PRAM (Parallel Random Access Machine)

1. PRAM(Parallel Random Access Machine) David Rodriguez-Velazquez Spring -09 CS-6260 Dr. Elise de Doncker

2. Overview • What is a machine model? • Why do we need a model? • RAM • PRAM • Steps in computation • Write conflict • Examples

3. A parallel Machine Model • What is a machine model? • Describes a “machine” • Puts a value to the operations on the machine • Why do we need a model? • Makes it easy to reason algorithms • Achieve complexity bounds • Analyzes maximum parallelism

4. RAM (Random Access Machine) • Unbounded number of local memory cells • Each memory cell can hold an integer of unbounded size • Instruction set included –simple operations, data operations, comparator, branches • All operations take unit time • Time complexity = number of instructions executed • Space complexity = number of memory cells used

5. PRAM(Parallel Random Access Machine) • Definition: • Is an abstract machine for designing the algorithms applicable to parallel computers • M’ is a system <M, X, Y, A> of infinitely many • RAM’s M1, M2, …, each Mi is called a processor of M’. All the processors are assumed to be identical. Each has ability to recognize its own index i • Input cells X(1), X(2),…, • Output cells Y(1), Y(2),…, • Shared memory cells A(1), A(2),…,

6. PRAM (Parallel RAM) • Unbounded collection of RAM processors P0, P1, …, • Processors don’t have tape • Each processor has unbounded registers • Unbounded collection of share memory cells • All processors can access all memory cells in unit time • All communication via shared memory

7. PRAM (step in a computation) • Consist of 5 phases (carried in parallel by all the processors) each processor: • Reads a value from one of the cells x(1),…, x(N) • Reads one of the shared memory cells A(1), A(2),… • Performs some internal computation • May write into one of the output cells y(1), y(2),… • May write into one of the shared memory cells A(1), A(2),… e.g. for all i, do A[i] = A[i-1] + 1; Read A[i-1] , compute add 1, write A[i] happened synchronously

8. PRAM (Parallel RAM) • Some subset of the processors can remain idle • Two or more processors may read simultaneously from the same cell • A write conflict occurs when two or more processors try to write simultaneously into the same cell P1 PN P0 P2 Shared Memory Cells

9. Share Memory Access Conflicts • PRAM are classified based on their Read/Write abilities (realistic and useful) • Exclusive Read(ER) : all processors can simultaneously read from distinct memory locations • Exclusive Write(EW) : all processors can simultaneously write to distinct memory locations • Concurrent Read(CR) : all processors can simultaneously read from any memory location • Concurrent Write(CW) : all processors can write to any memory location • EREW, CREW, CRCW

10. Concurrent Write (CW) • What value gets written finally? • Priority CW: processors have priority based on which value is decided, the highest priority is allowed to complete WRITE • Common CW: all processors are allowed to complete WRITE iff all the values to be written are equal. • Arbitrary/Random CW: one randomly chosen processor is allowed to complete WRITE

11. Strengths of PRAM • PRAM is attractive and important model for designers of parallel algorithms Why? • It is natural: the number of operations executed per one cycle on p processors is at most p • It is strong: any processor can read/write any shared memory cell in unit time • It is simple: it abstracts from any communication or synchronization overhead, which makes the complexity and correctness of PRAM algorithm easier • It can be used as a benchmark: If a problem has no feasible/efficient solution on PRAM, it has no feasible/efficient solution for any parallel machine

12. Computational power • Model A is computationally stronger than model B (A>=B) iffany algorithm written for B will run unchanged on A in the same parallel time and same basic properties. Priority >= Arbitrary >= Common >=CREW >= EREW Most Least powerful powerful Least Most realistic realistic

13. An initial example • How do you add N numbers residing in memory location M[0, 1, …, N] • Serial Algorithm = O(N) • PRAM Algorithm using N processors P0, P1, P2, …, PN ?

14. PRAM Algorithm (Parallel Addition) P0 + P1 + P2 + P3 + Step 1 Step 2 P0 + P2 + P0 + Step 3

15. PRAM Algorithm (Parallel Addition) • Log (n) steps = time needed • n / 2 processors needed • Speed-up = n / log(n) • Efficiency = 1 / log(n) • Applicable for other operations • +, *, <, >, etc.

16. Example 2 • p processor PRAM with n numbers (p<=n) • Does x exist within the n numbers? • P0 contains x and finally P0 has to know • Algorithm • Inform everyone what x is • Every processor checks [n/p] numbers and sets a flag • Check if any of the flags are set to 1

17. Example 2

18. Some variants of PRAM • Bounded number of shared memory cells. Small memory PRAM (input data set exceeds capacity of the share memory i/o values can be distributed evenly among the processors) • Bounded number of processor Small PRAM. If # of threads of execution is higher, processors may interleave several threads. • Bounded size of a machine word. Word size of PRAM • Handling access conflicts. Constraints on simultaneous access to share memory cells

19. Lemma • Assume p’<p. Any problem that can be solved for a p processor PRAM in t steps can be solved in a p’ processor PRAM in t’ = O(tp/p’) steps (assuming same size of shared memory) Proof: • Partition p is simulated processors into p’ groups of size p/p’ each • Associate each of the p’ simulating processors with one of these groups • Each of the simulating processors simulates one step of its group of processors by: • executing all their READ and local computation substeps first • executing their WRITE substeps then

20. Lemma • Assume m’<m. Any problem that can be solved for a p processor and m-cell PRAM in t steps can be solved on a max(p,m’)-processors m’-cell PRAM in O(tm/m’) steps Proof: • Partition m simulated shared memory cells into m’ continuous segments Si of size m/m’ each • Each simulating processor P’i 1<=i<=p, will simulate processor Pi of the original PRAM • Each simulating processor P’i 1<=i<=m’, stores the initial contents of Si into its local memory and will use M’[i] as an auxiliary memory cell for simulation of accesses to cell of Si • Simulation of one original READ operation Each P’ii=1,…,max(p,m’) repeats for k=1,…,m/m’ • write the value of the k-th cell of Si into M’[i] i=1…,m’, • read the value which the simulated processor Pii=1,…,,p, would read in this simulated substep, if it appeared in the shared memory • The local computation substep of Pii=1..,p is simulated in one step by P’i • Simulation of one original WRITE operation is analogous to that of READ

21. Conclusions • We need some model to reason, compare, analyze and design algorithms • PRAM is simple and easy to understand • Rich set of theoretical results • Over-simplistic and often not realistic • The programs written on these machines are, in general, of type MIMD. Certain special cases such as SIMD may also be handled in such a framework

22. Question • Why is PRAM attractive and important model for designers of parallel algorithms ? • It is natural: the number of operations executed per one cycle on p processors is at most p • It is strong: any processor can read/write any shared memory cell in unit time • It is simple: it abstracts from any communication or synchronization overhead, which makes the complexity and correctness of PRAM algorithm easier • It can be used as a benchmark: If a problem has no feasible/efficient solution on PRAM, it has no feasible/efficient solution for any parallel machine