Final Option 1: Primes!

generating primes quickly

Background

Recall that a prime number is a an integer \(p > 1\) that is not divisible by any number \(d\) strictly between \(1\) and \(p\). For example, the first few prime numbers are \(2, 3, 5, 7, 11, 13, 17, 19, 23, 27, 29,\ldots\). A number \(n > 1\) that is not prime is said to be composite. Prime numbers have many applications in computer science, notably in the RSA encryption scheme—currently the most widely used public key encryption scheme.

The Sieve of Eratosthenes

The problem of computing prime numbers has fascinated mathematicians for millenia. One of the earliest recorded methods for producing prime numbers (and indeed, one of the earliest recorded algorithms for any problem) is the Sieve of Eratosthenes (SoE). For any value of \(N\), the SoE produces all of the primes between \(1\) and \(N\). The basic idea is as follows:

  1. Write all numbers from \(2\) to \(N\) consecutively, and start reading the numbers in order.
  2. Whenever a new number is read:
    • if the number is marked as composite, continue
    • otherwise, if the number is not marked, mark it as prime; then mark all multiples of the number as composite

For example, we can generate the prime numbers up to 20:

1

02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20

In the first iteration of step 2 above, we’ll mark 2 as prime, and mark all of the multiples of 2 to be composite:

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02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20
PP    cc    cc    cc    cc    cc    cc    cc    cc    cc

In the next iteration, we see the 03 is the next unmarked number, so we mark it as prime, and remove all of its multiples:

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02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20
PP PP cc    cc    cc cc cc    cc    cc cc cc    cc    cc

Now 5 is the next unmarked number, so we mark it as prime, and mark its multiples as composite:

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02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20
PP PP cc PP cc    cc cc cc    cc    cc cc cc    cc    cc

Continuing in this way, we’ll reach the state

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02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20
PP PP cc PP cc PP cc cc cc PP cc PP cc cc cc PP cc PP cc

from which we can infer that the primes up to \(20\) are

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02 03 05 07 11 13 17 19
An Optimization

This process can be optimized somewhat. If a number \(n \leq N\) is composite, then it has a divisor that is at most \(\sqrt{N}\). (To see this, suppose \(n = a \cdot b\) with \(a, b > \sqrt{N}\). Then \(a \cdot b > \sqrt{N} \cdot \sqrt{N} = N\). Thus if \(n\) is composite—i.e., \(n = a \cdot b\) for some \(a, b\) satisfying \(1 < a, b < n\) and \(n \leq N\), then we must have either \(a \leq \sqrt{N}\) or \(b \leq \sqrt{N}\).) Therefore, if we have marked numbers with divisors at most \(\sqrt{N}\) as composite and some number \(n < N\) has not been marked as composite, then we can infer that \(n\) is prime.

Implementing the Sieve of Eratosthenes

To implement the SoE in Java, we can store which numbers are prime/composite as an array of Boolean values. Specifically, we can create an array boolean[] isPrime, with the interpretation that isPrime[i] should be true if i is prime, and false if i is composite. Using the optimization described above, we can implement the sieve as follows:

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        boolean[] isPrime = new boolean[MAX];
	
	// initialize isPrime[i] to true for i >= 2
	for (int i = 2; i < MAX; ++i) {
	    isPrime[i] = true;
	}

        // apply the Sieve of Eratosthenes
	for (int i = 0; i < MAX; ++i) {
	    if (isPrime[i]) {

		if (i < ROOT_MAX) {
		    int j = 2 * i;

		    while (true) {
			isPrime[j] = false;
			if (j >= MAX - i) {
			    break;
			}
			j += i;
		    }
		}
	    }
	}

When the method above completes, isPrime will have the property that isPrime[i] is true precisely when i is prime. It is then straightforward to use the array isPrime to list all of the primes up to MAX - 1: read through the array and print each index i with isPrime[i] == true.

Your Task

For this lab you will write a method optimizedPrimes(int[] primes) that fills the array primes with consective prime numbers. That is, after calling Primes.optimizedPrimes(primes), the array primes contains the first primes.length consecitve primes in order:

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primes[0] = 2;
primes[1] = 3;
primes[2] = 5;
...

To this end, you should complete an implementation of ParallelPrimes.java. To get started, download the following file:

You should modify only ParallelPrimes.java to perform your optimized implementation.

Suggestions

Optimization of SoE and Parallelization

The optimization of the SoE described above suggests a way to parallelize (some of) our computation. Suppose we’ve already computed all primes up to, say 100, and stored these primes in an array int[] smallPrimes. Since every composite integer up to 10,000 (= 100 x 100) has a prime factor that is at most 100, the SoE will find all primes up to 10,000 after we remove multiples of the primes in smallPrimes. Moreover, the tasks of finding primes between 1,000 and 1,999 (say) and finding primes between 2,000 and 2,999 can be done indpendently (given shared access to smallPrimes). This observation suggests a way of breaking down the problem into tasks: first find “sufficiently many” small primes using the sequential SoE, then use these small primes to find larger primes in parallel.

Coordination

Using the SoE to generate an array of primes consists of two logically distinct steps:

  1. Compute the array boolean[] isPrime.
  2. Use isPrimes to compute the actual array int[] primes of prime numbers.

The first step more computationally intensive, but it can be parallelized as suggested above (at least once some small primes are found). The second step is relatively faster, but it should be done sequentially to ensure that the primes appear in sorted order. This suggests using two distinct types of tasks: one type of task that performs step 1 (which can be parallelized), and another type of task that performs step 2. You might invoke many type-1 tasks, while a single type-2 runs concurrently to write primes.

You may find it useful that Executors in Java assign tasks to threads in the order the tasks are added (using the execute method). This does not guarantee that the first task submitted will be the first task completed. Thus you will need to ensure that the type-2 task waits until it has the “next” block of primes to write. To this end, you may find it helpful to use a concurrent data structure in which completed type-1 tasks can place their output while the type-2 task waits for the appropriate task to be completed. You can find a complete list of built-in concurrent data structures in the Java API documentation here.