The Sieve of Erastothenes
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1 The Sieve of Erastothenes Parallel and Distributed Computing Department of Computer Science and Engineering (DEI) Instituto Superior Técnico October 25, 2010 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
2 Outline The Sieve of Eratosthenes Data decomposition options Parallel algorithm development, analysis Benchmarking Optimizations José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
3 The Sieve of Eratosthenes The Sieve of Eratosthenes Algorithm for finding all primes up to n, proposed by Greek mathematician Erastothenes ( BC). 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 Θ(n log log n) José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
4 The Sieve of Eratosthenes The Sieve of Eratosthenes Algorithm for finding all primes up to n, proposed by Greek mathematician Erastothenes ( BC). 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 Complexity?Θ(n log log n) José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
5 The Sieve of Eratosthenes The Sieve of Eratosthenes Algorithm for finding all primes up to n, proposed by Greek mathematician Erastothenes ( BC). 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 Complexity: Θ(n log log n) José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
6 Example with n = 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 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
7 Example with n = 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 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
8 Example with n = 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 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
9 Example with n = 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 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
10 Example with n = 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 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
11 Example with n = 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 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
12 Foster s Methodology on the Sieve of Erastothenes Partitioning: José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
13 Foster s Methodology on the Sieve of Erastothenes Partitioning: Make the computation of each element of the list 2, 3,..., n a primitive task. Communication: José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
14 Foster s Methodology on the Sieve of Erastothenes Partitioning: Make the computation of each element of the list 2, 3,..., n a primitive task. Communication: Need to send value of k to all tasks. Agglomeration + Mapping: What s the best way to partition the list? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
15 Foster s Methodology on the Sieve of Erastothenes Partitioning: Make the computation of each element of the list 2, 3,..., n a primitive task. Communication: Need to send value of k to all tasks. Agglomeration + Mapping: What s the best way to partition the list? interleaved (cyclic) data decomposition block data decomposition José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
16 Foster s Methodology on the Sieve of Erastothenes Partitioning: Make the computation of each element of the list 2, 3,..., n a primitive task. Communication: Need to send value of k to all tasks. Agglomeration + Mapping: What s the best way to partition the list? interleaved (cyclic) data decomposition easy to determine owner of each index leads to load imbalance for this problem block data decomposition balances loads more complicated to determine owner if n not a multiple of p José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
17 Block Decomposition The devil is in the details What s the best way to distribute n elements over p tasks? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
18 Block Decomposition The devil is in the details What s the best way to distribute n elements over p tasks? Some tasks get n p elements, other get n p. Which task gets which size? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
19 Block Decomposition The devil is in the details What s the best way to distribute n elements over p tasks? Some tasks get n p elements, other get n p. Which task gets which size? One approach: group larger blocks at lower index tasks. Let r = n mod p. if r = 0, assign n p elements per task. Otherwise, n first r tasks get p remaining tasks get n p José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
20 Block Decomposition Range of elements that each task gets? First element of task i: José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
21 Block Decomposition Range of elements that each task gets? First element of task i: i n p Last element of task i: + min(i, r) José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
22 Block Decomposition Range of elements that each task gets? First element of task i: i n p + min(i, r) Last element of task i: (i + 1) n p + min(i + 1, r) 1 Task owner of element j: José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
23 Block Decomposition Range of elements that each task gets? First element of task i: i n p + min(i, r) Last element of task i: (i + 1) n p + min(i + 1, r) 1 ) Task owner of element j: max ( j/ n p, (j r)/ n p José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
24 Block Decomposition Grouped approach: group larger blocks at lower index tasks. First element of task i: i n p + min(i, r) Last element of task i: (i + 1) n p + min(i + 1, r) 1 ) Task owner of element j: max ( j/ n p, (j r)/ n p Distributed approach: distribute larger blocks evenly. José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
25 Block Decomposition Grouped approach: group larger blocks at lower index tasks. First element of task i: i n p + min(i, r) Last element of task i: (i + 1) n p + min(i + 1, r) 1 ) Task owner of element j: max ( j/ n p, (j r)/ n p Distributed approach: distribute larger blocks evenly. First element of task i: i n p Last element of task i: José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
26 Block Decomposition Grouped approach: group larger blocks at lower index tasks. First element of task i: i n p + min(i, r) Last element of task i: (i + 1) n p + min(i + 1, r) 1 ) Task owner of element j: max ( j/ n p, (j r)/ n p Distributed approach: distribute larger blocks evenly. First element of task i: i n p Last element of task i: (i + 1) n p 1 Task owner of element j: José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
27 Block Decomposition Grouped approach: group larger blocks at lower index tasks. First element of task i: i n p + min(i, r) Last element of task i: (i + 1) n p + min(i + 1, r) 1 ) Task owner of element j: max ( j/ n p, (j r)/ n p Distributed approach: distribute larger blocks evenly. First element of task i: i n p Last element of task i: (i + 1) n p 1 Task owner of element j: (p(j + 1) 1) /n Task 0 Task 1 Task 2 Task 3 Task 4 Grouped Distributed José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
28 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_HIGH(id,p,n)-BLOCK_LOW(id,p,n)+1) #define BLOCK_OWNER(index,p,n) (((p)*((index)+1)-1)/(n)) José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
29 Local vs Global Indexes Local Global José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
30 Local vs Global Indexes Local Global Sequential program for (i = 0; i < n; i++) {... } Parallel program size = BLOCK_SIZE (id, p, n); for (i = 0; i < size; i++) { gi = i + BLOCK_LOW(id,p,n); } José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
31 Parallel Implementation use distributed block decomposition José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
32 Parallel Implementation use distributed block decomposition first task responsible for selecting next sieving prime broadcast new sieving prime at end of each iteration first task has first n/p elements, make sure it includes the largest prime used to sieve, n José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
33 Parallel Implementation use distributed block decomposition first task responsible for selecting next sieving prime broadcast new sieving prime at end of each iteration first task has first n/p elements, make sure it includes the largest prime used to sieve, n to simplify, our program will return the number of primes up to n José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
34 Parallel Implementation 1. Create list of unmarked natural numbers 2, 3,..., n José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
35 Parallel Implementation 1. Create list of unmarked natural numbers 2, 3,..., n each process creates its share of the list 2. k 2 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
36 Parallel Implementation 1. Create list of unmarked natural numbers 2, 3,..., n each process creates its share of the list 2. k 2 all processes perform this 3. Repeat (a) Mark all multiples of k between k 2 and n José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
37 Parallel Implementation 1. Create list of unmarked natural numbers 2, 3,..., n each process creates its share of the list 2. k 2 all processes perform this 3. Repeat (a) Mark all multiples of k between k 2 and n each process marks its share of the list (b) k smallest unmarked number > k José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
38 Parallel Implementation 1. Create list of unmarked natural numbers 2, 3,..., n each process creates its share of the list 2. k 2 all processes perform this 3. Repeat (a) Mark all multiples of k between k 2 and n each process marks its share of the list (b) k smallest unmarked number > k process 0 only and broadcasts it until k 2 > n 4. The unmarked numbers are primes 5. Reduction to compute number of primes José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
39 Function MPI Bcast int MPI_Bcast ( void *buffer, /* Addr of 1st element */ int count, /* # elements to broadcast */ MPI_Datatype datatype, /* Type of elements */ int root, /* ID of root process */ MPI_Comm comm) /* Communicator */ MPI Bcast (&k, 1, MPI INT, 0, MPI COMM WORLD); José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
40 Task / Channel Model Input Output José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
41 Analysis of the Parallel Algorithm Let α be the time to mark a cell. Sequential execution time? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
42 Analysis of the Parallel Algorithm Let α be the time to mark a cell. Sequential execution time: αn log log n Computation time of parallel program? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
43 Analysis of the Parallel Algorithm Let α be the time to mark a cell. Sequential execution time: αn log log n Computation time of parallel program: αn log log n/p Number of broadcasts? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
44 Analysis of the Parallel Algorithm Let α be the time to mark a cell. Sequential execution time: αn log log n Computation time of parallel program: αn log log n/p Number of broadcasts: n/ log n Broadcast time? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
45 Analysis of the Parallel Algorithm Let α be the time to mark a cell. Sequential execution time: αn log log n Computation time of parallel program: αn log log n/p Number of broadcasts: n/ log n Broadcast time: λ log p Reduction time? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
46 Analysis of the Parallel Algorithm Let α be the time to mark a cell. Sequential execution time: αn log log n Computation time of parallel program: αn log log n/p Number of broadcasts: n/ log n Broadcast time: λ log p Reduction time: λ log p Expected parallel execution time? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
47 Analysis of the Parallel Algorithm Let α be the time to mark a cell. Sequential execution time: αn log log n Computation time of parallel program: αn log log n/p Number of broadcasts: n/ log n Broadcast time: λ log p Reduction time: λ log p Expected parallel execution time: n log log n n log p α + λ p log + λ log p n José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
48 Code (1/4) #include <mpi.h> #include <math.h> #include <stdio.h> #define BLOCK_LOW(id,p,n) ((i)*(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)-BLOCK_LOW(id)) int main (int argc, char *argv[]) {... MPI_Init (&argc, &argv); MPI_Barrier(MPI_COMM_WORLD); elapsed_time = -MPI_Wtime(); MPI_Comm_rank (MPI_COMM_WORLD, &id); MPI_Comm_size (MPI_COMM_WORLD, &p); if (argc!= 2) { if (!id) printf ("Command line: %s <m>\n", argv[0]); MPI_Finalize(); exit (1); } José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
49 Code (2/4) n = atoi(argv[1]); low_value = 2 + BLOCK_LOW(id,p,n-1); high_value = 2 + BLOCK_HIGH(id,p,n-1); size = BLOCK_SIZE(id,p,n-1); proc0_size = (n-1)/p; if ((2 + proc0_size) < (int) sqrt((double) n)) { if (!id) printf ("Too many processes\n"); MPI_Finalize(); exit (1); } marked = (char *) malloc (size); if (marked == NULL) { printf ("Cannot allocate enough memory\n"); MPI_Finalize(); exit (1); } for (i = 0; i < size; i++) marked[i] = 0; José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
50 Code (3/4) if (!id) index = 0; prime = 2; do { if (prime * prime > low_value) first = prime * prime - low_value; else { if (!(low_value % prime)) first = 0; else first = prime - (low_value % prime); } for (i = first; i < size; i += prime) marked[i] = 1; if (!id) { while (marked[++index]); prime = index + 2; } MPI_Bcast (&prime, 1, MPI_INT, 0, MPI_COMM_WORLD); } while (prime * prime <= n); José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
51 Code (4/4) count = 0; for (i = 0; i < size; i++) if (!marked[i]) count++; MPI_Reduce (&count, &global_count, 1, MPI_INT, MPI_SUM, 0, MPI_COMM_WORLD); elapsed_time += MPI_Wtime(); if (!id) { printf ("%d primes are less than or equal to %d\n", global_count, n); printf ("Total elapsed time: %10.6f\n", elapsed_time); } MPI_Finalize (); } return 0; José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
52 Benchmarking Expected parallel execution time: n log log n n log p α + λ p log + λ log p n José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
53 Benchmarking Expected parallel execution time: n log log n n log p α + λ p log + λ log p n Experimental estimation of α: with n = 10 8, runtime= 24, 9s α = 24, = 85, 47ns log 108 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
54 Benchmarking Expected parallel execution time: n log log n n log p α + λ p log + λ log p n Experimental estimation of α: with n = 10 8, runtime= 24, 9s α = 24, = 85, 47ns log 108 Experimental estimation of λ: sequence of broadcasts using p = 2, 3,..., 8 processors. λ = 250µs Using n = 10 8, execution times of the algorithm were measured and compared against the above formula, for p = 2, 3,..., 8 processors. José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
55 Experimental Results p Predicted Actual 1 24,9 24,9 2 12,7 13,0 3 8,8 9,0 4 6,8 7,1 5 5,8 6,0 6 5,0 5,2 7 4,4 4,7 8 3,9 4,2 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
56 Improvements to the Program - I Every other mark is on an even number: José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
57 Improvements to the Program - I Every other mark is on an even number: delete them! New expected parallel execution time: José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
58 Improvements to the Program - I Every other mark is on an even number: delete them! New expected parallel execution time: n log log n n log p α + λ 2p log + λ log p n José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
59 Improvements to the Program - I Every other mark is on an even number: delete them! New expected parallel execution time: n log log n n log p α + λ 2p log + λ log p n p Original Optimized 1 24,9 12,2 2 13,0 6,6 3 9,0 5,0 4 7,1 4,1 5 6,9 3,7 6 5,2 3,3 7 4,7 3,1 8 4,2 2,9 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
60 Improvements to the Program - II Minimize communication: José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
61 Improvements to the Program - II Minimize communication: duplicate sieve computation on all tasks to avoid broadcast replace λ log p n/ log n by α n log log n New expected parallel execution time: ( n log log n α + n log log ) n + λ log p 2p José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
62 Improvements to the Program - II Minimize communication: duplicate sieve computation on all tasks to avoid broadcast replace λ log p n/ log n by α n log log n New expected parallel execution time: ( n log log n α + n log log ) n + λ log p 2p p Original Optimized 1 24,9 12,5 2 13,0 6,4 3 9,0 4,3 4 7,1 3,2 5 6,0 2,6 6 5,2 2,1 7 4,7 1,8 8 4,2 1,6 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
63 Improvements to the Program - III Memory access pattern? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
64 Improvements to the Program - III Memory access pattern? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
65 Improvements to the Program - III Memory access pattern? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
66 Improvements to the Program - III Memory access pattern? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
67 Improvements to the Program - III Solution? José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
68 Improvements to the Program - III Solution: reorganize loops mark each element of the array for all sieves between 3 and n José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
69 Improvements to the Program - III p Original Opt.I Opt.II Opt.III 1 24,9 12,2 12,5 2,5 2 13,0 6,6 6,4 1,3 3 9,0 5,0 4,3 0,9 4 7,1 4,1 3,2 0,7 5 6,0 3,7 2,6 0,5 6 5,2 3,3 2,1 0,5 7 4,7 3,1 1,8 0,4 8 4,2 2,9 1,6 0,3 José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
70 Review The Sieve of Eratosthenes Data decomposition options interleaved block grouped distributed Parallel algorithm development, analysis Benchmarking Optimizations José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
71 Next Class A shortest path algorithm José Monteiro (DEI / IST) Parallel and Distributed Computing / 28
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