Pseudocode
1. Create list of unmarked natural numbers 2, 3, …, n
2. k 2
3. Repeat:
(a) Mark all multiples of k between k2 and n
(b) k smallest unmarked number > k
until k2 > n
4. The unmarked numbers are primes
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Pseudocode - Parallel and Distributed Computing - Lecture Slides, Slides of Parallel Computing and Programming
During the course of work of the Parallel and Distributed Computing we learn the core of the programming. The main points disucss in these lecture slides are:Pseudocode, Unmarked Natural Numbers, Sequential Algorithm, Complexity, Sources of Parallelism, Domain Decomposition, Array Element, Agglomeration Goals, Data Decomposition Options, Block Decomposition
Typology: Slides
2012/2013
1 / 32
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Pseudocode
1. Create list of unmarked natural numbers 2, 3, …, n
2. k 2
3. Repeat:
(a) Mark all multiples of k between k
2
and n
(b) k smallest unmarked number > k
until k
2
> n
4. The unmarked numbers are primes
Sequential Algorithm
Complexity: ( n ln ln n )
Making 3(a) Parallel
Mark all multiples of k between k
2
and n
for all j where k
2
j n do
if j mod k = 0 then
mark j (it is not a prime)
endif
endfor
Making 3(b) Parallel
Find smallest unmarked number > k
Min-reduction (to find smallest unmarked number > k )
Broadcast (to get result to all tasks)
Data Decomposition Options
• Interleaved (cyclic)
– Easy to determine “owner” of each index
– Leads to load imbalance for this problem
• Block
– Balances loads
– More complicated to determine owner if n not a
multiple of p
Block Decomposition Options
• Want to balance workload when n not a
multiple of p
• Each process gets either n/p or n/p
elements
• Seek simple expressions
– Find low, high indices given an owner
– Find owner given an index
Examples
17 elements divided among 7 processes 17 elements divided among 5 processes 17 elements divided among 3 processes
Method #1 Calculations
• First element controlled by process i
• Last element controlled by process i
• Process controlling element j
i n / p min( i , r )
( i 1 ) n / p min( i 1 , r ) 1
min( j /( n / p 1 ) , ( j r )/ n / p )
Examples
17 elements divided among 7 processes 17 elements divided among 5 processes 17 elements divided among 3 processes
Comparing Methods
Operations Method 1 Method 2
Low index 4 2
High index 6 4
Owner 7 4
Assuming no operations for “floor” function Our choice
Pop Quiz
• Illustrate how block decomposition method
would divide 13 elements among 5 processes.
Block Decomposition Macros #define BLOCK_LOW(id,p,n) ((id)(n)/(p)) #define BLOCK_HIGH(id,p,n)
(BLOCK_LOW((id)+1,p,n)-1) #define BLOCK_SIZE(id,p,n)
(BLOCK_LOW((id)+1,p,n)-BLOCK_LOW(id,p,n)) #define BLOCK_OWNER(index,p,n)
(((p)(index)+1)-1)/(n))**
Local vs. Global Indices
- L - L - L - L - L - G 0 1 G - G
- G 7 8 9 G
Decomposition Affects Implementation
- Largest prime used to sieve is n
- First process has n/p elements
- It has all sieving primes if p < n
- First process always broadcasts next sieving prime
- No reduction step needed