Parallel Computational Models and Algorithm Analysis

Lecture 2: Parallel computational models 1

Sequential computational models Turing machine RAM (Figure ) Logic circuit model Control Unit (contains algorithms) Data Memory (unlimited size) CPU RAM (Random Access Machine) Operations supposed to be executed in one unit time (1) Control operations such as for and while can be realized by for and goto. (2) I/O operations such as (3) Substitution operations such as (4) Arithmetic and logic operations such as +, -, AND. if goto print a = b 2

O-notation for computing complexity Definition Assume that f(n) is a positive function. If there are two positive constants c, n0 such that f(n) c g(n) for all n n0, then we say f(n) = O( g(n) ). For example, 3n2-5n = O(n2) n log n + n = O(n log n) 45 = O(1) The item which grows most quickly) 3

Algorithm analysis for sequential and parallel algorithms Sequential algorithms Parallel algorithms Models RAM Many types Data division Not necessary Most important Analysis Computing time Computing time Memory size Communicating time Number of processors 4

PRAM (Parallel RAM) model PRAM consists of a number of RAM (Random Access Machine) and a shared memory. Each RAM has a unique processor number. Processors act synchronously. Processor execute the same program. According to the condition fork based on processor numbers, it is possible to executed different operations.) Data communication between processors (RAMs) are held through the shared common memory. Each processor can write data to and read data from one memory cell in O(1) time. RAM 1 RAM 2 Shared Memory RAM m 5

Features of PRAM Merits Demerits (It is not realistic that one synchronized reading and writing can be done in one unit time.) Distributed memory is not considered. Parallelism of problems can be considered essentially. Algorithms an be easily designed. Algorithms can be changed easily to ones on other parallel computational models. Communicational cost is not considered. In the following, We use PRAM to discuss parallel algorithms. 6

Analysis of parallel algorithms on PRAM model Ts(n): Computing time of the optimal sequential algorithm Computing time T(n) Number of processors P(n) Cost P(n) T(n) Speed-up Ts(n)/T(n) 1. Cost optimal parallel algorithms The cost is the same as the computing time of the optimal sequential algorithm, i.e., speed-up is the same as the number of processors. 2. Time optimal parallel algorithms Fastest when using polynomial number of processors. 3. Optimal parallel algorithms Cost and time optimal. 7

Analysis of parallel algorithms on PRAM model NC-class and P-class World of sequential computation P problems the class of problems which can be solved in polynomial time (O(n )). NP problems the class of problems which can be solved non-determinately in polynomial time. NP-complete problems the class of NP problems which can be reduced to each other. P = NP ? World of parallel computation NC Problems: the class of problems which can be solved in log-polynomial time (O(lg n) ). P-complete problems the class of problems which are not NC problems and can be reduced to each other. Similarly, NC = P ? k k 8

An Example of PRAM Algorithms Problem Find the sum of n integers (x1, x2, ... , xn) - Assume that n integers are put in array A[1..n] on the shared memory. - To simplify the problem, let n = 2k(k is an integer). main () { for (h=1; h log n; h++) { if (index of processor i n/2h) processor i do { a = A[2i-1]; /* Reading from the shared memory*/ b = A[2i]; /* Reading from the shared memory*/ c = a + b; A[i] = c; /* Writing to the shared memory } } if (the number of processor == 1) printf("%d n", c); } */ 9

An Example of PRAM Algorithms Processor Pi reads A[2i-1], A[2i] from the shared memory, then writes their summation to A[i] of the shared memory. Pi P1 P2 A[1] A[2] A[3]A[4] A[2i-1]A[2i] A[n] A[i] 10

An Example of PRAM Algorithms Find the summation of 8 integers (x1, x2, ... , x8). Output P Step 3 Parallel algorithm Step 2 P P P4 P3 P1 P2 Step 1 x1x2 x3x4 x5x6 x7x8 Input Sequential algorithm 11

An Example of PRAM Algorithms Analysis of the algorithm Computing time for loop is repeated log n times, each loop can be executed in O(1) time O(log n) time Number of processors not larger than n/2 n/2 processors Cost: O(n log n) It is not cost optimal since the optimal sequential algorithm run in (n) time. 12

Classification of PRAM by the access restriction EREW (Exclusive read exclusive write) PRAM Both concurrent reading and concurrent writing are prohibited. P1 P1 P2 P2 CREW (Concurrent Read Exclusive write) PRAM Concurrent reading is allowed, but concurrent writing is prohibited. P1 P1 P1 P1 P2 P2 P2 P2 13

Classification of PRAM by the access restriction CRCW (Concurrent Read Concurrent write) PRAM Both concurrent reading and concurrent writing are allowed. P1 P1 P2 P2 It is classified furthermore: - common CRCW PRAM Concurrent writing happens is only if the writing data are the same. - arbitrary CRCW PRAM An arbitrary data is written. - priority CRCW PRAM The processor with the smallest number writes the data. 14

Algorithms on different PRAM models Algorithms for calculating and of n bits (Input is put in array A[1..n]) Algorithm on EREW PRAM model main (){ for (h=1; h log n; h++) { if (index of processor i n/2h) processor i do { a = A[2i-1]; b = A[2i]; if ((a==1) and (b==1)) a[i] = 1; }}} Algorithm on common CRCW PRAM model main (){ if (A[index of processor i] == 1) processor i do A[1] = 1; } O(log n) time n/2 processors O(1) time n processors Abilities of PRAM models: EREW < CREW < CRCW 15

Exercise 1. Suppose n n matrix A and matrix B are saved in two dimension arrays. Design a PRAM algorithm for A+B using n and n n processors, respectively. Answer the following questions: (1) What is the PRAM model that you use in your algorithm? (2) What is the running time? (3) Is your algorithm cost optimal? (4) Is your algorithm time optimal? 2. Design a PRAM algorithm for A+B using n processors. Answer the same questions. 3. Design a PRAM algorithm for A+B using k (k n n) processors. Answer the same questions. 16