franck petit inria / lip lyon stabilization, synchronization of logical clocks and applications
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Franck Petit INRIA / LIP
Lyon
Stabilization, Synchronization of Logical Clocks and Applications
o G=(V,E) is a connected networko V: set of n nodes/processeso E: set of m bidirectional linkso Np: set of neighboring nodes of p o Memory shared between neighboring nodes
2
Preliminaries
Bidirectional links
o In each computation step: According to its variables and the variables
of its neighbors, a node/process is either enabled to execute an action or not
Synchronous system Every enabled nodes execute an action atomically
Asynchronous system Some enabled nodes are chosen by an unfair
adversary The chosen nodes execute an action atomically
3
Preliminaries, Distributed Algorithm
Logical Clock Synchronization
o Synchronous/Asynchronous Environment
4
0 0 0 0 0 0 0 0 0
+4
+3
+2
+1
+0
1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3
o Each node p maintains a logical clock register rp
rp := rp + 1
Logical Clock Synchronization
o Synchronous/Asynchronous Environment
5
0 0 0 0 0 0 0 0 0
+4
+3
+2
+1
+0
o Each node p maintains a logical clock register rp
rp := rp + 1
0 0
1
0 0 0
1
0 00 0
2
0 0 0
2
0 0
+ Synchronizer
Global Synchronizer (asynchronous) distributed systems
o Wave Algorithms, e.g., Propagation of Information with Feedback
(PIF) Depth-First Token Circulation
o Global Synchronization, e.g., (Group) Mutual Exclusion Leader Election Reset Logical Clock Synchronization Rooted Spanning Tree …
6
In systems with unique IDs or a particular node (root, leader, main server…)?
r
Propagation of Information with Feedback
Propagation of Information with Feedback
r
r
Propagation of Information with Feedback
r
Leaf processors
Propagation of Information with Feedback
r
A slow processor
Propagation of Information with Feedback
r
A slow processor
Propagation of Information with Feedback
r
Propagation of Information with Feedback
r
Propagation of Information with Feedback
r
Propagation of Information with Feedback
r
Propagation of Information with Feedback
r
Propagation of Information with Feedback
r
Propagation of Information with Feedback
r
Propagation of Information with Feedback
o For One Puse:o Best Case: O(D)o Worst Case: O(N)Can we provide better complexities?
Logical Clock Synchronization
o Synchronous/Asynchronous Environment
20
o Each node p maintains a phase clock register rp
0 0 0 0 0 0 0 0 00 0
1
0 0 0
1
0 00
1 1 1
0 0
1
0 00
1
2
1 1 1 1
0 00
1
2 2 2
1 1 1
0
1 1
2
3
2
1
2
1
0
2 2 2
3
2 2 2
1
0
2
3 3 3 3 3
2
1
0
2
3
4 4 4
3
2
1
0
+4
+3
+2
+1
+0
If ∀q N∈ p : rp ≤ rq then rp := rp + 1
Logical Clock Synchronization
21
o Each node maintains a phase clock registerr in [0,…,K-1]=ZK
o Safety: The gap between the phase clock of two neighboring nodes is at most equal to 1 (mod K).
o No starvation (Vivacity): Each phase clock r is incremented by 1 (mod K) infinitely often.
Unison
Logical Clock Synchronization
22
Unison
2
3
4
0
4
3
2
1 1
0
+4
+3
+2
+1
+0
K=5
0 0 0 0 0 0 0 0 00 0
1
0 0 0
1
0 00
1 1 1
0 0
1
0 00
1
2
1 1 1 1
0 00
1
2 2 2
1 1 1
0
1 1
2
3
2
1
2
1
0
2 2 2
3
2 2 2
1
0
2
3 3 3 3 3
2
1
0
2
3
4 4 4
3
2
1
0
Minimality of K ?
Logical Clock Synchronization
23
o K=2?o No possible order among the clocks
0 0 0 0 0 0 0 0 0
+1
+0
1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1
Synchronous
0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1
Logical Clock Synchronization
24
o K=2?o No possible order among the clocks
0 0 0 0 0 0 0 0 0
+1
+0
Asynchronous
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Logical Clock Synchronization
25
o Successor Function and Predecessor Function possible if K≥3
o K=3o Local total order ≤l over Z3={0,1,2}
a ≤l b iff 0≤ b-a mod 3 ≤ 1
0 ≤l 1 ; 1 ≤l2 ; 2 ≤l 0 ;
Logical Clock Synchronization
26
o K=3, Lifting
0
0
0 0
0 0
00
0
0
0
0
p.R, virtual register, it countsthe number of increments of p
0
0
State of the virtualp.R lifted from
Projection mod K
0
State of p.r
Logical Clock Synchronization
27
o K=3, Lifting
0
1
1 1
1 0
01
1
1
1
0
0 1
D0
0 1
D0
Logical Clock Synchronization
28
o K=3, Lifting
0
1
1 1
2 1
01
1
2
1
1
0 1 2
D0 D1
0 1 2
D0 D1
Logical Clock Synchronization
29
o K=3, Lifting
0
2
1 1
2 2
01
1
2
2
2
0 1 2…D0 D1 D2
0 1 2…D0 D1 D2
Logical Clock Synchronization
30
o K=3, Lifting
0
0
1 1
2 2
01
1
2
3
2
0 1 2…D0 D1 D2
0 1 2…D0 D1 D2
Logical Clock Synchronization
31
The lifting is compatible with the local ordering of ZK
0 1 2…D0 D1 D2
0 1 2…D0 D1 D2
q.r
p.rq.R
p.R
p.R ≤ q.R
p.r ≤l q.r
Lifted execution
Execution
The lifting defines a global preorder over the nodes
32
1
0K-1
K-2
Generalization over ZK
o (Local) total order ≤l over ZK={0,1,2,…,K}o Let M ≥ 1 and K ≥ 2M+1
a ≤l b iff 0≤ b-a mod K ≤ M
With M=2 and K=5, then for 14 ≤l 1 ; 0 ≤l1 ; 1 ≤l 1 ; 1 ≤l 2 ; 1 ≤l 3 ;
33
Complexities
o Space : O(1)o Time, for one pulse:
o Best Case: O(1)o Worst Case: O(D) ← [DELAY]
34
Fault Tolerance
Starting from an arbitrary configuration ?
Self-stabilization
35
A self-stabilizing system, regardless of its initial state, is guaranteed to converge to the intended behavior in finite time. [Dijkstra 74]
A self-stabilizing system, regardless of its initial state, is guaranteed to converge to the intended behavior in finite time. [Dijkstra 74]
correct correctstabilization phase
Tra
nsi
ent
Fau
lt
Transient Faults
After a fault or an arbitrary initialization
36
o K=3
0 1 2…D0 D1 D2
0 1 2…D0 D1 D2
0
0
1 1
2 2
After a fault or an arbitrary initialization
37
o K=3
0 1 2…D0 D1 D2
0 1 2…D0 D1 D2
0
0
2 1
1 2
01
5
4
3
2
DEADLOCK!
DEADLOCK!
After a fault or an arbitrary initialization
38
o K=60
1
2
3
0
5
4
01
2
34
5
6
After a fault or an arbitrary initialization
39
o K=61
1
2
3
0
5
4
01
2
34
5
6
After a fault or an arbitrary initialization
40
o K=61
1
2
3
0
5
4
01
2
34
5
6
MUTUAL
EXCLUSION!MUTUAL
EXCLUSION!
41
What are the conditions to be able to (re)-synchronize the system ?
After a fault or an arbitrary initialization
42
System Configurations
: The register values are arbitrary
3 0
1
1
21 10
No Deadlock
1 : The register of two neighboring process is less than or equal to 1
01 : There is no deadlock (there exists a compatible lifting to
the configuration)
1 3
1
2
3
1 2
1
2
0
K=4
43
System Configurations
o CG (CG n), the cyclomatic characteristic of
the graph: Equal to the size of the greatest cycle in one of the cycle basis of G where the size of the greatest cycle is minimum (equal to 2 if G is acyclic)[Boulinier, Petit, and Villain, PODC 2004]
THEOREM:
K>CG═> 1 =10
: The register values are arbitrary
1 : The register of two neighboring process is less than or equal to 1
01 : There is no deadlock (there exists a compatible lifting to
the configuration)
10/04/23 44
Cyclomatic characteristic of G
CG=4 in meches and tories
10/04/23 45
CG=6 in honeycombs
Cyclomatic characteristic of G
10/04/23 46
CG=4 in hypercubes
Cyclomatic characteristic of G
10/04/23 47
CG=n on rings
Cyclomatic characteristic of G
48
Logical Clock Synchronization
o To avoid deadlock due arbitrary initial values, K must be greater than CG (CG
n), the cyclomatic characteristic of the graph
1
0K-2
K>CG
Values of Register r
49
System Configurations
1
= 10
No Deadlock
?
: The register values are arbitrary
1 : The register of two neighboring process is less than or equal to 1
01 : There is no deadlock (there exists a compatible lifting to
the configuration)
Stabilization
o Global Reset (Stabilizing PIF), implies a root or IDso Local Reset (also woks in anonymous networks)
QUESTION: What is the motivation behind anonymity?
“Only a few amount of bits allows to distinguish a huge number of nodes!”
o Lack of (underlying) infrastructure
No unique identifier assignation or no central process
No maintenance of any distributed structure
o Economic advantages
o User privacy preserved [Delporte-Gallet, Fauconnier, Guerraoui, and Ruppert, OPODIS 2007]
o No one-to-one routing
51
Advantages of Anonymous Solutions
52
Self-Stabilizing Logical Clock Synchronization
o To avoid starvations due arbitrary initial values, K must be greater than CG (CG
n), the cyclomatic characteristic of the graph
o Safety eventually guaranteed using a reset mechanism: Register r is set in [-,…,0]
o To avoid starvations due arbitrary initial values, must be greater than than HG+1 (HG
n), the greatest chordless cycle of the graph [Boulinier, Petit, and Villain, PODC 2004]
-
K-1-1
-2
-3
> HG+1
1
0K-2
K>CG
Values of Register r
53
Self-Stabilizing Logical Clock Synchronization
-
K-1-1
-2
-3
> HG+1
1
0K-2
K>CG
Values of Register r
SSSyncAN: qNp: Correctp(q) <<application>>;rp:=(rp);AR: (qNp: Correctp(q) ) (rptail) rp:=-; //resetAC: rp init*( qNp: rqinit*rp≤(rq) rp:=(rp):
54
Asynchronous, Anonymous Logical Clock, Related Works
Gouda, Couvreur, Francez, 1992
# of states Stabilizing Time
O(n²) O(nd)
o SSSync(K,α,M)
O(nd) O(nd)α=0, K>M.CG, M>HG-2
O(n) O(n)α>HG-2, M=1, K>CG
Sur un arbre
Dolev, 2000 O(n²) O(d)
O(1) O(d)
55
Wave Algorithms in Anonymous Networks
o Anonymous global ( = Diam) waves for non fault-tolerant systems [Tel, Concurrency’88]
o All existing solutions for coordination problems (either global, local or at distance ) are based on: Existence of unique IDs (or a root) Underlying wave mechanism
QUESTION: Can we provide wave algorithms using neither IDS nor root, i.e., working in an anonymous system?
56
Logical Clock Synchronization in Asynchronous Settings
o Let us observe a possible execution of the self-stabilizing phase clock synchronization again
o The system (a chain of 4 nodes) is stabilizedo K=5
1
1
1
1
2
2
2
3
3
3
2
4
4
4
3 4
0
0
0
0
P0 P1 P2 P3 P4
0
0
time
During the execution, coherent phase cuts Pi are built defined by the same value i on each clock register.
57
Logical Clock Synchronization in Asynchronous Settings
o Let us observe a possible execution of the self-stabilizing phase clock synchronization again
o The system (a chain of 4 nodes) is stabilizedo K=5
1
1
1
1
2
2
2
3
3
3
2
4
4
4
3 4
0
0
0
0
P0 P1 P2 P3 P4
0
0
time
During the execution, coherent phase cuts Pi are built defined by the same value i on each clock register.
58
Phase Clock Synchronization in Asynchronous Settings
o Observe that:
1. After stabilization, the phase clock synchronization algorithm provides a wave stream
0
0
0
0 1
1
1
1
2
2
2
2 D
D
D
D
D+1
D+1
D+1
D+1
D+2
D+2
D+2
D+2
Stabilization(Convergence)
2. In the future of a coherent phase Pi , the dth phase clock incrementation causally depends on nodes at distance ≤d.
o When the clock register r is equal to a particular value, e.g., 0, add a particular event “DECIDE”
Self-Stabilizing -Wavelets
THEOREM: The self-stabilizing phase clock synchronization provides a self-stabilizing -wavelet mechanism
60
Self-Stabilizing -Wavelets
0
0
0
0
D
D
D
D
2D
2D
2D
2D
Propagation Feedback
o For instance, if =Diam, the above protocol provides a Propagation of Information with Feedback over an anonymous network
61
Self-Stabilizing -Local Infimum Computation
0
0
0
0
D
D
D
D
2D
2D
2D
2D
Propagation Feedback
For instance, =
{a,e}
{b}
{a,b,c,f}
{d}
{a,b,c,d,e,f}
o An infimum over a set S is an associative, commutative, and idempotent (xx=x) binary operator.
62
Self-Stabilizing -Local Infimum Computation
0
0
0
0
D
D
D
D
2D
2D
2D
2D
Propagation Feedback
For instance, = min
{9}
{2}
{6}
{0}
{0}
o An infimum over a set S is an associative, commutative, and idempotent (xx=x) binary operator.
63
Self-Stabilizing -Barrier Synchronization
o A two-layer phase clock algorithm
o The upper layer clock is driven by the lower layer clock
o In other words, every node moves the upper layer clock rUP from i to i+1 (mod KUP) each time the lower layer clock incremented the lower layer clock, rLOW, times
o The upper layer implements a self-stabilizing -Barrier Synchronization, i.e., Phase i+1 (mod KUP) is executed by each node only after all nodes at distance less than or equal to have completed Phase i (mod KUP)
64
Self-Stabilizing -Local Resource Allocation
o Let be a reflexive binary compatibility relationship over A, a set of resources.
o If (a,b) , then a and b are compatible, otherwise a and b are conflicting.
o The -Local Resource Allocation is specified as follows:
o Safety: If two nodes at distance less than or equal to execute their critical section simultaneously, then both are using compatible resources
o No starvation (Vivacity): If a node requests a resource to enter the critical section, then its request is eventually satisfied and it enters the critical section
65
An example : -Local Mutual Exclusion
o Let A be a set of IDs s.t. for every pair (a,b) in A, a (resp. b) is conflicting with b (resp.a) (i.e., =)
o Given two neighboring nodes P,Q, rP (resp. rQ) being the upper clock of P (resp. Q):
(rP ,IDP) (rQ ,IDQ)iff
(rP <(mod K) rQ) or ((rP =(mod K) rQ) and (IDP<IDQ))
o Mutual Exclusion at distance is provided by applying the order relation in each phase transition
o The stabilization time is in O(n) time unitso The space complexity is in O(log Diam)
Enabling Snap-Stabilization in Anonymous Networks
QUESTION: Can we design snap-stabilizing waves algorithms in anonymous networks?
Self-stabilization
67
A self-stabilizing system, regardless of its initial state, is guaranteed to converge to the intended behavior in finite time. [Dijkstra 74]
A self-stabilizing system, regardless of its initial state, is guaranteed to converge to the intended behavior in finite time. [Dijkstra 74]
correct correctstabilization phase
Tra
nsi
ent
Fau
lt
Stabilization and Safety
A self-stabilizing system, regardless of the initial state of the processors, is guaranteed to converge to the intended behavior in finite time.
Dijkstra 1974
A priori, Safety is not guaranted during the faults
A snap-stabilizing system, regardless of the initial state of the processors, is guaranteed to converge to the intended behavior in 0 times.
Bui, Datta, Petit, Villain 1999
Safety is lost during faults and system stabilization.
Safety is lost during faults only.
User’s Point of View
X F(X)
Network
F(.)
X F(X) ?F(.)
User’s Point of View
1.X ≠ F(X)F(.)2.X ≠ F(X)3.X ≠ F(X)4.X ≠ F(X)N.X F(X)
Self-Stabilizing Systems
User’s Point of View
1.X F(X)F(.)
Snap-Stabilizing Systems
User’s Point of View
correctstabilization phase
illeg
itim
ate
leg
itim
ate
correctT
ran
sien
t F
ault
A general technique to deal with transient faults
Self-Stabilization
correct correctstabilization phase
illeg
itim
ate
leg
itim
ate
stabilization time
enabled starting actions legitimate initial configuration
Request Request Request Request
Self-Stabilization
Tra
nsi
ent
Fau
lt
correct
abn
orm
aln
orm
al
first enabled starting action
always correct
Quality of Service
Request
can be abnormal
NORMAL initial configuration
Request
but service is ok
Snap-Stabilization
Tra
nsi
ent
Fau
lt
76
Coherent Cuts Detection
0 1 2
10 2
If I execute 3 consecutive normal
actions, my neighbors execute
at least one
77
If I execute x consecutive normal
actions, then my neighbors execute
at least x-2
0 1 2
10 2
0 1 2
10 n
n
n-12
Generalization
Coherent Cuts Detection
78
I can extend the phenomenon to my
neighbors
0 1
1
nn-1
n-1
2 n-2
3 n-3
Coherent Cuts Detection
79
10 11 12
11
20
19
12 18
13 17
14 15 16
13 14 15 16 17 18 19
15
15
15
[Cu, Cu+D]
16
16
16
14
14
14
After an enought long normal action sequence (NA), a coherent phase is guaranteed in the past.
Coherent Cuts Detection
80
11 0 -α
Reset1
2
…0
0 can mask a reset.If I want to know the
behavior of the network, I must avoid this value
Coherent Cuts Detection
81
Snap-Stabilization Wave Protocol for Anonynous Networks
After an enought long normal action sequence (NA), a coherent phase is guaranteed in the past.
A wave can computed after a sequence of 2D+3 normal actions passing through 0 and with an initial value greater than D.
82
A wave is requestedby an application
D < clock < K-2D-4
2D+3 consecutiveincrementations DECIDE
No, a reset occured
Yes
No
Yes
Snap-Stabilization Wave Protocol for Anonynous Networks
K > 3D+5
83
Enabling Snap-Stabilization in Anonymous Networks
Snap-Wave Toolbox
Underlayer Unison
Application layer
Request Answer
Some references
84
• UNISON and its applications• Boulinier, Petit, Villain [PODC 04]
When Graph Theory helps Self-Stabilization• Boulinier, Petit, Villain, [Algorithmica 08]
Synchronous vs. Asynchronous Unison• Boulinier, Petit, [APDCM 08]
Self-Stabilizing Wavelets and -Hop Coordination
• Boulinier, Levert, Petit : [ICDCN 08]Snap-Stabilizing Waves in Anonymous Networks
• Snap-Stabilization• Bui, Datta, Petit, Villain [Distributed Computing 07]
Snap-stabilization and PIF in Tree Networks
http://graal.ens-lyon.fr/~fpetit
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