CPSC 689: Discrete Algorithms
for Mobile and Wireless Systems
Spring 2009
Prof. Jennifer Welch
Lecture 28
 Topic:
 Distributed Primitives Using Link Reversal
 Sources:




Walter, Welch & Vaidya
Malpani, Welch & Vaidya
Ingram, Shields, Walter & Welch
MIT 6.885 Fall 2008 slides
Discrete Algs for Mobile Wireless Sys
2
Mutual Exclusion in MANETs [WWV]
 Problem statement:
 Assume a single connected component
 No two nodes are ever in the critical section
(CS) simultaneously
 If topology changes eventually cease, then
every request to enter the CS is eventually
granted.
Discrete Algs for Mobile Wireless Sys
3
Mutual Exclusion
 Raymond's (1989) ME algorithm:
 unique token that traverses the network over a
spanning tree
 node with token can enter CS
 nodes that want to enter CS send requests toward the
token holder
 What happens if links disappear?
 Give a logical direction to all the links, not just those in
a spanning tree, to get more redundancy
 Use link reversal ideas with a twist: token-holder is
destination (of request messages), but different nodes
are token-holder at different times
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ME Algorithm [WWV]



Heights are triples (partial reversal, as in [GB])
Each node keeps a queue of ids of neighbors that have sent it requests
Each node identifies one of its outgoing links as its next link.
1 3
• node 1 requests
3
0
3 2
1
1
2
2
• node 3 requests
• node 2 requests
• node 0 leaves the CS,
forwards the token to 3,
and sends a req. to 3 on
behalf of 2. When 3
gets the token, it lowers
its height…
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5
ME Algorithm with No Link Failures
130
3
30
0
2
2
1
1
node 0
forwards
token to 2,
etc.
1
3
node 3 forwards
token to 1, follows
it with a request
0
2
2
1
1 3
2
3
0
2
2
2
node 3 enters
CS since
it is at head of
queue; when
done, forwards
token to 0
2
node 1 enters CS since
it is at head of queue; when
done, forwards token to 3
30
3
0
2
1
2
2
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ME Algorithm with Failures
1 3
3
0
3 2
1
1
2
2
0 removes 3
from queue;
3 raises ht.,
purges queue,
sends req. on
new next to 1
3
3
0
2
2
1
1 3
A node that loses its last outgoing
link either due to link failure or due
to nbr’s height change does a
partial reversal to raise its height.
Queues must be adjusted.
2
1 raises ht., sends req.
on new next to 2
3
0
2
1
1 3
Discrete Algs for Mobile Wireless Sys
3
2
2 1
7
Correctness of ME Algorithm
 Safety: There is never more than one node
in the CS at a time.
 The unique token ensures this.
 Liveness: Eventually every request to
enter the CS is granted.
 To prove this, we use a "variant function"
argument…
Discrete Algs for Mobile Wireless Sys
8
Request Chain and Variant Function
In this state, node 1’s request chain is:
3
0
node 1, node 3, node 0.
I.e., the red path.
3 2
Node 1’s variant function is [1,1,1].
1
2
Node 2’s variant function is [1,2].
1
2
Node 3’s variant function is [2,1].
Given a node p1’s request chain, define a variant function:
1 3
p1
p2
,
pos. of p1
in p1’s
queue
p3
,
pos. of p1
in p2’s
queue
pm
pos. of p2
in p3’s
queue
,
,
pos. of pm-1
in pm’s
queue
Discrete Algs for Mobile Wireless Sys
If a request
message is
in transit
from pm,
then append
n+1.
9
Correctness of ME Algorithm
Assume finite number of link changes.
 finite number of calls to RaiseHt
 eventually DAG is token-oriented and stays that
way
 eventually node i’s request chain includes the
token holder
 variant function for a particular request keeps
decreasing, since nodes don’t stay in CS
forever and thus token is sent and received
 eventually variant function reaches [1] and
request is granted
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10
Leader Election in MANETs
 Problem Statement:
 Any connected component whose topology is static
sufficiently long eventually contains a unique leader
 [MWV] paper modified TORA:
 Key observation: when a node detects a partition from
the destination (current leader), instead of initiating
procedure to quiesce, it elects itself
 Add a 6th component to the height tuple of TORA
 But resulting algorithm does not handle concurrent
topology changes
 [ISWW] uses a 7-tuple for height…
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LE Algorithm: Height Data Structure
 reference level (RL):
 tau: either 0 or time when current search for alternative
path to leader was initiated
 oid: either 0 or id of node that started current search
 r: either 0 (nonreflected) or 1 (reflected)
 delta: integer to ensure links are directed
appropriately to neighbors with same RL
 leader pair (LP):
 nlts: negative of time when current leader was elected
 lid: id of current leader
 id: node's unique identifier
Discrete Algs for Mobile Wireless Sys
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LE Algorithm: Topology Changes
 When link to j goes down:
 if no neighbors then elect self
 else if sink then
 start new RL
 send new height to all neighbors
 When link to j comes up:
 send height to j
Discrete Algs for Mobile Wireless Sys
13
LE Algorithm: Response to New
Height Message
 if LP is same then
 if sink then
 if all neighbors have same RL then
 if common RL has r = 0 then reflect RL
 else if common RL has r = 1 and oid= i then
elect self
 else start new RL
 else propagate largest RL
 else // LPs are not same
 adopt LP if priority
 send new height to all neighbors
Discrete Algs for Mobile Wireless Sys
14
LE Algorithm: Subroutines
 elect self:
 set height to ((0,0,0),0,(-t,i),i), where t is current time
 reflect RL:
 set r to 1 and delta to 0
 propagate largest RL (partial reversal):
 set RL to largest neighboring RL, set delta to 1 less than
smallest neighboring delta associated with the new RL
 start new RL:
 set RL to (t,id,0) and delta to 0, where t is current time
 adopt LP if priority:
 if new LP has newer time or (equal time and smaller lid) then
set RL to new RL, delta to one larger than new delta, and LP
to new LP
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Example Execution
((0,0,0),4,(-1,H),A)
((0,0,0),3,(-1,H),B)
B
A
((0,0,0),2,(-1,H),D)
C ((0,0,0),3,(-1,H),C)
D
((0,0,0),2,(-1,H),E) E
((0,0,0),0,(-1,H),H)
H
F
G
((0,0,0),2,(-1,H),F)
((0,0,0),1,(-1,H),G)
at time 5
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Example Execution: Start New RL
((0,0,0),4,(-1,H),A)
((0,0,0),3,(-1,H),B)
B
A
((0,0,0),2,(-1,H),D)
C ((0,0,0),3,(-1,H),C)
D
((0,0,0),2,(-1,H),E) E
((0,0,0),0,(-1,H),H)
H
F
G
((0,0,0),2,(-1,H),F)
((5,G,0),0,(-1,H),G)
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Example Execution: Propagate
Largest RL
((0,0,0),4,(-1,H),A)
((0,0,0),3,(-1,H),B)
B
A
((5,G,0),-1,(-1,H),D)
C ((0,0,0),3,(-1,H),C)
D
((5,G,0),-1,(-1,H),E) E
((0,0,0),0,(-1,H),H)
H
F
G
((5,G,0),-1,(-1,H),F)
((5,G,0),0,(-1,H),G)
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Example Execution: Propagate
Largest RL
((0,0,0),4,(-1,H),A)
((5,G,0),-2,(-1,H),B)
B
A
((5,G,0),-1,(-1,H),D)
C ((5,G,0),-2,(-1,H),C)
D
((5,G,0),-1,(-1,H),E) E
((0,0,0),0,(-1,H),H)
H
F
G
((5,G,0),-1,(-1,H),F)
((5,G,0),0,(-1,H),G)
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Example Execution: Reflect RL
((5,G,1),0,(-1,H),A)
((5,G,0),-2,(-1,H),B)
B
A
((5,G,0),-1,(-1,H),D)
C ((5,G,0),-2,(-1,H),C)
D
((5,G,0),-1,(-1,H),E) E
((0,0,0),0,(-1,H),H)
H
F
G
((5,G,0),-1,(-1,H),F)
((5,G,0),0,(-1,H),G)
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Example Execution: Propagate
Largest RL
((5,G,1),0,(-1,H),A)
((5,G,1),-1,(-1,H),B)
B
A
((5,G,0),-1,(-1,H),D)
C ((5,G,1),-1,(-1,H),C)
D
((5,G,0),-1,(-1,H),E) E
((0,0,0),0,(-1,H),H)
H
F
G
((5,G,0),-1,(-1,H),F)
((5,G,0),0,(-1,H),G)
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Example Execution: Propagate
Largest RL
((5,G,1),0,(-1,H),A)
((5,G,1),-1,(-1,H),B)
B
A
((5,G,1),-2,(-1,H),D)
C ((5,G,1),-1,(-1,H),C)
D
((5,G,1),-2,(-1,H),E) E
((0,0,0),0,(-1,H),H)
H
F
G
((5,G,1),-2,(-1,H),F)
((5,G,0),0,(-1,H),G)
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Example Execution: Elect Self
((5,G,1),0,(-1,H),A)
((5,G,1),-1,(-1,H),B)
B
A
((5,G,1),-2,(-1,H),D)
C ((5,G,1),-1,(-1,H),C)
D
((5,G,1),-2,(-1,H),E) E
((0,0,0),0,(-1,H),H)
H
F
G
((5,G,1),-2,(-1,H),F)
((0,0,0),0,(-14,G),G)
at time 14
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Example Execution: Adopt New LP
((0,0,0),3,(-14,G),A)
((0,0,0),2,(-14,G),B)
B
A
((0,0,0),1,(-14,G),D)
C ((0,0,0),2,(-14,G),C)
D
((0,0,0),1,(-14,G),E) E
((0,0,0),0,(-1,H),H)
H
F
G
((0,0,0),1,(-14,G),F)
((0,0,0),0,(-14,G),G)
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Correctness Proof
 Relies on several invariant predicates:
 properties of the global state of the system that are
always true
 Show that if topology changes cease, then
eventually a leader-oriented DAG is obtained in
each connected component
 Show that a new leader is not elected
"unnecessarily":
 if a component is a leader-oriented DAG and one
topology change occurs, then no node remaining in
same component as the old leader elects itself
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Leader-Oriented DAG
 Lemma 1: Any node that adopts a LP with
timestamp after the last topology change never
subsequently becomes a sink.
 Lemma 2: Any node that elects itself after the last
topology change never subsequently becomes a
sink.
 Lemma 3: No node elects itself more than once
after the last topology change.
 Lemma 4: Every node starts a finite number of
new RLs after the last topology change.
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Leader-Oriented DAG
 Lemma 5: Eventually every node in the
component has the same LP.
 Lemma 6: Eventually no messages are in
transit.
 Lemma 7: Eventually every node has an
accurate view of its neighbors' heights.
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Leader-Oriented DAG
 Theorem 8: Eventually the component is a leader-oriented
DAG.
 Proof Sketch:
 Eventually all nodes have same LP (-s,L).
 Eventually every node has an accurate view of its neighbors'
heights.
 Properties tell us that





Node L must be in the component.
L and only L has RL (0,0,0) and 0 delta
no node has a negative RL
L has smallest height, so no outgoing links
every node other than L has an outgoing link
 Since cycles are impossible, L-oriented DAG.
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Leader Stability
 Theorem 9: Suppose G' is an L-oriented
DAG with no messages in transit and a link
goes down at time t. Let resulting
component containing L be G. Then, as
long as no further topology changes occur,
no node in G elects itself.
 Proof Sketch: Suppose node u becomes a
sink at time t and starts new RL (t,u,0).
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Leader Stability
 Suppose in contradiction some node elects itself
at time te.
 Claim 2: No new RL prefix (oid and timestamp) is
started between t and te.
 Thus the node elected at te must be u.
 Let A be nodes in G with a directed path to L at
time t (after the link goes down) and B be the
remaining nodes.
 L is in A and u is in B.
 Claim 3: No node in A becomes a sink between t
and te.
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Leader Stability
 At te, all neighbors of u have RL (t,u,1).
 By a property, there are no "height tokens" in the
system at te with RL (t,u,0)
 a node cannot have an unreflected RL while one of its
ancestors is reflected
 By another property, every node with RL (t,u,1)
has all its neighbors with RL (t,u,1).
 But then some node with RL (t,u,1) is a neighbor
of some node in A, which cannot have RL (t,u,1)
since it never becomes a sink. Contradiction.
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