[PPT] - Rep eated Computation of a Global F unction 1 Goals of the PowerPoint Presentation

SLIDE 1 Rep eated Computation

f

a Global F unction 1 Goals

f

the lecture Rep eated Computation

f

a Global F unction

Deadlo

ck Detection

Clo

ck Synchronization

Distributed

Branch and Bound Sea rch

Distributed

Debugging c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 2 Rep eated Computation

f

a Global F unction 2 Desirable Cha racteristics

Light

Load

not

mo re than k messages/time step

High

Concurrency

log

k N time steps

Symmetry

(Equitable W

rkload)
load

balancin g

fairness

c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 3 Rep eated Computation

f

a Global F unction 3 Some P

ssible

App roaches

Centralized

f f f f f f f f f

?

@ @ @ R

6

@ @ @ I

Ring-based

f f f f f f f f

:
X

X X z C C C W

9

X X X y C C C O

Hiera

rchical h h h h h h h A A K

@

@ I A A K

@

@ @

A

A A

A

A A All links a re logical connections c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 4 Rep eated Computation

f

a Global F unction 4 Message Flo w T able Static Hiera rchy

Numb

er

f

no des (p ro cesse) = 7 time step Messages 1 1; 3 ! 2 5; 7 ! 6 2 1; 3 ! 2 5; 7 ! 6 2; 6 ! 4 3 1; 3 ! 2 5; 7 ! 6 2; 6 ! 4

A

A A K

@

@ @ I A A A K

@

@ @ @ @

A

A A A A A

A

A A A A A 4 2 6 3 1 7 5 c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 5 Rep eated Computation

f

a Global F unction 5 Overlapping T ress

@

@ @ @ @ @ @ @ @ @ @ @

@

@ @ @ @ @ @ @ @ @ @ @

@

@ @ @ @ @ @ @ @ @

@

@ @ @

7

@ @ @ @ @ @

@

@ @ @ 6 3 2 7 1 4 5 4 5 2 6 2 6 1 3 1 3 5 c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 6 Rep eated Computation

f

a Global F unction 6 Message Flo w T able

Revolving

Hiera rchy

numb

er

f

no des = 7 time step Messages idle 1 2 1; 3 6 5; 7 4 2 4 2; 6 5 1; 3 7 3 7 4; 5 1 2; 6 3 4 3 7; 1 2 4; 5 6

Reo

rganization

f

Hiera rchy

Reuse
f

messages B B @ 1 2 3 4 5 6 7 5 1 7 2 6 3 4 1 C C A c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 7 Rep eated Computation

f

a Global F unction 7 Requirements fo r Desired P ermutation

Gather

tree constraints

interio

r no des

f

T i = subtree

f

T i+1

F

airness constraints

No

cycle

f

size less than N .

4

2 6 1 3 5 7 c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 8 Rep eated Computation

f

a Global F unction 8 Interesting but ..

Do

es there alw a ys exist such a p ermutation ?

Is

there a systematic metho d to nd it ?

Is

there an ecient implementation fo r it ? c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 9 Rep eated Computation

f

a Global F unction 9 Metho d to Generate the P ermutation 3 2 5 7 6 4 9 11 10 13 15 14 12 8 1

@

@ @ @ @ @

@

@ @ @ @ @ @ @ H H H H H H H H next(x) : [ ev en(x) ! x := x=2; (* gather tree constraint *) 2

dd(x)

^ (x < 2 n1 ) ! x := x + 2 n1 ; (*fairness constraint *) 2

dd(x)

^ (x > 2 n1 ) ! [ x = N ! x := (N

1)=2;

2 x 6= N ! y := x

2

n1 + 2 x := y

2

dlog 2 n y 1e ] ] c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 10 Rep eated Computation

f

a Global F unction 10 Implem entati

n

1 Q: Who should I send message to at time t ? msg (x; t) = next t (par ent(next t (x))); if next t (x) is

dd

= nil ;

therwise

x is in-o rder lab el next is the new p

sition

function par ent is the pa rent function fo r in-o rder lab elin g pa rent

f

x = x with last t w

bits

changed to 10 c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 11 Rep eated Computation

f

a Global F unction 11 Implem etati

n

2 3 2 5 7 6 4 9 11 10 13 15 14 12 8 1

@

@ @ @ @ @

@

@ @ @ @ @ @ @ H H H H H H H H 12 13 2 14 3 6 9 4 7 10 1 5 8 11 msg (x; t) = new par ent(x + t)

t;

if (x + t) is a leaf-no de = nil ;

therwise
Just

need to sto re new par ent a rra y c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 12 Rep eated Computation

f

a Global F unction 12 Comm unicatio n Required

Communication

distance set (CDS) = fnew par ent(j )

j

j j a leaf no de g

p

ro cess x will send a message to p ro cess y i y

x

2 C D S .

fo

r N = 15 C D S = f1; 5; 8; 10; 13; 14g:

CDS

dep ends

n

the next function. c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 13 Rep eated Computation

f

a Global F unction 13 Data Gathering and Broadcasting

a

p ro cess can send/receive

nly
ne

message p er time step

require

that the same set

f

messages is used fo r data gathering and b roadcasting.

Constraints

: 1. fairness contraints

equal

load 2. gather tree constraints.

G(t)

available at t + log N time step at

ne

no de. 3. b roadcast constraints.

G(t)

available at t + 2 log N time step at all no des. c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 14 Rep eated Computation

f

a Global F unction 14 Message Flo w T able time step Messages ! 7 4 ! 6 1 ! 3 2 ! 5 1 7 ! 6 3 ! 5 ! 2 1 ! 4 2 6 ! 5 2 ! 4 7 ! 1 ! 3 3 5 ! 4 1 ! 3 6 ! 7 ! 2 4 4 ! 3 ! 2 5 ! 7 6 ! 1

fairness

in w

rkload
four

times less messages than static hiera rchy c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 15 Rep eated Computation

f

a Global F unction 15 Metho d to Generate the P ermutation bcnext(x) :: [ b = 1 ! x := R S (x) (* gather tree *) 2 (b = 0) ^ (b 1 = 0) ! x := R S 1 (x) (* b roadcast *) 2 (b = 0) ^ (b 1 = 1) ! x := LS a 1

LS

b (x) + 2

mo

d 2 n1

;

(* fairness *) ] b n1

b

= x R S p = Right shift with p as m.s.b LS p = Left shift with p as l.s.b. a = numb er

f

leading zeros b = numb er

f

leading

nes

c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 16 Rep eated Computation

f

a Global F unction 16 Algo rithm to nd Current Minimum in the Net w

rk
Distributed

b ranch and b

und
Distributed

simulation

p

ro cess x : step = [ dest msg (x; step) 6= nil ! send msg (dest msg (x; step); my min) step := step + 1 sr c msg (x; step) 6= nil ! r ecv msg (sr c msg (x; step); hismin) recompute my min step := step + 1 ] c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 17 Rep eated Computation

f

a Global F unction 17 P erfo rmance

f

the Algo rithm

at

most k messages handled b y a no de/time step

the

global function G(t) is available at t + dlog N e time steps.

a

throughput

f
ne

global function p er times step.

numb

er

f

messages required

half
f

that fo r static hiera rcy .

equal

w

rkload

distribution c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 18 Rep eated Computation

f

a Global F unction 18 Extensions

General

N

use

virtual no des

General

k

metho

ds to generate p ermutation s fo r bina ry trees generaliz e to k

a

ry trees.

asynchronous

messages

can

b e used instead

d

synchronous messages. No des synchronize d due to \receives". c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 19 Rep eated Computation

f

a Global F unction 19 Conclusions

Useful

fo r algo rithms that

use

hiera rchi ca l control

run

fo r long time

main

advantages

equal

w

rkload

distribution .

reduction

in numb er

f

messages due to their reuse

main

disadvantages

requires

that the communicat ion net w

rk

has mo re edges than static hiera rchy . c Vija y K. Ga rg Distributed Systems F all 94