SieveOfErastothenes

PD_ParallelDecomposition PD_ParallelArchitecture PD_ParallelPerformance TCPP_Programming TCPP_Architecture CS2 DSA Systems ProgLang movement visual

Originally described by Gregory F. Bachelis, David A. James, Bruce R. Maxim and Quentin F. Stout (Maxim1990, Bachelis1994). Also described by Michelle Moore (Moore2000) and (Kitchen1992).

No web-link to independent description available. See papers (Bachelis1994, Moore2000) for additional details.

Other activities by authors

(Bachelis1994): ParallelAddition, FindSmallestCard, OddEvenTranspositionSort

(Moore2000): CardSorting, OddEvenTranspositionSort, MatrixAddition


Details

Suppose the goal is to find all the primes less than n = 1,000. In the serial case, all the numbers from 2 .. 1000 would be written up on the board. The sieve of eratosthenes algorithm is then as follows:

  1. Designated the first unmarked number as prime.
  2. Mark (cross out) all multiples of that number from the list.
  3. Repeats steps 1 and 2, designating the next unmarked number as prime each time.

The question is next on how best to accomplish this algorithm in parallel.

First attempt

Observe first that all even numbers are a multiple of 2. Therefore, it suffices to consider odd primes (and odd numbers) going forward. Next, determine the set of prime numbers that are less than sqrt(n), and then cross out the multiples in the range of sqrt(n) .. n. In this case, sqrt(1000) is about 32, and the odd prime numbers between 3 .. 32 are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. Each student processor is then handed a subset of primes, and is asked to cross out the multiples of their primes. Thus, assuming there are 10 student processors, each student process is handed a separate prime, and crosses out the multiples of their prime in the matrix.

An issue with this outlined approach is that larger primes necessarily have fewer multiples in the range. Therefore, student processors assigned the larger prime numbers have less work to perform than students assigned smaller prime numbers. This highlights a load-balancing issue with this algorithm.

Second attempt

The second version of the algorithm takes a data parallel approach. Each student is assigned a subset of centuries. Thus, if there are 10 student processors, each student is handed one century (shown below, from Bachelis1994):

Century _00 to _99

_01  _03  _05  _07  _09 
_11  _13  _15  _17  _19 
_21  _23  _25  _27  _29 
_31  _33  _35  _37  _39 
_41  _43  _45  _47  _49 
_51  _53  _55  _57  _59 
_61  _63  _65  _67  _69 
_71  _73  _75  _77  _79 
_81  _83  _85  _87  _89 
_91  _93  _95  _97  _99 

So, student 0 is assigned the century 01 .. 99, while student 1 is assigned the century 101 .. 199, student 2 is assigned the century 201 .. 299 and so forth. Note that student 0 will necessarily eliminate 1 from their matrix, since 1 is not prime.

Students find the prime numbers in their century by following the algorithm below:

for each number p in set of prime numbers
   divide the first number in the century by p and round up to the next highest integer. Call this number q.
   Now multiply this number by p (i.e., q = q * p)
   if q is even, add p to it. This is the first odd multiple of p in the century.
   cross out q (unless it is p).
   continue to add p to q, crossing out each multiple.

As an example, consider student 1 who is assigned the century 101 .. 199.

It should be pointed out to students that they are all running the same algorithm on different parts of data. This is an example of the SPMD (data paralllel) approach to parallel computing. It should also be pointed out that this approach is must better load balanced than the first attempt.

Students should now be able to clearly see how to extend the activity to find the set of odd primes less than 1 million.

Variation (Moore2000)

Moore describes an example exercise that she presents in a parallel processing lab to illustrate data parallelism. Originally, the exercise is meant to distinguish between task parallelism and data parallelism. The “First attempt” in (Bachelis1994) is shown as an example of “control” (task) parallelism. The data parallel component of the exercise as shown in (Moore2000) is extrapolated below:

Prior to the start of the exercise, the instructor starts by writing as many integers as possible on the board (1-30 is recommended at the very least). To illustrate the data parallel model:

Variation (Kitchen1992)

Kitchen et al. uses Sieve of Eratoshenes to distinguish between shared memory and distribted memory. For shared memory, the instructor initially writes all the numbers on a shared blackboard. Student threads each get assigned a component of the blackboard, perform the sieve on thier component and leaves. At the end, the blackboard will contain all the prime numbers.

The distributed memory description in (Kitchen1992) is ambiguous. Here is another way to show the distributed memory case. For distributed memory, the instructor (representing the boss process) assigns each student process a component of the matrix by sending them a pair of numbers (indicated the range). This can be done by having the instructor throw piece of paper at the students (scatter), or walk to each individual desk to deliver message (send). Students can’t start their work until they receive the message. Once students receive the message, they enumerate the numbers in their assigned range on a piece of paper at their desks (local memory), and determine the primes in that range. They then “send” messages back to the instructor by either throwing a paper with the numbers into a waste-paper basket (“gather”), or handing their papers individually to the instructor (“send/receive”). The instructor then writes the received primes on the board.


CS2013 Knowledge Unit Coverage

PD/Parallel Architecture (Core Tier 1)

1. Explain the differences between shared and distributed memory. [Familiarity]

PD/Parallel Decomposition (Core Tier 2)

5. Parallelize an algorithm by applying data-parallel decomposition.

PD/Parallel Performance

  1. Detect and correct a load imbalance. [Usage]

TCPP Topics Coverage

Architecture Topics

TCPP Programming Topics



Accessibility

Since this activity requires students to mark values on a board in front of the classroom, it may be challenging for mobility-impaired students or blind students.


Assessment

(Kitchen1992) mentions the shared vs. distributed memory aspect of this exercise was used effectively at various invited talks, in a classroom at RIT, and at least two workshops at Colgate University. However, formal evaluation is not provided. (Moore2000) discusses the impact of the parallel computing labs in her course. In general, students responded to the labs positively, and felt that labs increased their knowledge significantly.


Citations