Distributed algorithm for efficient construction and maintenance of connected k-hop dominating sets in mobile ad hoc networks 1

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 2 m o n b a h q p g ij cd e f l k 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 3 m o n b a h q p g ij cd e f l k 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 4

5 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} m o n b a h q p g ij cd e f l k ∞ ∞ ∞ e j f (j.num,j.id)>(e.mum,e.id) 6 (f.num,f.id)>(e.num,e.id)

m o n b a h q p g ij cd e f l k ∞ ∞ ∞ e j f i o p q 7 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 connected 1) It suffices to show there is a path in VBB between any two nodes I and j in VBB x y i j a b c b ∞ ∞ ac 8 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 9

m o n b a h q p g ji cd e f l k 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) m o n b a h q p g ij cd e f l k a b h e ji f “An Extended Localized Algorithm for Connected Dominating Set Formation in Ad Hoc Networks,” IEEE TPDS,

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 m o n b a h q p g ij cd e f l k ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 11 e e.num=(1-1)*255+5=5

m o n b a h q p g ij cd e f l k ∞ ∞ ∞ A connected 2-hop DS 12 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 CKDS 1) Nodes pruned in round t 1 has a smaller chosen number than nodes pruned in round t 2 if t 1 <t 2 2) P(x) are x’s neighbors in round t having a larger ID than x if x is pruned in round t 3) 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 m o n b a h q p g ji cd e f l k ∞ ∞ ∞ i d e p j g 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 CKDS 3.1) 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) m o n b a h q p g ij cd e f l k ∞ ∞ ∞ b o

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 CKDS 3.2) 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) m o n b a h q p g ij cd e f l k ∞ ∞ ∞ i p j

 Sufficient Condition  Construction of CKDS  Maintenance of CKDS  Routing Based on CKDS  Analyses  Simulations 16

 Keep each node not in CKDS is reached within k hops from at least one node in CKDS  Keep the connectivity of CKDS 17

m o n b a h q p g ij cd e f l k ∞ ∞ ∞ 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 18 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=1 a.down=0 x.down=1+max{y.down|y.parent=x} b x.up =1+y.up if y=x.parent o x.down+z.up < k, where z=x.parent

r i 19 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 m o n b a h q p g j cd e f l k ∞ ∞ ∞ b.up=1 o.up=0 b.down=1 a.down=0 r r.up=1 r.down=0 e e.down=0 e.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 20

10 s i 21 Sufficient Condition: each node x not in CKDS keeps P(x) induces a connected subgraph Switch-on Switch-off Movement m o n b a h q p g j cd e f l k ∞ ∞ ∞ r 267 m q n ∞ r s r p o ∞ s q

i 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 i m o n b a h q p g j cd e f l k ∞ ∞ ∞ 265 ∞ i 266 o.pri=5 q.pri=6 p.pri=3 p.pri=7 p

 Sufficient Condition  Construction of CKDS  Maintenance of CKDS  Routing Based on CKDS  Analyses  Simulations 23

j 24 m o n b a h q p g i cd e f l k 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 Source Destination a l r n’s descendant list={l}

 Sufficient Condition  Construction of CKDS  Maintenance of CKDS  Routing Based on CKDS  Analyses  Simulations 25

26 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 network 1) Each node x needs O(∆ 2 ) time to verify if P(x) induces a connected subgraph using depth-first search 2) 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 m o n b a h q p g ji cd e f l k ∞ ∞ ∞

27 Theorem 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 network 1) 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, a 2,…,a k 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 p k,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, a 2,…,a i. 2) p k,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 28 P k,3 Under a 1, a 2, a 3 each has at least one node the probability of a 4 has a least one node i nodes in cell a 1, a 2,…,a i j-i nodes at least one in cell a i+1 A akak a2a2 a1a1

Q(y): the set of y’s neighbor having larger ID than y P t,1 : the probability that |Q(y)|is t P t,2 : the probability that no node exists in all square cells with x P t,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) 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 29 Theorem 3.4. The probability that node y is pruned by K-Pruning is at least Where and y

30 Theorem 3.4. The probability that node y is pruned by K-Pruning is at least Where and

31 Lemma 3.6. The expected overlapped area of two circles each containing the center of the other and having radius r is xy r “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 ∞ 32 m o n b a h q p g ji cd e f l k ∞ ∞ ∞ r ∞ 1 g b S(r)={b,g} Lemma 3.7. The expected number of nodes in S(x) is at most where and ∞ ∞

33 Lemma 3.7. The expected number of nodes in S(x) is at most, if d<<|V| where and y1y1 x ∞ ydyd Let (y i.num.y i.id)>(y i+1.num.y i+1.id) P i,1 : the probability of y i.num≠∞ P i,2 : the probability that P(y i ) does not induce a connected subgraph of G after x.num is reset to ∞ under the condition y i.num≠∞ The probability of y 1, y 2, …,y i-1 are not the neighbor of y i (not in the overlap area of the transmission ranges of y i and x) y3y3 y3y3 p’ 0 : no neighbor in CKDS p’ 1 : one neighbor in CKDS x y2y2 y2y2

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 reformation 2) Switch-on: 3) Switch-off: 4) Movement: 5) Removed from CKDS:0 34

 Sufficient Condition  Construction of CKDS  Maintenance of CKDS  Routing Based on CKDS  Analyses  Simulations 35

36 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)

37 (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-MCDS 2-MCDS 3-MCDS 1-CKDS, Dai 2-CKDS 3-CKDS Alzoubi

38 (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) Dai 1-CKDS 2-CKDS 3-CKDS Alzoubi

39 (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

40 (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-CKDS 2-CKDS 3-CKDS Alzoubi