Transcript Presentation

Sieve
of
Eratosthenes
by
Fola Olagbemi
Outline
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What is the sieve of Eratosthenes?
Algorithm used
Parallelizing the algorithm
Data decomposition options for the parallel
algorithm
• Block decomposition (to minimize interprocess communication)
• The program (returns count of primes)
• References
What is the sieve of Eratosthenes?
• An algorithm designed by the Greek
mathematician Eratosthenes: used to
find the prime numbers between two
and another specified number.
• It works by gradually eliminating
multiples of the smallest unmarked
number (x) in the given interval, till 𝑥 2 >
the specified number.
Algorithm
1. Create a list of natural numbers in the specified
range e.g. 2, 3, 4, ….. , n, none of which is
marked
2. Set k to 2 (or smallest number in list) where k is
the smallest unmarked number in the list
3. Repeat
1. Mark all multiples of k between square of k and n
2. Find smallest number greater than k that is
unmarked. Set k to the new value, until 𝑘 2 > n
4. The unmarked numbers are prime.
Parallelizing the algorithm
• Step 3a – key parallel computation step
• Reduction needed in each iteration to
determine new value of k (process 0 –
controller - may not have the next k)
• Broadcast needed after reduction to inform all
processes of new k.
• The goal in parallelizing is to reduce interprocess communication as much as possible.
Data decomposition options
• Goal of data decomposition approach chosen
– each process should have roughly the same
number of values to work on (optimal load
balancing)
• Decomposition options:
– Interleaved data decomposition
– Block data decomposition:
• Grouped
• Distributed
Data decomposition options (contd.)
• Interleaved data decomposition (e.g.):
– Process 0 processes numbers 2, 2+p, 2+2p, …
– Process 1 processes numbers 3, 3+p, 3+2p, …
• Advantage
– easy to know which process is working on the value in any
array index i
• Disadvantages
– It can lead to significant load imbalance among processes
(depending on how array elements are divided up among
processes)
– May require a significant amount of reduction/broadcast
(communication overheads)
Data decomposition options (contd.)
• Block data decomposition:
– Grouped: first few processes have more values
than the last couple of processes
Task 0
Task 1
Task 2
Task 3
Grouped
– Distributed: First process has less, second has
more, third has less etc. (this is the option used in
the program)
Task 0
Distributed
Task 1
Task 2
Task 3
Data decomposition (contd.)
• Grouped data decomposition:
– Let r = n mod p (n – number of values in the list, p –
number of processes)
– If r = 0, all processes get block of values of size n/p
– Else, first r processes get blocks of size ceil(n/p),
remaining processes blocks of size (floor(n/p)
• Distributed data decomposition:
– First element controlled by process i is floor(i*n/p)
– Last element controlled by process I is
floor((i+1)n/p) - 1
Data decomposition (Contd.)
• To reduce communication (i.e. reduce number
of MPI_Reduce calls to 1), all the primes used
for sieving must be held by process 0.
• This requirement places a limitation on the
number of processes that can be used in this
approach.
References
• Michael J. Quinn, Parallel Programming in C
with MPI and OpenMP; McGraw Hill; 2004
Questions?