[PPT] - Clo cks 1 Goals of the lecture Logical Clo cks (Lamp o PowerPoint Presentation

SLIDE 1 Clo cks 1 Goals

f

the lecture

Logical

Clo cks (Lamp

rt's

clo cks)

Concurrency

vs Simultaneit y

T
tal

Ordering

Physical

Clo cks

V

ecto r Clo cks c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 2 Clo cks 2 Logical Clo cks A glob al clo ck C : S ! N that satises: 8s; t 2 S : s

1

t _ s ; t ) C (s) < C (t) C : the set

f

all global clo cks Equivalent to : 8s; t 2 S : s ! t ) 8C 2 C : C (s) < C (t) (CC)

Lemma:

C is non-empt y i (S; !) is an irreexive pa rtial

rder.
happ

ened-b efo re relation c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 3 Clo cks 3 Concurrency

simul

taneit y fo r some

bserver

8u; v 2 S : ujjv ) 9C 2 C : (C (u) = C (v )) If t w

lo

cal states a re concurrent, ) there exists a global clo ck such that b

th

states a re assigned the same timestamp. This will sho w the converse

f

(CC), i.e., 8s; t 2 S : s 6! t ) 9C 2 C : :(C (s) < C (t)) 3 7 9 12 2 10 13 15 m m u m m

m

m m v m

3

7 10 13 2 10 13 16 m m u m m

m

m m v m

T

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

SLIDE 4 Clo cks 4 Logical Clo ck

Useful

fo r va rious algo rithms

Actions

tak en fo r each event t yp e: F

r

any initial state s: s:c = 0; Rule fo r a send event (s; snd; t): /* s.c is sent as pa rt

f

msg */ t:c := s:c + 1; Rule fo r a receive event (s; r cv (u); t): t:c := max (s:c; u:c) + 1; Rule fo r an internal event (s; int; t): t:c := s:c + 1; The follo wing claim is easy to verify: (Converse ?) 8s; t 2 S : s ! t ) s:c < t:c c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 5 Clo cks 5 Ordering the events totally

Extend

the logical clo ck with p ro cess numb er

the

timestamp

f

any event is a tuple < e:c; e:p >

the

total

rder

< is

btained

as: (e:c; e:p) < (f :c; f :p) , (e:c < f :c) _ ((e:c = f :c) ^ (e:p < f :p)): c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 6 Clo cks 6 Physical Clo cks

What

if some messages do not follo w the algo rithm ?

Given

app ro ximately co rrect physical clo cks,

ne

can syn- chronize clo cks such that u ! v implies C (u) < C (v ).

=

upp er b

und
n

the drift rate

f

any clo ck

=

minimum transmission time fo r any message

t

= physical time at which the message is sent W e require C i (t + ) > C j (t) fo r all i; j; t: F rom the b

und
n

the drift w e kno w that C i (t + ) > C i (t) + (1

):

Thus, w e need C i (t) + (1

)

> C j (t). That is, C j (t)

C

i (t) < (1

).

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

SLIDE 7 Clo cks 7 Clo ck Synchronization Algo rithm The synchronization constant () < (1

).
Algo

rithm:

send
ut

a timestamp ed message along its

utgoing

link at least every

seconds.
Every

message tak es time b et w een

and
+
.
On

receipt

f

a message timestamp e d with T m , the clo ck is up dated as maximum

f

the p revious value and T m + .

Let

the net w

rk

b e strongly connected with d as the diam- eter. Then, it can b e sho wn that

=

d(2 +

)

fo r all t > t +

d

assuming that

+
<<
.

h h h h h h h h

A

A A A A A U

@

@ @ R 6 @ @ @ I

6
c

Vija y K. Ga rg Distributed Systems F all 94

SLIDE 8 Clo cks 8 V ecto r Clo cks

Logical

clo cks satisfy s ! t ) s:c < t:c: Ho w ever, the converse is not true.

V

ecto r clo ck satisfy: s ! t , s:v < t:v : c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 9 Clo cks 9 Consistent Cuts

(E

; )

do

wn-set Y in this pa rtial

rder

will b e called a p rex.

The

set

f

all p rexes is a lattice.

supY

fo r any p rex Y is called a cut.

(E

; !) where ! is the causal-p recedes.

A

do wn-set Y in this pa rtial

rder

is called a consistent p rex.

Simila

rly , sup Y is called a consistent cut.

The

set

f

all consistent p rexes is also a lattice. F

E

is a consistent cut i 8e; f 2 F : :(e ! f ).

H

H H H H H j H H H H H H j

1
*

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

SLIDE 10 Clo cks 10 V ecto r Algo rithm

Let

there b e N p ro cesses

Algo

rithm: F

r

any initial state s: (8i : i 6= s:p : s:v [i] = 0) ^ (s:v [s:p] = 1) Rule fo r an internal event (s; int; t): t:v := s:v ; t:v [t:p] + +; s B B B @ 1 1 C C C A s B B B @ 1 1 1 C C C A s B B B @ 1 1 1 C C C A s B B B @ 2 1 C C C A s B B B @ 2 1 1 C C C A s B B B @ 2 1 2 1 1 C C C A s B B B @ 2 1 C C C A s B B B @ 2 1 3 1 1 C C C A s B B B @ 2 3 3 1 1 C C C A s B B B @ 1 1 C C C A

A

A A A A A A A A A A A U

P

1 P 2 P 3 P 4 c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 11 Clo cks 11 V ecto r Algo rithm [Contd.] Rule fo r a send event (s; snd; t): t:v := s:v ; t:v [t:p] + +; Rule fo r a receive event (s; r cv (u); t): fo r i := 1 to N t:v [i] := max (s:v [i]; u:v [i]); t:v [t:p] + +; s B B B @ 1 1 C C C A s B B B @ 1 1 1 C C C A s B B B @ 1 1 1 C C C A s B B B @ 2 1 C C C A s B B B @ 2 1 1 C C C A s B B B @ 2 1 2 1 1 C C C A s B B B @ 2 1 C C C A s B B B @ 2 1 3 1 1 C C C A s B B B @ 2 3 3 1 1 C C C A s B B B @ 1 1 C C C A

A

A A A A A A A A A A A U

P

1 P 2 P 3 P 4 c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 12 Clo cks 12 Prop erties

f

the V ecto r Clo ck Algo rithm Lemma 1 L et s 6= t. Then, s 6! t ) t:v [s:p] < s:v [s:p] Pro

f:
t:p

= s:p: then it follo ws that t

s.
s:p

6= t:p. Since s:v [s:p] is the lo cal clo ck

f

P s:p and P t:p could not have seen this value as s 6! t Theorem 1 s ! t i s:v < t:v . Pro

f:

(s ! t) ) (s:v < t:v )

s

! t: there is a message path from s to t. Therefo re, 8k : s:v [k ]

t:v

[k ]. F urthermo re, since t 6! s, from lemma 1 t:v [j ] > s:v [j ].

The

converse follo ws from Lemma 1. c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 13 Clo cks 13 Optimizatio n Recall x < y if and

nly

if (8i : x[i]

y

[i]) ^ (9j : x[j ] < y [j ]). If w e kno w the p ro cesses the vecto rs came from, the compa rison b et w een t w

states

can b e made in constant time. Lemma 2 s ! t i (s:v [s:p]

t:v

[s:p]) ^ (s:v [t:p] < t:v [t:p]) s B B B @ 1 1 C C C A s B B B @ 1 1 1 C C C A s B B B @ 1 1 1 C C C A s B B B @ 2 1 C C C A s B B B @ 2 1 1 C C C A s B B B @ 2 1 2 1 1 C C C A s B B B @ 2 1 C C C A s B B B @ 2 1 3 1 1 C C C A s B B B @ 2 3 3 1 1 C C C A s B B B @ 1 1 C C C A

A

A A A A A A A A A A A U

P

1 P 2 P 3 P 4 c Vija y K. Ga rg Distributed Systems F all 94