (u=255-128,j= 59,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add(101); for (u=255-  1,j= 97,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add(103); for (u=255-  2,j= 89,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add(107); for (u=255-  4,j=109,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add(109); for (u=255-  8,j= 79,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add(113); for (u=255- 16,j= 73,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add(127); for (u=255- 32,j= 71,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add(131); for (u=255- 64,j= 31,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        primes_i32.add(137); for (u=255-128,j= 29,i=j>>1;i<primes_map_sz;i+=j) primes_map[i]&=u;
        }

    if (!primes_i32_alloc)
        {
        if (p  <=1) return 0;               // ignore too small walues
        if (p&1==0) return 0;               // not prime if even
        if (p>primes_map_p0)
            {
            j=p>>1;
            i=j; if (i>=primes_map_N[0]) i%=primes_map_N[0]; if (!(primes_map[i]&  1)) return 0;
            i=j; if (i>=primes_map_N[1]) i%=primes_map_N[1]; if (!(primes_map[i]&  2)) return 0;
            i=j; if (i>=primes_map_N[2]) i%=primes_map_N[2]; if (!(primes_map[i]&  4)) return 0;
            i=j; if (i>=primes_map_N[3]) i%=primes_map_N[3]; if (!(primes_map[i]&  8)) return 0;
            i=j; if (i>=primes_map_N[4]) i%=primes_map_N[4]; if (!(primes_map[i]& 16)) return 0;
            i=j; if (i>=primes_map_N[5]) i%=primes_map_N[5]; if (!(primes_map[i]& 32)) return 0;
            i=j; if (i>=primes_map_N[6]) i%=primes_map_N[6]; if (!(primes_map[i]& 64)) return 0;
            i=j; if (i>=primes_map_N[7]) i%=primes_map_N[7]; if (!(primes_map[i]&128)) return 0;
            }
        }

    an=primes_i32[primes_i32.num-1];
    if (an>=p)
        {
        // linear table search
        if (p<127)  // 31st prime
            {
            if (an>=p) for (i=0;i<primes_i32.num;i++)
                {
                a=primes_i32[i];
                if (a==p) return 1;
                if (a> p) return 0;
                }
            }
        // approximation table search
        else{
            for (j=1,i=primes_i32.num-1;j<i;j<<=1); j>>=1; if (!j) j=1;
            for (i=0;j;j>>=1)
                {
                i|=j;
                if (i>=primes_i32.num) { i-=j; continue; }
                a=primes_i32[i];
                if (a==p) return 1;
                if (a> p) i-=j;
                }
            return 0;
            }
        }
    a=an; a+=2;
    for (j=a>>1,i=0;i<8;i++) im[i]=j%primes_map_N[i];
    an=(1<<((bits(p)>>1)+1))-1; if (an<=0) an=1;
    an=an+an;
    for (;a<=p;a+=2)
        {
        for (j=1,i=0;i<8;i++,j<<=1)                     // check if map is set
         if (!(primes_map[im[i]]&j)) { j=-1; break; }   // if not dont bother with division
        for (i=0;i<8;i++) { im[i]++; if (im[i]>=primes_map_N[i]) im[i]-=primes_map_N[i]; }
        if (j<0) continue;
        for (i=primes_map_i0;i<primes_i32.num;i++)
            {
            b=primes_i32[i];
            if (b>an) break;
            if ((a%b)==0) { i=-1; break; }
            }
        if (i<0) continue;
        primes_i32.add(a);
        if (a==p) return 1;
        if (a> p) return 0;
        }
    return 0;
    }
//---------------------------------------------------------------------------
void getprimes(int p)                       // compute and allocate primes up to p
    {
    if (!primes_i32.num) isprime(3);
    int p0=primes_i32[primes_i32.num-1];    // biggest prime computed yet
    if (p>p0+10000)                         // if too big difference use sieves to fast precompute
        {
        // T((0.3516+0.5756*log10(n))*n) -> O(n.log(n))
        // sieves N/16 bytes p=100 000 000 t=1903.031 ms
        //  ------------------------------
        //   0  1  2  3  4  5  6  7 bit
        //  ------------------------------
        //   1  3  5  7  9 11 13 15 +-> +2
        //  17 19 21 23 25 27 29 31 |
        //  33 35 37 39 41 43 45 47 V +16
        //  ------------------------------
        int N=(p|15),M=(N>>4);              // store only odd values 1,3,5,7,... each bit ...
        char *m=new char[M+1];              // m[i] ->  is 1+i+i prime? (factors map)
        int i,j,k,n;
        // init sieves
        m[0]=254; for (i=1;i<=M;i++) m[i]=255;
        for(n=sqrt(p),i=1;i<=n;)
            {
            int a=m[i>>4];
            if (int(a&  1)!=0) for(k=i+i,j=i+k;j<=N;j+=k) m[j>>4]&=255-(1<<((j>>1)&7)); i+=2;
            if (int(a&  2)!=0) for(k=i+i,j=i+k;j<=N;j+=k) m[j>>4]&=255-(1<<((j>>1)&7)); i+=2;
            if (int(a&  4)!=0) for(k=i+i,j=i+k;j<=N;j+=k) m[j>>4]&=255-(1<<((j>>1)&7)); i+=2;
            if (int(a&  8)!=0) for(k=i+i,j=i+k;j<=N;j+=k) m[j>>4]&=255-(1<<((j>>1)&7)); i+=2;
            if (int(a& 16)!=0) for(k=i+i,j=i+k;j<=N;j+=k) m[j>>4]&=255-(1<<((j>>1)&7)); i+=2;
            if (int(a& 32)!=0) for(k=i+i,j=i+k;j<=N;j+=k) m[j>>4]&=255-(1<<((j>>1)&7)); i+=2;
            if (int(a& 64)!=0) for(k=i+i,j=i+k;j<=N;j+=k) m[j>>4]&=255-(1<<((j>>1)&7)); i+=2;
            if (int(a&128)!=0) for(k=i+i,j=i+k;j<=N;j+=k) m[j>>4]&=255-(1<<((j>>1)&7)); i+=2;
            }
        // compute primes
        i=p0&0xFFFFFFF1; k=m[i>>4]; // start after last found prime in list
        if ((int(k&  1)!=0)&&(i>p0)) primes_i32.add(i); i+=2;
        if ((int(k&  2)!=0)&&(i>p0)) primes_i32.add(i); i+=2;
        if ((int(k&  4)!=0)&&(i>p0)) primes_i32.add(i); i+=2;
        if ((int(k&  8)!=0)&&(i>p0)) primes_i32.add(i); i+=2;
        if ((int(k& 16)!=0)&&(i>p0)) primes_i32.add(i); i+=2;
        if ((int(k& 32)!=0)&&(i>p0)) primes_i32.add(i); i+=2;
        if ((int(k& 64)!=0)&&(i>p0)) primes_i32.add(i); i+=2;
        if ((int(k&128)!=0)&&(i>p0)) primes_i32.add(i); i+=2;
        for(j=i>>4;j<M;i+=16,j++)   // continue with 16-blocks
            {
            k=m[j];
            if (!k) continue;
            if (int(k&  1)!=0) primes_i32.add(i   );
            if (int(k&  2)!=0) primes_i32.add(i+ 2);
            if (int(k&  4)!=0) primes_i32.add(i+ 4);
            if (int(k&  8)!=0) primes_i32.add(i+ 6);
            if (int(k& 16)!=0) primes_i32.add(i+ 8);
            if (int(k& 32)!=0) primes_i32.add(i+10);
            if (int(k& 64)!=0) primes_i32.add(i+12);
            if (int(k&128)!=0) primes_i32.add(i+14);
            }
        k=m[j]; // do the last primes
        if ((int(k&  1)!=0)&&(i<=p)) primes_i32.add(i); i+=2;
        if ((int(k&  2)!=0)&&(i<=p)) primes_i32.add(i); i+=2;
        if ((int(k&  4)!=0)&&(i<=p)) primes_i32.add(i); i+=2;
        if ((int(k&  8)!=0)&&(i<=p)) primes_i32.add(i); i+=2;
        if ((int(k& 16)!=0)&&(i<=p)) primes_i32.add(i); i+=2;
        if ((int(k& 32)!=0)&&(i<=p)) primes_i32.add(i); i+=2;
        if ((int(k& 64)!=0)&&(i<=p)) primes_i32.add(i); i+=2;
        if ((int(k&128)!=0)&&(i<=p)) primes_i32.add(i); i+=2;
        delete[] m;
        }
    else{
        bool b0=primes_i32_alloc;
        primes_i32_alloc=true;
        isprime(p);
        primes_i32_alloc=false;
        primes_i32_alloc=b0;
        }
    }
//---------------------------------------------------------------------------

• solved some overflow bugs in mine code (periodicity of sieves ...)
• also some further optimizations
• added getprimes(p) function which add all primes<=p to the list fast as it can if they are not there yet
• tested correctness on first 1 000 000 primes (up to 15 485 863)
• getprimes(15 485 863) solves it on 175.563 ms on mine setup

• isprime is way slower for this of coarse

• primes_i32 is a dynamic list of int s

• primes_i32.num is the number of int s in the list
• primes_i32[i] is the i -th int i = <0,primes_i32.num-1>
• primes_i32.add(x) add x to the end of list
• primes_i32.allocate(N) preallocates space for N items in list to avoid reallocation slowdowns

[notes]

I have used this non parallel version for Euler problem 10 (sum of all primes below 2000000)

    ----------------------------------------------------------------------------------
           Time         ID      Reference    | My solution   | Note                     
    ----------------------------------------------------------------------------------
    [  35.639 ms] Problem010. 142913828922 | 142913828922  | 64_bit

• The sieve tabs are each one as a bit slice in the primes_map[] array and only the odd values are used (no need to remember even sieves).
• if you want maximize speed for all primes found then just call isprime(max value) and read the contents of primes_i32[]
• I use half of the bit-size instead of sqrt for speed

Hope I did not forget to copy something here