[PPT] - Clo cks [Contd.] 1 Goals of the lecture Direct dep PowerPoint Presentation

SLIDE 1 Clo cks [Contd.] 1 Goals

f

the lecture

Direct

dep endency clo cks

Pred

and Succ functions

Matrix

clo cks

Prop

erties

f

matrix clo cks c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 2 Clo cks [Contd.] 2 Overhead

f

the vecto r clo ck algo rithm m = #

f

messages sent/recd b y any p ro cess n = #

f

p ro cesses

Space
verhead

: n log m

Time
verhead

: O (n)

Communication
verhead

: n log m c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 3 Clo cks [Contd.] 3 Direct Dep endency Clo cks

Main

dra wback

f

the vecto r clo ck algo rithm

O

(N ) integers with every message

Direct

Dep endency Clo cks require

nly
ne

integer to b e app ended to each message

Used

in Lamp

rt's

algo rithm fo r mutual exclusion c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 4 Clo cks [Contd.] 4 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 a send event (s; snd; t)

r

an internal (s; int; t): t:v [t:p] := s:v [t:p] + 1; Rule fo r a receive event (s; r cv (u); t): t:v [t:p] := max(s:v [t:p]; u:v [u:p]) + 1; t:v [u:p] := max (u:v [u:p]; s:v [u:p]);

1

1 1 @ @ @ @ @ @ R 1 1 2 @ @ @ @ @ @ R 2 2 3 c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 5 Clo cks [Contd.] 5 Prop erties

f

Direct Dep endency Clo cks

p

rojection

n

ith comp

nent

results in the logical clo ck algo rithm. Lemma 1 s ! t ) s:v [s:p] < t:v [t:p] Converse ? Lemma 2 (8s; t :: s 6! t ) :(s:v [s:p]

t:v

[s:p]) Converse ? c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 6 Clo cks [Contd.] 6 Direct Dep endency s ! d t

(s
t)

_ (9 q ; r : s

q

^ q ; r ^ r

t)
@

@ @ @ @ R s s s t Lemma 3 8 s; t : s:p 6= t:p : :(s ! d t) ) :(s:v [s:p]

t:v

[s:p]) ! ! d c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 7 Clo cks [Contd.] 7 Higher Dimensional Clo cks W e describ e a matrix clo ck and its p rop erties b elo w. T

initialize:

M k [; ] := 0; M k [k ; k ] := M k [k ; k ] + 1; T

send

a message: T ag message with M k [; ]; M k [k ; k ] := M k [k ; k ] + 1; incr ement lo c al clo ck Up

n

receipt

f

a message tagged with W [; ]: for i := 1 to n do if (M k [i; i] < W [i; i]) then M k [i; ] := W [i; ]; c

py

ve ctor clo ck for (i; W [i; i]) M k [k ; k ] := M k [k ; k ] + 1; incr ement lo c al clo ck M k [k ; ] := diag

nal

(M k ); c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 8 Clo cks [Contd.] 8 c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 9 Clo cks [Contd.] 9 Interval

e

t f t @ @ R g t h t e

f

i there is no communication b et w een the state e and f .

is

an equivalence relation. F urther,

is

a congruence w.r.t. !. That is, s

s

) 8u : u:p 6= s:p : (s ! u

s

! u)^(u ! s

u

! s ) Congruence exploited b y assigning same identier to all states in the same equivalence class (interval). c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 10 Clo cks [Contd.] 10 Interval [contd.]

@

@ @ @ @ @ @ @ R A A A A A A A A A A A A A A A U

(1,1)
(2,1)

(1,4) (3,1) (3,2) (3,3) (2,2) (1,2) (1,3) c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 11 Clo cks [Contd.] 11 Pred and Succ functions

S

i = set

f

lo cal states at P i

Defn
f

Pred : pr ed:u:i = max fv j v 2 S i ^ v ! ug returns ? if the ab

ve

set is empt y . Simila r defn fo r Succ Because

f

congruence, can use intervals instead

f

states. Lemma 4 (p; i) = pr ed:(q ; j ):p ) (8 k : k > i : (p; k ) 6! (q ; j )) c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 12 Clo cks [Contd.] 12 F

r

V ecto r Clo cks V n k = vecto r clo ck in the interval (k ; n) 8i; k ; n : k 6= i : pr ed:(k ; n):i = V n k [i]

(1;

1) [1; 0; 0] (1; 2) [2; 0; 0] (2; 1) [0; 1; 0] (3; 1) [0; 0; 1] @ @ @ @ @ @ R (2; 2) [1; 2; 0] (2; 3) [1; 3; 0] @ @ @ @ @ @ R (3; 2) [1; 2; 2] c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 13 Clo cks [Contd.] 13 Prop erties

f

Matrix Clo ck

F
r

P i ; i th ro w implements the vecto r clo ck

M

n k [k ; k ] = n

Diagonal

is same as the i th ro w c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 14 Clo cks [Contd.] 14 Prop erties

f

Matrix Clo ck [Contd.]

i

6= j ) (j; M n k [i; j ]) = pr ed:(i; M n k [i; i]):j

i

6= j ) pr ed:(pr ed:(k ; n):i):j = (j; M n k [i; j ]) h h

h

P i P j P k (k ; n) pr ed:(k ; n):i = M n k [k ; i] pr ed:(pr ed:(k ; n):i):j c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 15 Clo cks [Contd.] 15 Disca rding Obsolete Info rmation If p ro cess s:p nds that its s:p column is unifo rmly bigger than r :v [s:p; s:p] (that is, its lo cal clo ck in some p revious state r ), then the info rmation at r has b een received b y all p ro cesses. Lemma 5 L et s:p = r :p. If (8i : s:v [i; s:p]

r

:v [r :p; r :p]), then 8t : tks : r ! t. Pro

f:
P

s:p P i s r s s

@

@ @ @ @ R

t
@

@ @ @ @ R

tks

) pr ed(s):i ! t (1) (8i : s:v [i; s:p]

r

:v [r :p; r :p]) ) r

pr

ed:(pr ed(s):i):(s:p) (2) F rom the ab

ve

t w

equations:

r ! t. c Vija y K. Ga rg Distributed Systems Sp ring 96