1. A Unified View to Greedy Routing Algorithms in Ad-Hoc Networks
○Truong Minh Tien
Joint work with
Jinhee Chun, Akiyoshi Shioura, and Takeshi Tokuyama
Tohoku University
Japan

2. Our Problem and Results
Problem: Geometric routing in ad-hoc network.
Main Results:
Give unified view to known greedy-type routing algorithms.
Propose new routing algorithms that works on Delaunay graphs.
Compare previous/new algorithms from the viewpoint of guaranteed delivery, fast transmission & power consumption.

3. Contents Ad-hoc network and geometric routing
Previous geometric routing algorithms
Desirable properties of routing algorithms
Comparison of algorithms
Generalized greedy routing algorithm
New greedy-type algorithms
Sufficient condition for guaranteed packet delivery

4. Ad-hoc Network
Self-organizing network without fixed pre-existing infrastructure
Communication between nodes are achieved by multi-hop links
Decentralized, mobility-adaptive operation
Network topology can be represented by undirected graph G=(V, E)
First, I will explain briefly about wireless ad hoc network.

5. Geometric Routing on Ad-hoc Network
Geometric Routing on Ad-hoc network G=(V,E)
Send packet from source node S to destination node T (position of T is known in advance) .
Packet is repeatedly sent from a node to its neighboring node.
No information of entire network; only local information around current node.
The routing algorithm used in the ad hoc network is geometric routing. More specifically, geometric routing is a routing algorithm from the source S to the destination T that satisfies: T S V

6. Greedy Approach for Routing Algorithms
Geometric Routing on Ad-hoc network G=(V,E)
Greedy approach is often useful:
Choose “closer” neighbor to destination in each iteration
Which neighbor to choose?
 Greedy Routing, Compass Routing, Midpoint Routing, etc.
The routing algorithm used in the ad hoc network is geometric routing. More specifically, geometric routing is a routing algorithm from the source S to the destination T that satisfies: T S V

7. Contents Ad-hoc network and geometric routing
Previous geometric routing algorithms
Desirable properties of routing algorithms
Comparison of algorithms
Generalized greedy routing algorithm
New greedy-type algorithms
Sufficient condition for guaranteed packet delivery

8. Greedy Routing v t
Well, the first and most popular routing algorithm is greedy routing , that was proposed by Finn in The greedy routing algorithm is based on the idea that we want to get as closer to the destination as possible. Therefore when the packet arrives at V, the next neighbor that we want to send the packet to is the node nearest to the destination. In the figure, the next neighbor is w4
Finn, 1987
The next neighbor w is the node nearest to t
smallest 𝑑𝑖𝑠𝑡(𝑤,𝑡)

9. Compass Routing v t Kranakis, Singh, Urrutia, 1999
smallest ∠𝑤𝑣𝑡 v t
The next routing algorithm is compass routing proposed in In the compass routing, among all the neighbors of V, the packet will be sent to w if the line vw form with the line vt the smallest angle.
Kranakis, Singh, Urrutia, 1999
Packet will be sent to w if the line vw forms with vt the smallest angle.

10. Midpoint Routing v m t Si, Zomaya, 2010
smallest 𝑑𝑖𝑠𝑡(𝑤,𝑚) v m t
Next is midpoint routing, proposed recently by Si and Zomaya in Here we will choose the next neighbor w that is closest to the midpoint m, the midpoint between v and t.
Si, Zomaya, 2010
Choose next neighbor w that is closest to midpoint m between v an t

11. Modified Midpoint Routingv m p t
smallest 𝑑𝑖𝑠𝑡(𝑤,𝑝)
Finally, we consider the Modified midpoint routing. Let p be any node between m and t and the algorithms will choose the next neighbor that is closest to p. We notice that if p coincide t, the algorithm will become greedy routing. If p coincide m, the algorithm will become midpoint routing.
Si, Zomaya, 2010
The next node w closest to p
p = t : Greedy routing
p = m : Midpoint routing

12. Contents Ad-hoc network and geometric routing
Previous geometric routing algorithms
Desirable properties of routing algorithms
Comparison of algorithms
Generalized greedy routing algorithm
New greedy-type algorithms
Sufficient condition for guaranteed packet delivery

13. Desirable Properties of
Routing Algorithms
Guaranteed Delivery:
It is guaranteed that a packet is delivered from source to destination.
Fast Transmission:
Each packet should be sent with a small number of hops. S T

14. Desirable Properties of
Routing Algorithms
Power Consumption:
Long edges should not be used as much as possible.

15. Comparison of Routing Algorithms
Guaranteed delivery
Number of hops
Power Consumption
Greedy
guaranteed on Delaunay graph
very small
very large
Midpoint
small
large
Modified Midpoint
Compass
average
Need appropriate routing algorithm satisfying desirable properties in response to the request of applications.

16. Contents Ad-hoc network and geometric routing
Previous geometric routing algorithms
Desirable properties of routing algorithms
Comparison of algorithms
Generalized greedy routing algorithm
New greedy-type algorithms
Sufficient condition for guaranteed packet delivery

17. Generalized Greedy Routing
Unify greedy-type routing algorithms using general objective function.
Obtain better understanding of previous algorithms.
Propose new algorithms.
T = {(w ,v ,t) | w ,v ,t: distinct nodes}
(w: next node, v: current node, t: terminal node)
General objective function
Generalized greedy routing:
Choose a neighbor w of v that minimizes f (w, v, t) in each iteration
Let T be the set of triplets of w, v and t, where w, v, and t are distinct nodes. We define the general objective function f to be function from T to the union of R and infinity. Here we can unify greedy routing algorithms by generalized greedy routing with objective functions.

18. Generalized Greedy Routing: Example
Choose next node w that minimize f (w, v, t)
Example: 7 3 v t 2
The algorithm is that: we choose the next node w that minimize the function value f(w,v,t). In the example here, the function values are 7, 3, 2, and infinity, so we choose w3 to be next node on the route to the destination. +∞

19. Congruence-Invariant Functionw t v w’ v’
In particular, we consider a restricted class of objective functions, that is congruence-invariant function. We say that f is congruence-invariant if the function value f(w ,v ,t) depends only on the shape and size of the triangle wvt.
In other words, the function value does not change if we transform the triangle wvt by a combination of translation, rotation and reflection.
f is congruence-invariant if function value f (w ,v ,t) depends only on shape and size of

20. Congruence-Invariant Function
f is congruence-invariant function if there exists a function h such that: w
More precisely, f is said to be congruence-invariant function if there exists a function h such that … Now we show that many existing greedy routing algorithms that I explained earlier are special cases of generalized greedy routing. t v

21. w t w t v v w w t t p m v v Greedy Routing: Min d(w, t) function
Compass Routing: Min
function w t w t v v
Midpoint Routing: Min d(w, m)
function
M. Midpoint Routing: Min d(w, p)
function w w t t p m v v

22. New routing algorithms

23. New Greedy I New Greedy II New Greedy III w t v w t v w t v max
function w t v
New Greedy II
min
function w t v
New Greedy III
min
function w t v

24. Contour Map GREEDY - concentric circles about t
COMPASS – rays with same endpoint v
Figures here show the contour maps of the corresponding functions of these four routings. For the greedy routing, it’s a set of concentric circles about t. For compass routing it’s a set of rays with same endpoint v. For the midpoint routing, it’s a set of concentric circles about m. For the modified midpoint routing, it’s a set of concentric circles about p.
Next we will propose 3 new routing algorithms that are also special cases of generalized greedy routing.
MIDPOINT - concentric circles about m
MODIFIED MIDPOINT - concentric circles about p

25. Contour Map of New Routings
New Greedy I – curves with same chord vt
Figures here show the contour maps of the corresponding functions of these four routings. For the greedy routing, it’s a set of concentric circles about t. For compass routing it’s a set of rays with same endpoint v. For the midpoint routing, it’s a set of concentric circles about m. For the modified midpoint routing, it’s a set of concentric circles about p.
Next we will propose 3 new routing algorithms that are also special cases of generalized greedy routing.
New Greedy II – circles tangent at v
New Greedy III – circles tangent at t

26. Comparison of Routing Algorithms
Guaranteed Delivery
Number of hops
Power Consumption
Greedy
guaranteed on Delaunay graph
Very small
Very large
Midpoint
Small
Large
Modified Midpoint
Compass
New Greedy I
Average
New Greedy II
New Greedy III

27. Properties of New Greedy II, III
If graph G contains Delaunay graph.
New Greedy II : always selects Delaunay edge without calculating which edge is Delaunay edge.
New Greedy III : always selects Delaunay neighbor of t if there is a two-hop path from v to t .
Desired by many occasions.
New Greedy II –
circles tangent at v
Let T be the set of triplets of w, v and t, where w, v, and t are distinct nodes. We define the general objective function f to be function from T to the union of R and infinity. Here we can unify greedy routing algorithms by generalized greedy routing with objective functions.
New Greedy III –
circles tangent at t

28. Comparison of Routing Algorithms
Guaranteed Delivery
Number of hops
Power Consumption
Greedy
guaranteed on Delaunay graph
Very small
Very large
Midpoint
Small
Large
Modified Midpoint
Compass
New Greedy I
Average
New Greedy II
New Greedy III

29. Contents Ad-hoc network and geometric routing
Previous geometric routing algorithms
Desirable properties of routing algorithms
Comparison of algorithms
Generalized greedy routing algorithm
New greedy-type algorithms
Sufficient condition for guaranteed packet delivery

30. Delivery on Delaunay graph
Known results: Each of greedy, compass, midpoint and modified midpoint routing guarantee delivery of packet on Delaunay graph.
Our result: Sufficient condition for guaranteed delivery of generalized greedy routing on Delaunay graph.

31. Delaunay Delivery Guarantee Condition
(DDG) ∀distinct nodes w, v, t ∈ P,
if f(w ,v ,t) ≤ max{ f(u ,v ,t) | u ∈D(v ,t)},
then d(w ,t) < d(v ,t) holds
d(a ,b) : distance between a and b
D(v ,t): open disk of diameter d(v,t)
The DGS condition is that : For every distinct nodes w, v and t of P such that w and v are adjacent and t and v are not adjacent, if … , then the distance between w and t is less than the distance between v and t. Here c is the midpoint between v and t.

32. DDG Condition D C : open disk with diameter vt
D : open disk with radius tv
about t
A = max{f(u, v, t) | u ∈ C}
DDG Condition :
For all w with f(w, v, t) ≤ A ;
w ∈ D w C v t u
We can see how the DGS condition works in this figure.

33. Strong DDG Condition w C : open disk with diameter vt
For all 　　　　　and
Strong DDG implies DDG C v t u
We can see how the DGS condition works in this figure.

34. Delivery Guarantee on Delaunay triangulations
Theorem. f is a function satisfying (strong) DDG condition.  The algorithm with function f guarantees packet delivery on Delaunay triangulations. w v t u

35. Routing Algorithms and
DDG Condition
Theorem. Greedy Routing, Midpoint Routing and Modified Midpoint Routing satisfy DDG condition
Theorem. New Greedy Routing I, II, III satisfy Strong DDG condition
 Guarantee delivery of packet on Delaunay graphs

36. Example: New Greedy I on Delaunay triangulationS
Or by using midpoint routing
And it can also be delivered by using modified midpoint routing, and New greedy I, II and III T

37. Hybrid of algorithms Theorem. If f and g satisfy (strong) DDG
condition, af+bg (a,b>0) also satisfies (strong)
DDG condition.
 Corresponding algorithm guarantees delivery on Delaunay triangulation
Possible to design appropriate hybrid of algorithms based on requirement of application.
Now we move on to the second part of finding the sufficient condition for delivery guarantee on the Delaunay triangulation.

38. Conclusion Our Problem: Geometric routing in Ad-hoc network
Our Results:
We gave unified view to known greedy-type routing algorithms.
We proposed new routing algorithms that works on Delaunay graphs.
We compared previous/new algorithms from the viewpoint of guaranteed delivery, fast transmission, & power consumption.

39. Future Work
Consider a metric space with the existence of obstacles and other natural/social conditions in real ad hoc network design.
That concludes my talk. Thank you for your attention

40. Thank You