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Parallel Algorithms

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Parallel Algorithms

Sung Yong Shin

TC Lab

CS Dept. KAIST

1. Background

2. Parallel Computers

3. PRAM

4. Parallel Algorithms

- Von Neumann Machines
- sequential
- executing one instruction at a time
- Inherent limitation“ not faster than electrical signals ” 1 ft / 1 nanosecond ( 10-9 sec )

- Parallelism or Concurrency Carrying out many operations simultaneously
- partition a complex problem in such a way that various parts of the work can be carried out independently and in parallel, and combine the results when all subcomputation are complete.
- need parallel computers to support this approach.

- Hardware-oriented
- A parallel architecture of a specific kind is built.
- The parallel algorithms for solving different problems are developed to make use of these hardware features to the best advantage.

- Problem-oriented
- Whether the parallel algorithms can truly enhance the speed of obtaining a solution to a given problem, or not.
- If so, how much ?

(i) The usefulness of parallel computers depends greatly on :

- suitable parallel algorithms
- parallel computer languages
“ A major rethinking needed”

(ii) Practical limitations by parallel computers“ too many factors to be considered” How to abstract ingredient from complex reality !!!

- Nicholas Pippenger (1976)
“ NC-class problems” ( Nick’s Class )

“ ultra-fast on a parallel computer with feasible amount of hardware”

( independent of the particular parallel model chosen )

- Inherent Parallelism
probably not possible now but for the future !!!

“ fascinating research topics”

P

P(n) processors

NC

(log n)m

P-complete

P = NC ?

- Computer vision / Image processing
- Computer Graphics
- Searching huge databases
- Artificial Intelligence
· · · · · · · ·

SIMD ( Single Instruction Multiple Data Stream )

MIMD ( Multiple Instruction Multiple Data Stream )

What does SISD stand for ?

Program

Result x+y

Data

Source

x

Function

Unit

y

SISD

Program

Result x+y

x

Add

Function

Unit

y

Data

Source

Result s+q

s

Add

Function

Unit

q

Result v+w

Add

v

Function

Unit

w

- SIMD
- array processors
- vector processors (pipelining)

Process3

Process4

Process1

Process2

Data

Source

Branch

Function

Unit

NO

Result x · y

YES

x

Multiply

Function

Unit

Data

Source

y

Result w+v

w

Add

Function

Unit

Data

Source

v

Result s/q

s

Divide

Function

Unit

Data

Source

q

MIMD

Array Processors

instructions (for multiple data)

master

slave

slave

slave

Control

Processor

Arithmetic

Processor

Arithmetic

Processor

Arithmetic

Processor

PE

Memory

Memory

Memory

Memory

Communication network

Identical

Processors

· · ·

P

P

P

Interconnection network

· · ·

M

M

M

tightly coupled multiprocessors

P

P

P

Identical

Processing

Elements

( PEs )

· · ·

M

M

M

Interconnection network

loosely coupled multiprocessors

Vector ( pipe-line ) processors

functional unit

Operand one

Add

exponents

and multiply

mantissas

Compare

components

Align

operands

accordingly

Determine

normalization

factor

Result

Operand two

Normalize

results

Stage 1

Stage 2

Stage 3

Stage 4

Stage 5

A simplified pipeline for floating-point multiplication

Processors

(i) p general-purpose processors

(ii) Each processor is connected to a large shared, random access memory M.

(iii) Each processor has a private (or local) memory for its own computation.

(iv) All communications among processors take place via the shared memory.

(v) The input for an algorithm is assumed to be the 1stn memory cells,

and output is to be placed in the 1st cell.

(vi) All memory cells are initialized to be “0”.

P1

P2

P3

Pp

· · ·

Interconnection

· · ·

M

· · ·

1

m

Memory

[A PRAM]

(vii) All processors run the same program.

(viii) Each processor knows its own index.

(ix) A PRAM program may instruct processors to do different things

depending on their indices.

write

read

computation

three phases

(i) PRAM processors are synchronized !!!

(1) processors begin each step at the same time.

(2) All the processors that write at any step write at the same time.

(ii) Any number of processors may read the same memory cell

concurrently !!!

CREW ( Concurrent Read Exclusive Write )

CRCW ( Concurrent Read Concurrent Write )

– Common-write

– Priority-write

Why not EREW ?

yes, if you want !!!

[Other parallel architectures]

(a) A hypercube (dimension = 3)

(b) A bounded degree network (degree = 4)

···

···

· · · · · · · · · · · · · · · ·

(c) Octree model

- Binary Fan-in Technique
- Matrix multiplication
- Handling write conflicts
- Merging & Sorting

Binary Fan-in Technique

P7

P5

P3

P1

x1

Read

Read

Compute

Write

Read

Compute

Write

Read

Compute

Write

x7

x3

x5

read

x8

x2

x4

x6

comparison

write

save

M[1] = max

[A parallel tournament]

( finding Max )

Processors:

Step 0

read M[i] into big

Step 1

read M[i+1] into temp

big := max (big, temp)

write big

Step 2

read M[i+2] into temp

big := max (big, temp)

write big

Step 3

read M[i+4] into temp

big := max (big, temp)

write big

P1 P2 P3 P4 P5 P6 P7 P8

M

16 12 1 17 23 19 4 8

16 12 1 17 23 19 4 8

M

16 12 1 17 23 19 4 8

12 1 17 23 19 4 8 –

16 12 17 23 23 19 8 8

M

16 12 17 23 23 19 8 8

17 23 23 19 8 8 – –

17 23 23 23 23 19 8 8

M

17 23 23 23 23 19 8 8

23 19 8 8 – – – –

23 23 23 23 23 19 8 8

M

23 23 23 23 23 19 8 8

max

[A tournament example showing the activity of all the processors.]

read M[i] into big ;

incr := 1 ;

write – { some very small value } into M[n+i] ;

for step := 1 to lg n do

read M[i+incr] into temp ;

big := max (big, temp) ;

incr := 2 * incr ;

write big into M[i]

end { for }

O( log n ) using n/2 processors

no write conflicts

O(n) using n2 processors

What if using n3 processors ?

O( log n )

Why ?

Algorithm:Computing the or of n Bits

Input : Bits x1, · · · · ,xn in M[1],· · · ·, M[n].

Output : x1 · · · · xn in M[1].

Pireads xi from M[i] ;

If xi=1, then Pi writes 1 in M[1].

O(1) using n processors

write conflict !!!

CRCW

– Common-write

– Priority-write

Fast algorithm for finding Max

Initial memory contents (n = 4).

Input

loser

2 7 3 6 0 0 0 0

8

1

P14

After Step 2

P13

P12

P23

P24

P34

1

1

1

1

1

1

2 7 3 6 1 0 1 1

P23

After Step 3

7

7

[Example for the fast max-finding algorithm]

O(1) using processors

common-write

Input : n keys x1, x2,···, xn, initially in memory cells M[1], M[2],···, M[n] (n>2).

Output : The largest key will be left in M[1].

Comment : For clarity, the processors will be numbered Pi.j for 1 i j n.

Step 1

Pi.j reads xi (from M[i]).

Step 2

Pi.j reads xj (from M[j]).

Pi.j compares xi and xj.

Let k be the index of the smaller key.

(If the keys are equal, let k be the smaller index.)

Pi.j writes 1 in loser[k].

{At this point, every key other than the largest has lost a comparison. }

Step 3

Pi.i+1 reads loser[i] ( and P1.n reads loser[n]) ;

Any processor that read a 0 writes xi in M[1]. (P1.n would write xn.)

{ Pi.i+1 already has xi in its local memory ; P1.n has xn. }

Merging and Sorting

merging

P1

Pn

Pn/2+1

Pn/2

x1

xn/2

yn

y1

· · ·

· · ·

M[1]

M[n]

M[n/2]

(a) Assignment of processors to keys.

Pi

yj

xi

>yj

>xi

<xi

<yj

(b) Binary search steps; Pi finds j such that yj-1<xi<yj.

binary search

Pi

x1,…, xi-1 and y1,…, yj-1 (merged)

xi

M[i+j-1]

(c) Output step.

[Parallel merging]

O(log n) using n processors

no write conflict

Algorithm :Parallel Merging

Input : Two sorted lists of n/2 keys each, in the first n cells of memory.

Output : The merged list, in the first n cells of memory.

Comment : Each processor Pihas a local variable x (if in/2) or y (if i>n/2) and other local variables for conducting its binary search. Each processor has a local variable position that will indicate where to write its key.

Initialization :

Pi reads M[i] into x (if in/2) or into y (if i>n/2).

Pi does initialization for its binary search.

Binary search steps :

Processors Pi, for 1in/2, do binary search in M[n/2+1],…, M[n]

to find the smallest j such that x<M[n/2+j], and assign i+j–1 to

position. If there is no such j, Piassigns n/2+i to position.

Processors Pn/2+i, for 1in/2, do binary search in M[1],…, M[n/2]

to find the smallest j such that y<M[j], and assign i+j–1 to position.

If there is no such j, Pi assigns n/2+i to position.

Output step :

Each Pi(for 1in) writes its key (x or y) in M[position].

Break the list into two halves.

Sort the two halves (recursively).

Merge the two sorted halves.

Algorithm : Sorting by Merging

Input : A list of n keys in M[1],…,M[n].

Output : The n key sorted in nondecreasing order in M[1],…,M[n].

Comment : The indexing in the algorithm is easier if the number of keys is a power of 2, so the first step will “pad” the input with large keys at the end. We still use only n processors.

Piwrites (some large key) in M[n+i] ;

for t := 1 to lg n do

k := 2t-1 ; { the size of the lists being merged }

Pi,…, Pi+2k-1 merge the two sorted lists of size k beginning at M[i];

end { for }

O((log n)2) using n processors