by j. burns and j. pachl

by J. Burns and J. by J. Burns and J. PachlPachl

Prepared by:

Irina Shapira

Julia MosinModified by Mordo Shalom

Uniform Self-Uniform Self-StabilizationStabilization

Self-Stabilizing System: Self-Stabilizing System: S = (G, R, S = (G, R, , , ))• G = (V,E) G = (V,E)

• = = 0 0 x 1 1 x . . . x n-1 n-1 - - set of configurationsset of configurations

wherewhere i i is a finite set of states of processor Pis a finite set of states of processor Pi i

• R - R - orientation of Gorientation of G

specifies a total ordering on the neighbors for each v specifies a total ordering on the neighbors for each v in Vin V

• = = 0 0 , 1 1 , . . . , n-1 n-1 - - sequence of transition sequence of transition

relationsrelations

wherewhere i i is a transition relation of processor Pis a transition relation of processor Pi i

from from i i x v1 v1 x . . . x vk vk to to i i ( R(i) = (v( R(i) = (v11, . . ., , . . .,

vvkk))))

For configuration For configuration ((ss00, s, s11, … , s, … , sn-1n-1) ) wherewhere

• ssjj is a state of P is a state of Pj j (for all j). (for all j).

Self-Stabilizing System: Self-Stabilizing System: S = (G, R, S = (G, R, , , ))

• x - possible next state of Px - possible next state of Pii , ,

i.e. if i.e. if R(vR(vii) = (v) = (v11, . . ., v, . . ., vkk ) then ( ) then (ssii,s,sv1v1, …, s, …, svk vk , x, x) )

iiwe define we define i i (())

as a set of as a set of allall configurations configurations ’’

such that such that ’’ss00,s,s11, …, s, …, si-1i-1, x, s, x, si+1i+1, …, s, …, sn-1n-1) )

(for all possible values of x)(for all possible values of x)

• PPi i is enabled (privileged) at is enabled (privileged) at if if i i (() is not ) is not

emptyemptyi

• ’’is a step of Pis a step of Pii if if ’’ i i (() )

• ’’is a step of Pis a step of Pii for some i for some i

• is a computation of the system is a computation of the system

where where

j-1j-1jj for all j for all j

Only one processor takes a step at a time -Only one processor takes a step at a time -

central daemoncentral daemon

Self-Stabilizing System: Self-Stabilizing System: S = (G, R, S = (G, R, , , ))

P0

is a system S = (G, R, is a system S = (G, R, , , ) where) where

An n-processor uniform ringAn n-processor uniform ring

• G is a cycleG is a cycle

• processors are denoted by 0,1, …, n-1 in processors are denoted by 0,1, …, n-1 in order order around the cycle around the cycle• R(i) = ( (i-1) mod n, (i+1) mod n )R(i) = ( (i-1) mod n, (i+1) mod n )

• 0 0 = 1 1 = . . . = n-1 n-1

• 0 0 = 1 1 = . . . = n-1n-1

P1

P2

P3P4

P5

Definition 1:Definition 1:A system S = (G, R, A system S = (G, R, , , ) is ) is self-stabilizingself-stabilizing iff iff there is a set there is a set , , called the called the legitimate configurationslegitimate configurations of S, such that: of S, such that:

For every For every there is a there is a ‘‘ such that such that ’’ (1) No Deadlock (1) No Deadlock

For every For every , every , every ’’ such that such that ’’ is in is in (2) Closure (2) Closure

Every infinite computation of S contains a configuration Every infinite computation of S contains a configuration

in in

(3) No Livelock (3) No Livelock

For every For every , exactly one processor is enabled, exactly one processor is enabled (4) Mutual Exclusion (4) Mutual Exclusion

For every processor For every processor PPii, every infinite computation , every infinite computation

consisting of configurations in consisting of configurations in contains an infinite contains an infinite

number of steps by number of steps by PPii

(5) Fairness (5) Fairness

Theorem 1:Theorem 1:There is no uniform n-processor self-stabilizing There is no uniform n-processor self-stabilizing ring ring if n is composite. if n is composite.

Proof:Proof:

P0

P1

P2

P3

P4

P5

• [i] = {P[i] = {Pjj | j = i mod | j = i mod p}p}• Sym(Sym() ) all the P all the Pjj in [i] are in the same state in in [i] are in the same state in

...2

...1

...0

• if Sym(if Sym(00) holds, we can construct an infinite ) holds, we can construct an infinite computation computation for which Sym(for which Sym(ii) holds for every i) holds for every i

00::

s0

s0

s0

s0s0

s0

11::

s1

s1s1

s2

s2

s2

22::

If the size of the ring is If the size of the ring is prime:prime:

• no distinguished processorno distinguished processor

• ( n( n2 2 ) states) states

• deterministic algorithmdeterministic algorithm

• central demoncentral demon

The Protocol:The Protocol:

• deterministicdeterministic

• unidirectional unidirectional (the value of (the value of ii((ssii, , ssi-1i-1, , ssi+1i+1) depends only on ) depends only on ssii and and ssi-1i-1))

s si-1i-1 ssii -> x means that x -> x means that x ii(s(sii,s,si-1i-1,s,si+1i+1) for any s) for any si+1i+1

A First AttemptA First Attempt

• 0 0 = 1 1 = . . . = n-1 n-1 = { 0, 1, …, n-2 }

• arithmetic modulo n-1arithmetic modulo n-1

The rule: if The rule: if ssii ssi-i-11+1 then +1 then ssi-i-11 ssii ssi-i-

11+1+1

P0

P2P3

P4 P1

00

00

1122

33

0 0 00 1 2 3 1 2 3

11P1

P2

0 1 0 1 11 2 3 2 3

PP0 0 PP1 1 PP2 2 PP3 3 PP44

0 1 2 0 1 2 22 3 3

P2

22P3

0 1 2 3 0 1 2 3 33

33P3

P4

00 1 2 3 0 1 2 3 0

P400

P0

...

0 0 00 1 2 3 1 2 3

0 0 00 1 2 3 1 2 3

P0

P1

22

33

11

00

The cycle of the intended The cycle of the intended legitimate configurations legitimate configurations

The rule: if sThe rule: if sii s si-i-11+1 then s+1 then si-i-11 ssii s si-i-

11+1+1

P2P3

P4

00

11

2200

11

00 1 2 1 2 00 1 1

PP0 0 PP1 1 PP2 2 PP3 3 PP44

...

00 1 2 1 2 00 1 1

The cycle which does not The cycle which does not intersect the cycle of the intersect the cycle of the legitimate configurations legitimate configurations

P1

P0

P3

00 1 2 3 1 2 3 11

P3

P0

P4

33

22 11 2 3 2 3 11 P0

22

P1P4

22 11 2 3 0 2 3 0 00 P4

P0

P1 22 3 3 22 3 0 3 0

P0

P1 33

P2

11 33 22 3 3 00

P0

11

P1

P2

11 33 0 0 33 0 0

P1

P2

00P3

1 21 2 00 33 0 0

P1 22

P2P3

1 21 2 00 1 1 00 P2P3

11

P4

00 1 2 1 2 00 1 1 00

P0

11

22P2P3

00

P411

livelocklivelock

The rule: if sThe rule: if sii s si-i-11+1 then s+1 then si-i-11 ssii s si-i-

11+1+1

The Protocol:The Protocol:• n n 3 3

• states are states are label.tag (Llabel.tag (Lii.T.Tii))

• labels range over { 0, 1, …, n-2 }labels range over { 0, 1, …, n-2 }

• tags range over { 0 } U { 2, 3, …, n-2 }tags range over { 0 } U { 2, 3, …, n-2 }

Rule A: Rule A: if if LLi i LLi-i-11+1 and +1 and ( (LLii0 or T0 or Ti-i-11==0 or T0 or Ti-i-11LLii--LLi-i-11 or T or Ti-i-11<<TTii) ) then then LLi-i-11.T.Ti-i-11 LLii.T.Tii ( (LLi-i-11+1).(+1).(LLii--LLi-i-11))

Rule B: Rule B: if if LLi i = = LLi-i-11+1 +1 and and LLi i 0 and T 0 and Ti-i-11 T Tii then then LLi-i-11.T.Ti-i-1 1 LLii.T.Tii LLii.T.Ti-i-11

The Proof of Self-The Proof of Self-Stabilization:Stabilization:LetLet be the set of all cyclic permutations of be the set of all cyclic permutations of configurations of the formconfigurations of the form

0.0 1.0 … a-1.0 a.0 0.0 1.0 … a-1.0 a.0 a.0a.0 a+1.0 … n-2.0 a+1.0 … n-2.0

for a=0, 1, …, n-2for a=0, 1, …, n-2

Theorem 3:Theorem 3:If n is prime, then the ring is self-If n is prime, then the ring is self-stabilizingstabilizing(and (and is a set of legitimate is a set of legitimate configurations) configurations)

S = (G, R, S = (G, R, , , ))

Rule A: if Rule A: if LLi i L Li-i-11+1 and (L+1 and (Lii0 or T0 or Ti-i-11==0 or T0 or Ti-i-11LLii-L-Li-i-11 or T or Ti-i-11<<TTii) ) then L then Li-i-11.T.Ti-i-11 LLii.T.Tii (L (Li-i-11+1).(L+1).(Lii-L-Li-i-11))

Rule B: if Rule B: if LLi i = L= Li-i-11+1 +1 and Land Lii0 0 and and TTi-i-11 T Tii then L then Li-i-11.T.Ti-i-11 LLii.T.Tii L Lii.T.Ti-i-11

Definition 2:Definition 2:Let Let be a configurationbe a configuration

• if Lif Li+i+11 LLii+1 (mod n-1), then P+1 (mod n-1), then Pi i and Pand Pi+i+1 1 formform a a gapgap of of • the the gap sizegap size (of (of PPi i and and ) ) g(i,g(i,LLi+i+11- - LLi i (mod (mod n-1)n-1)• a a segmentsegment of of is a maximal cyclically is a maximal cyclically contiguous contiguous sequence s=(P sequence s=(Pii, P, Pi+i+11, …, P, …, Pj-j-11, P, Pj j ) ) which contains which contains no gapsno gaps• the the gap size of sgap size of s g(s g(s,, is the gap size of its is the gap size of its rightright end (g(j, end (g(j,))))

• PPi i is is the the left endleft end of s, of s, PPjj is the is the right endright end of s of s

• a segment is a segment is zero basedzero based if its left end is 0 ( if its left end is 0 (LLii = 0)= 0)ExampleExample::

(P6 P0 P1 P2) zero based segmentgap of size 2 gap of size 5

PP00 P P11 P P22 P P3 3 PP44 P P55 P P66

1.3 2.3 3.1 5.2 0.2 1.1 0.41.3 2.3 3.1 5.2 0.2 1.1 0.4

PP00 P P11 P P22 P P3 3 PP44 P P55 P P66

1.3 2.3 3.1 1.3 2.3 3.1 5.2 0.2 1.1 5.2 0.2 1.1 0.4 0.4

1.01.0

2.02.0

3.33.32.32.3

3.03.0

1.01.0 2.0 2.0 3.33.3 2.32.3 3.03.0

PP0 0 PP1 1 PP2 2 PP3 3 PP44

P1

P0

P3P2

0.20.2 2.02.0 3.33.3 2.32.3 3.03.0

P4



0.20.2

P0

P1

0.20.2 1.21.2 3.33.3 2.32.3 3.03.0

1.21.2P1P4

0.20.2 1.21.2 3.33.3 2.32.3 3.3 3.3

3.33.3 P4

P3

0.20.2 1.21.2 3.33.3 0.30.3 3.33.3

0.30.3P3

P4 0.20.2 1.21.2 3.33.3 0.30.3 1.31.3 1.31.3 P4

P2

0.20.2 1.21.2 2.22.2 0.30.3 1.31.3

2.22.2P2P3

0.20.2 1.21.2 2.22.2 3.2 3.2 1.31.3

3.23.2P3

P4

0.20.2 1.21.2 2.22.2 3.2 0.23.2 0.2

0.20.2 P4

P0

1.01.0 1.21.2 2.22.2 3.2 0.23.2 0.2

1.01.0

P1

1.01.0 2.02.0 2.22.2 3.2 0.23.2 0.2

P1 2.02.0

P2

1.01.0 2.02.0 3.03.0 3.23.2 0.2 0.2

P2

3.03.0P3

1.01.0 2.02.0 3.03.0 0.0 0.0 0.20.2

0.00.0P3

P4

1.01.0 2.02.0 3.03.0 0.0 1.00.0 1.0

1.01.0 P4

P0P0

Definition 1:Definition 1:A system S = (G, R, A system S = (G, R, , , ) is ) is self-stabilizingself-stabilizing iff iff there is a set there is a set , , called the called the legitimate configurationslegitimate configurations of S, such that: of S, such that:

(1) No Deadlock (1) No Deadlock

(2) Closure (2) Closure

(3) No Livelock (3) No Livelock

(4) Mutual Exclusion (4) Mutual Exclusion

(5) Fairness (5) Fairness

Lemma 1:Lemma 1:Conditions 2 (Closure), 4 (Mutual Exclusion), Conditions 2 (Closure), 4 (Mutual Exclusion), and 5 (Fairness) of Definition 1 hold for and 5 (Fairness) of Definition 1 hold for and and S.S.Proof:Proof:• all configurations in all configurations in are cyclic permutations of the formare cyclic permutations of the form

= 0.0 1.0 … a-1.0 a.0 = 0.0 1.0 … a-1.0 a.0 a.0a.0 a+1.0 … n-2.0 a+1.0 … n-2.0• TTii = T = Tjj for all i and j, hence rule B does not apply for all i and j, hence rule B does not apply

• rule A applies to exactly one processor - rule A applies to exactly one processor - Mutual ExclusionMutual Exclusion

• ’’by rule A by rule A ’’ = 0.0 1.0 … a-1.0 a.0 a+1.0 = 0.0 1.0 … a-1.0 a.0 a+1.0 a+1.0a+1.0 … n-2.0 … n-2.0

’’- - ClosureClosure

• if Pif Pii is enabled in is enabled in , then P, then Pi+i+1(mod n)1(mod n) will be enabled in will be enabled in ’’- - FairnessFairness



Lemma 2:Lemma 2:

Proof:Proof:

If n is prime, then some processor is enabled for every If n is prime, then some processor is enabled for every configuration in configuration in , and therefore Condition 1 (No , and therefore Condition 1 (No Deadlock) holdsDeadlock) holds

• if rule A doesn’t apply, then all the segments if rule A doesn’t apply, then all the segments are zero-are zero-basedbased

• Case 1: one zero-based segment of length nCase 1: one zero-based segment of length n

if Tif Ti-1 i-1 = 0 rule A applies = 0 rule A applies

else Telse Ti-1 i-1 0 and so T 0 and so Ti-1 i-1 LLi i - L- Li-1 i-1 and again rule A appliesand again rule A applies

… … LLi-1i-1.T.Ti-i-

11 0.T0.Ti . i . … …

… 0.Ti-1

0.Ti ...



• Case 2: there is more than one segment Case 2: there is more than one segment

0.t0.t1 1 1.t 1.t11 … … mm11.t.t11

0.t0.t22 1.t 1.t22 … … mm22.t.t2 2

0.t0.tll 1.t 1.tll … m … mll.t.tll

...

all the gaps and all the segments have the same sizeall the gaps and all the segments have the same size

Since n is prime, all the segment lengths must Since n is prime, all the segment lengths must be 1be 1

0.00.0 ...0.00.0 0.00.0 0.00.0

and rule A applies for every gapand rule A applies for every gap



rule B is not applicable - all tags within each segment are rule B is not applicable - all tags within each segment are equalequal

rule A doesn’t apply - Trule A doesn’t apply - Ti-1i-1= L= Lii- L- Li-1i-1 for each gap for each gap ttjj = 0 - m = 0 - mjj

rule A doesn’t apply - trule A doesn’t apply - tj j t tj+1j+1 for each gap for each gap

Lemma 3:Lemma 3:

Proof:Proof:

In every infinite computation of S, every processor takes a In every infinite computation of S, every processor takes a step by rule A infinitely many times step by rule A infinitely many times

• suppose that there is no such Psuppose that there is no such Pii

• if someif some PPii takes an infinite number of steps by rule A, so takes an infinite number of steps by rule A, so do alldo all

• all labels are fixed and so are all segmentsall labels are fixed and so are all segments

• the left end of the segment cannot take a step by the left end of the segment cannot take a step by rule Brule B• PPii which is not the left end can take at most one step more which is not the left end can take at most one step more than Pthan Pi-1i-1

... P... Pi-i-

11

PPi i ……



Lemma 4:Lemma 4:

Proof:Proof:

For any computation of S, the number of segments is For any computation of S, the number of segments is nonincreasing. nonincreasing.

• rule B does not change the number of segments rule B does not change the number of segments

• rule A cannot increase the number of segmentsrule A cannot increase the number of segments

Pi … Pi-1 Pi

… Pi-1 Pi Pi+1 … Pi+1 …

Pi+1 … … Pi-1 Pi+1 …

… Pi-1 Pi



Definition:Definition:

A computation in which the number of A computation in which the number of segments is constant is called segments is constant is called quietquiet..



Definition :Definition :Let C =Let C =00 1 1 … be a quiet computation… be a quiet computation

• a dynamic gapa dynamic gap is a function z from indices of is a function z from indices of computation computation into indices of processors, satisfying the following: into indices of processors, satisfying the following: (1) processor z(0) is the right end of the segment in (1) processor z(0) is the right end of the segment in 00

(2) if (2) if j-1j-1-> -> jj is a step of P is a step of Pii by rule A and z(j-1) = i-1 by rule A and z(j-1) = i-1 then z(j) = i, else z(j)=z(j-1) then z(j) = i, else z(j)=z(j-1)

... P... Pi-i-

11

z(j-1)

j-1j-1:: PPii P Pi+1i+1 ... ...

... P... Pi-1 i-1

PPii

z(j)

jj:: PPi+1i+1 ... ...



Definition (cont):Definition (cont):

• a a dynamic segmentdynamic segment is a function is a function sszz from indices of computation into segments, from indices of computation into segments, such that such that sszz(j) (j) is a segment inis a segment in jj whose right end is z(j). whose right end is z(j).

... ... PPii

sszz(j)(j)

z(j)

jj::



Lemma 5:Lemma 5:

Proof:Proof:

Let C =Let C =00 1 1 … be a quiet computation and z a dynamic … be a quiet computation and z a dynamic gap of C. Then g(z(j-1),gap of C. Then g(z(j-1),j-1j-1) = g(z(j),) = g(z(j),jj) for 0 < j < ) for 0 < j < length(C).length(C).

• PPii = = z(j-1)+1 takes a step z(j-1)+1 takes a step

... P... Pi-i-

11

z(j-1)

j-1j-1:: PPii P Pi+1i+1 ... ...

... P... Pi-1 i-1

PPii

z(j)

jj:: PPi+1i+1 ... ...

length > 1length > 1

• PPii z(j-1)+1z(j-1)+1 takes a step - does not affect the gaptakes a step - does not affect the gap



• PPii = z(j-1) = z(j-1) takes a step - the segment disappears-takes a step - the segment disappears-contradictioncontradiction

PPii

z(j-1)

g =Lg =Lii-L-Li-1i-1

L’L’i+1i+1== LLi i +1+1

L’L’ii=L=Li-1i-1+1+1

Lemma 6:Lemma 6:

Proof:Proof:

Let C =Let C =00 1 1 … be an infinite quiet computation and z a dynamic … be an infinite quiet computation and z a dynamic gap of C. Then gap of C. Then • there is jthere is j0, such that the state of processor z(j) in 0, such that the state of processor z(j) in j j is 0.g(z) andis 0.g(z) and• there is kthere is k0, such that the state of processor z(k) +1 in 0, such that the state of processor z(k) +1 in k k is 0.u is 0.u for for some u some u

... ... a.*a.*

sz

jj11:: ......

... a.* ... a.* (a+1).g(z)(a+1).g(z)

z(j1)

jj22:: ...... z(j2)

… … 0.g(z)0.g(z)

jj:: ...... z(j)

... at most n-1 times



Lemma 6 (b):Lemma 6 (b):

Proof:Proof:

Let C =Let C =00 1 1 … be an infinite quiet computation and z a dynamic … be an infinite quiet computation and z a dynamic gap of C. gap of C. Then Then

there is jthere is j0, such that the state of processor z(j) in 0, such that the state of processor z(j) in j j is 0.g(z) is 0.g(z) andand

• there is kthere is k0, such that the state of processor z(k) +1 in 0, such that the state of processor z(k) +1 in k k is 0.u is 0.u

for some u for some u ... ...

sz

jj:: … … PPii 0.g(z’)0.g(z’)

... ... PPii

z’(j)=Pi+1

kk:: 0.g(z’) ...0.g(z’) ... z(k)

sz’

Pi+

1

...sz sz’



Definition:Definition:

A segment is well formed if all its tags are A segment is well formed if all its tags are equal equal to its gap size.to its gap size.



Lemma 7:Lemma 7:

Proof:Proof:

Let C =Let C =00 1 1 … be an infinite quiet computation. Then there is j… be an infinite quiet computation. Then there is j0 0

0, 0,such that for all j such that for all j j j0 0 all segments in all segments in j j are well formedare well formed

... ... 0.g(z)0.g(z)

sz

jj11::

... 0.g(z) ... 0.g(z) 1.g(z)1.g(z)

z(j1)=Pi

j’j’11:: z(j’1)=Pi

+1

… … 0.g(z) 1.g(z) … 0.g(z) 1.g(z) … m.g(z)m.g(z)

j’j’mm::

...

jj22::

... 0.g(z) 1.g(z) … 0.g(z) 1.g(z) … *.g(z)*.g(z)

z(j’m)=Pi

+m

z(j2)





Lemma 8:Lemma 8:

Proof:Proof:

There is a point in every infinite computation of S after which There is a point in every infinite computation of S after which every configuration has exactly one segment.every configuration has exactly one segment.

If for all dynamic gaps of C’ g(z) were 0, then there would be only If for all dynamic gaps of C’ g(z) were 0, then there would be only one segmentone segment

… a.0

a.0 ...

If there is some dynamic gap of C’ such that its size is greater If there is some dynamic gap of C’ such that its size is greater than 0:than 0:

... ...

sz

...... z(j1)=Pi

sz’

Pi+1

g(z) g(z) g(z’) and g(z) g(z’) and g(z) 0 0

a.g(z) a.g(z)

0.g(z’) 0.g(z’)

C’ - quiet computation with well formed segments - rule B is voidC’ - quiet computation with well formed segments - rule B is void

PPi+1i+1 can make a step only after P can make a step only after Pii, , but such a step would destroy sbut such a step would destroy szz contradiction contradiction

Conclusion:Conclusion:

Proof:Proof:

The system never livelocks.The system never livelocks.

… a-1.0 a.0

a.0 a+1.0 ...

one well formed segment, g(z) = 0, all tags are 0one well formed segment, g(z) = 0, all tags are 0

After finitely many step the system reaches the following After finitely many step the system reaches the following configuration configuration ::

Hence, Hence, is a legitimate configuration is a legitimate configuration

RecallRecall:: the set of legitimate configurationsthe set of legitimate configurations is the set of all is the set of all cyclic permutations of configurations of the formcyclic permutations of configurations of the form

0.0 1.0 … a-1.0 a.0 0.0 1.0 … a-1.0 a.0 a.0a.0 a+1.0 … n-2.0 for a=0, 1, …, n- a+1.0 … n-2.0 for a=0, 1, …, n-22

::



by j. burns and j. pachl

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