virtual backbone
DESCRIPTION
Distributed algorithm for efficient construction and maintenance of connected k-hop dominating sets in mobile ad hoc networks. Virtual Backbone. The virtual backbone (VBB) is usually formed by a connected dominating set - PowerPoint PPT PresentationTRANSCRIPT
Distributed algorithm for efficient construction and
maintenance of connected k-hop dominating sets in mobile
ad hoc networks
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The virtual backbone (VBB) is usually formed by a connected dominating set Each node not in the dominating set is a neighbor of
a node in the dominating set The nodes in the dominating set induce a connected
subgraph
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A connected dominating set
A connected k-hop dominating set (CKDS) can further reduce the flooding search space Each node not in CKDS can be reached within k
hops from at least one node in CKDS The nodes in CKDS induce a connected subgraph
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A connected 2-hop dominating set
Require a distributed algorithm to construct and maintain a CKDS !
Sufficient Condition Construction of CKDS Maintenance of CKDS Routing Based on CKDS Analyses Simulations
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bb
Each node knowing the disappearance of node b has the same two hop information in two cases
VBB
x.num: each node x is assigned a chosen number
P(x): x’s neighbor with larger chosen number or equal chosen number and larger ID EX: P(e)={i, j, f}
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(j.num,j.id)>(e.mum,e.id)
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(f.num,f.id)>(e.num,e.id)
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Theorem 3.1. Assume that G is connected graph in which each node x has a chosen number such that x.num= ∞ if and only if x belongs to VBB. If P(x) induces a connected subgraph of G for all x does not belong to VBB, then VBB is connected
x.num=∞ if and only if x belongs to VBB o, p, q are in VBB
If P(x) induces a connected subgraph of G for all x does not belong to VBB, then VBB is connected
Theorem 3.1. Assume that G is connected graph in which each node x has a chosen number such that x.num= ∞ if and only if x belongs to VBB. If P(x) induces a connected subgraph of G for all x does not belong to VBB, then VBB is connected1) It suffices to show there is a path in VBB between any two nodes I and j in VBB
x yi ja
b
c
b
∞ ∞
a c
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Select a node x such that (x.num,x.id)<(y.num,y.id) for all nodes y in the path
Nodes a and c must be in P(b) The path is updated to i,a,…,c,j
b no longer appears in the updated path
X
Sufficient Condition Construction of CKDS Maintenance of CKDS Routing Based on CKDS Analyses Simulations
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Node x is pruned by Dai and Wu’s method if 1) All x’s neighbors are directly connected (Marking
Process)2) x’s neighbors are covered by a connected subgraph
induced by x’s neighbors having larger ID than x (K-Pruning)
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“An Extended Localized Algorithm for Connected Dominating Set Formation in Ad Hoc Networks,” IEEE TPDS, 2004.
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Assume max_id=255 Assume a.id=1, b.id=2, and so on
Use Dai and Wu’s method to prune nodes in k rounds
if x is pruned in round t, x’s chosen number is assigned to (t-1)max_id+x.id
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∞ ∞
∞
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∞∞
∞
∞
∞ ∞
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e
e.num=(1-1)*255+5=5
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A connected 2-hop DS
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Use Dai and Wu’s method to prune nodes in k rounds
if x is pruned in round t, x’s chosen number is assigned to (t-1)max_id+x.id
Theorem 3.2. At the end of the construction process, CKDS={x|x.num= ∞} is a connected k-hop dominating set and P(x) induces a connected subgraph of G for all x does not belong to CKDS1) Nodes pruned in round t1 has a smaller chosen number than nodes pruned in round t2 if t1<t2
2) P(x) are x’s neighbors in round t having a larger ID than x if x is pruned in round t3) It suffices to show P(x) induces a connected subgraph of G if Case 3.1: x is pruned by Marking Process in round t Case 3.2: x is pruned by K-Pruning in round t
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P(i)={p,j}
Theorem 3.2. At the end of the construction process, CKDS={x|x.num= ∞} is a connected k-hop dominating set and P(x) induces a connected subgraph of G for all x does not belong to CKDS3.1) x is pruned by Marking Process in round t
x’s neighbors in round t are directly connected P(x) are x’s neighbors in round t having a larger ID than x if x is pruned in round t (by 2)
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Theorem 3.2. At the end of the construction process, CKDS={x|x.num= ∞} is a connected k-hop dominating set and P(x) induces a connected subgraph of G for all x does not belong to CKDS3.2) x is pruned by K-Pruning in round t
x’s neighbors in round t are covered by a connected subgraph induced by x’s neighbors in round t having larger ID than x P(x) are x’s neighbors in round t having a larger ID than x if x is pruned in round t (by 2)
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Sufficient Condition Construction of CKDS Maintenance of CKDS Routing Based on CKDS Analyses Simulations
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Keep each node not in CKDS is reached within k hops from at least one node in CKDS
Keep the connectivity of CKDS
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a
x.parent: a node in P(x) x.down: the maximum hop distance from x to its
descendants x.up: the hop distance from x to its dominator
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x.down=0 if x is not the parent of any node
x.up=0 if x is in CKDS
b.up=1
o.up=0
b.down=1a.down=0
x.down=1+max{y.down|y.parent=x}
b
x.up =1+y.up if y=x.parent
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x.down+z.up < k, where z=x.parent
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x.parent: a node in P(x) x.down: the maximum hop distance from x to its
descendants x.up: the hop distance from x to its dominator
Example of switch-on
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b.up=1
o.up=0
b.down=1a.down=0
r r.up=1r.down=0
e e.down=0e.up=1+1=2
Example of switches-off
Keep each node not in CKDS is reached within k hops from at least one node in CKDS
Keep the connectivity of CKDS
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Sufficient Condition: each node x not in CKDS keeps P(x) induces a connected subgraph Switch-on Switch-off Movement
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Each node in CKDS is expected to be removed from CKDS CKDS may become disconnected due to the simultaneous
removal of two neighboring nodes in CKDS
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o.pri=5
q.pri=6p.pri=3p.pri=7
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Sufficient Condition Construction of CKDS Maintenance of CKDS Routing Based on CKDS Analyses Simulations
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Each node keeps a descendant list Each node in CKDS keeps a dominator routing table
p (c,f,g,i) …
q (l,n,r) …
j (d,e) … q’s descendant list={l,n,r}
o’s dominator routing table
SourceDestination
a
l
r n’s descendant list={l}
Sufficient Condition Construction of CKDS Maintenance of CKDS Routing Based on CKDS Analyses Simulations
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Theorem 3.9. The computation complexity of the maintenance of a CKDS for each node is O(∆2), where ∆ is and the maximum degree of a node in the network1) Each node x needs O(∆2) time to verify if P(x) induces a connected subgraph using depth-first search2) Each node x needs O(∆) time to 2.1) Set a maximum or minimum chosen number among neighbors 2.2) Find a parent 2.3) Update down and priority numbers 2.4) …etc
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Theorem 3.10. In the maintenance of a CKDS, the node sends O(logn) and O(∆logn) bits in a round if each node has and has not the position information, respectively, where n denotes the number of nodes in the network1) To check if P(x) induces a connected subgraph 1.1) x.id: O(logn) 1.2) x.num O(logn) 1.3.1) x’s neighbor set if x has not location information: O(∆ logn) 1.3.2) x’s position if x has location information: O(1)2) To keep the distance property 2.1) x.parent: O(logn) 2.2) x.up: O(1) 2.3) x.down: O(1)3) To prevent the simultaneous removal of two neighboring nodes in CKDS 3.1) x.pri: O(1)
Lemma 3.3. Given an area A having measure equal to |A| and k cells in A a 1, a2,…,ak each having measure equal to |a|. If j nodes are randomly distributed in A, the probability that at least one node exists in each of k cell is at least
1) Let pk,i be the probability that at least one node exists in cell a i+1 under the condition that at least one node exists in each of cells a 1, a2,…,ai.
2) pk,i is not less than the probability that at least one node exists in cell ai+1 as j-I nodes are randomly distributed in A
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1
0
| |(1 (1 ) )
| |
kj i
i
a
A
1
,0
k
k k ii
p p
,
| | | |1 ( )
| |j i
k i
A ap
A
Pk,3
Under a1, a2, a3 each has at least one node
the probability of a4 has a least one node
1
0
| |(1 (1 ) )
| |
kj i
ki
ap
A
i nodes in cell a1, a2,…,ai
j-i nodes at least one in cell ai+1
A
ak
a2a1
Q(y): the set of y’s neighbor having larger ID than y Pt,1: the probability that |Q(y)|is t
Pt,2: the probability that no node exists in all square cells with x
Pt,3: the probability that each square cell with has at least one node in Q(y) as |Q(y)|=j and no node exists in all square cells with x (by Lemma 3.3)
1
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| |(1 (1 ) )
| | | |
kt i
ti
ap
A u a
d
,2
| | | |( )
| |
where | | is circle size
and | | is cell size
t
A u ap
A
A
a
y is located on the center of the circle with radius R A node in a square cell can communicate with any node in the neighboring
square cells The square cell with contains at least one node having larger ID than y The square cell with х contains no node
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Theorem 3.4. The probability that node y is pruned by K-Pruning is at least
Where and (0,8) 8 (1,8) 28 (2,8) 34 (3,8) 4 (3.75,6) 8 (2.38,7) 8 (3.38,7),f f f f f f f
| ( ) |d N y
1
0
1(1 ) (1 (1 ) )
8 8( , )
1
kdd j i
j k i
u
f u kd
y
,1
1
d 1tp
,1 ,2 ,30
( , )d
t t tt
f u k p p p
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Theorem 3.4. The probability that node y is pruned by K-Pruning is at least
Where and (0,8) 8 (1,8) 28 (2,8) 34 (3,8) 4 (3.75,6) 8 (2.38,7) 8 (3.38,7),f f f f f f f
| ( ) |d N y
1
0
1(1 ) (1 (1 ) )
8 8( , )
1
kdd j i
j k i
u
f u kd
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Lemma 3.6. The expected overlapped area of two circles each containing
the center of the other and having radius r is
23 3( )
4r
x yr
“Computing Subgraph Probability of Random Geometric Graphs: Quantitative Analysis of Wireless Ad Hoc Networks,” FORTE, 2005
S(x): the set of x’s neighbors entering CKDS after x.num is reset to ∞ y is x’s neighbor P(y) induces a connected subgraph of G before x.num is
reset to ∞ P(y) does not induce a connected subgraph of G after x.num
is reset to ∞
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22∞ ∞
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4 1 1
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∞
1
gb S(r)={b,g
}
Lemma 3.7. The expected number of nodes in S(x) is at most where and
4 3 3(1 ) ,
44 3 3dp p
| |
| |
CKDSp
V | ( ) |d N x
∞
∞
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Lemma 3.7. The expected number of nodes in S(x) is at most , if d<<|V| where and
4 3 3(1 ) ,
44 3 3dp p
y1
x
∞
yd
Let (yi.num.yi.id)>(yi+1.num.yi+1.id)
Pi,1: the probability of yi.num≠∞
Pi,2: the probability that P(yi) does not induce a connected subgraph of G after x.num is reset to ∞ under the condition yi.num≠∞
The probability of y1, y2, …,yi-1 are not the neighbor of yi (not in the overlap area of the transmission ranges of yi and x)
y3y3
2 2
1,2 2
3 3( )
4( )ii
r rp
r
,1 ,21
d
i ii
p p
1
,10
'i
i kk
p p
p’0 : no neighbor in CKDS
p’1 : one neighbor in CKDS
x
y2y2
' (1 )d d k kk kp C P P
| |
| |
CKDSp
V | ( ) |d N x
Theorem 3.8. Assume that nodes are uniformly distributed in the network with degree d<<|V|. If node x switches on, switches off, moves, and is removed from CKDS, the expected numbers of reformation rounds are at most , , , and equal to 0, respectively, if α < 1, where
and
1) In each round, at least one node enters CKDS if CKDS needs reformation2) Switch-on:
3) Switch-off:
4) Movement:
5) Removed from CKDS:0
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1 1i
i
2 1
d
1 2 1
d
1
12 2
d
( 2)1
2 2
d
4 3 3(1 )
44 3 3dp p
| |
| |
CKDSp
V
Sufficient Condition Construction of CKDS Maintenance of CKDS Routing Based on CKDS Analyses Simulations
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Random-Walk Mobility and Gauss-Markov Mobility Models 0~100 km/hr (0~27.7m/s)
A message-passing synchronous network Communication proceed in synchronous rounds Every node exchanged message with its neighbors and did some
computation in each round Compared to
MCDS (Algorithmica, 1998) Dai and Wu’s method (TPDS, 2004) Alzoubi’s method (MobiHoc, 2004)
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(a) Number of nodes:50~150, mobile nodes:0, (α=1,β=1,γ=0.5) lower density
(b) Number of nodes:50~150, mobile nodes:0, (α=0.5,β=0.1,γ=1) larger density
1-MCDS < 1-CKDS = Dai and Wu’s method < Alzoubi’s method The size of VBB decreases as
The density increases k increases
(a) (b)
1-MCDS2-MCDS3-MCDS
1-CKDS, Dai
2-CKDS
3-CKDS
Alzoubi
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(a) Number of nodes:100, Random-Walk, mobile nodes:5~25, lower density
(b) Number of nodes:100, Random-Walk, mobile nodes:5~25, larger density
K-CKDS < Dai and Wu’s method < Alzoubi’s method
(a) (b)
Dai1-CKDS
2-CKDS3-CKDS
Alzoubi
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(a) Number of nodes:100, Random-Walk, mobile nodes:5~25, lower density
(b) Number of nodes:100, Random-Walk, mobile nodes:5~25, larger density
Dai and Wu’s method < K-CKDS << Alzoubi’s method The number of reformation rounds increases as
The number of mobile nodes increases The density decreases k increases
(a) (b)
Dai, K-CKDS
Alzoubi
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(a) Number of nodes:100, Random-Walk, mobile nodes:5~25, lower density
(b) Number of nodes:100, Random-Walk, mobile nodes:5~25, larger density
K-CKDS < Dai and Wu’s method < Alzoubi’s method The number of changed nodes increases as
The number of mobile nodes increases The density decreases k increases
(a) (b)
Dai
1-CKDS2-CKDS3-CKDS
Alzoubi