Asymptotic Critical Transmission Radius for Greedy Forward Routing in Wireless

Ad Hoc Networks

Chih-Wei Yi

Submitted to INFOCOM 2006

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Wireless Ad Hoc Networks

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Greedy Forward Routing

• What is greedy forward routing?– Packets are discarded if there is no neighbor which is nearer to the

destination node than the current node; otherwise, packets are forwarded to the neighbor which is nearest to the destination node.

– Each node needs to know the locations of itself, its 1-hop neighbors and destination node.

• Pros: easy implement

• Cons: deliverability

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Examples

uw4

v

w1 w2 w3

w5 w6w6 is a local

minimum w.r.t. v

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• Each node has an omni-directional antenna, and all nodes have the same transmission radii.

• CTR for GFR:

Critical Transmission Radius for Greedy Forward Routing

wuV vuvBwVvu ,, minmax

vu

w

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Random Deployment

• Deterministic deployment at some regular pattern is prohibited due to – Large network size– Harsh environment– Mobility

• Random deployment– Nodes are independently and

uniformly distributed in the deployment region

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r-Disk Graphs

• D: deployment region of unit-area

• Vn: a random point process with rate n over D

• r: transmission radius (a function of n)

• Gr(Vn): r-disk graph over Vn

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Relative Works: Critical Transmission Radius for Connectivity

• D is a unit-area square or disk.

• Vn is a uniform point process or Poisson point process.

. expconnected is Prlim

Then .ln

Let

e

n

nr

nrn

n

nVG

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Relative Works: the Longest Edge of the Gabriel Graph

• D is a unit-area disk.

• Vn is a Poisson point process.

• Let

• A Gabriel edge is called long if its length is larger than rn .

• The number of long Gabriel edges is asymptotically Poisson with mean 2e-ξ .

• The probability of the event that the length of the longest edge is less than rn is asymptotically equal to exp(-2e-ξ) .

. ln

2

n

nrn

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Main Results

• D is a convex unit-area region.

• Vn is a Poisson point process with rate n over D, denoted by Pn .

• Let

• Suppose nrn2 = (+o(1))lnn for some .

– If > 0, then (Pn) ≤ rn is a.a.s..

– If < 0, then (Pn) rn is a.a.s..

• Let Luv denote the lune area B(u,||u-v||)B(v,||u-v||).– If ||u-v|| = (1/)1/2, |Luv| = 1/0 .

– |Luv| = (||u-v||2)/0 .

. 6.1/1 22

332

0

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A Sufficient Condition

vu w

If some node exists on Luv, packets can be forwarded from u toward v.

Assume u needs to forward packets to v. Let w denote the intersection point of the ray uv and circle B(u,r).

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Minimum Scan Statistic

• Minimum scan statistic– D : the deployment region– C : the scanning set– VD : a point set– The minimum scan statistic for V (with

scanning set C) is the smallest number of points of V covered by a copy of C.

• Assume Cn=B(o,rn) and nrn2

=lnn.Let Sm(Vn,Cn) = minxD|Vn∩(Cn+x)|.

=1 is a threshold.– If >1, Sm(Vn,Cn)>0 is a.a.s..– If <1, Sm(Vn,Cn)=0 is a.a.s..

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Upper Bounds of the CTR

• For any > 0 and ||u-w|| = rn, we have

• Since /0 > 1, according to minimum scan statistics, there almost surely exist nodes on Luw. Therefore, u can forward packets toward v.

. ln

00

2

0

2

n

nrwuL n

uw

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A Necessary Condition

u v

Assume ||u-v||>r. If u can forward packets to v, there must exit nodes in Luv.

If we can find a pair of nodes u and v such that there doesn’t exist node in Luv, it implies ρ(Pn)rn.

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Lower Bounds of the CTR

uv

• For any <0 , we can find a pair of nodes u and v whose distance is larger than rn such that there is no other node on the lune Luv.

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Conclusion

• Threshold of critical transmission radius for greedy forward routing

• Future works– Critical transmission radius for other geographic routing heuristics

– Relation between the length of the longest edge of the relative neighbor graph and the critical transmission radius for the greedy forward routing