a unified view to greedy routing algorithms in ad-hoc networks ○truong minh tien joint work with...
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A Unified View to Greedy Routing
Algorithms in Ad-Hoc Networks
○Truong Minh TienJoint work with
Jinhee Chun, Akiyoshi Shioura, and Takeshi Tokuyama
Tohoku University Japan
Our Problem and ResultsProblem: 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.
Contents1. Ad-hoc network and geometric routing
2. Previous geometric routing algorithms
3. Desirable properties of routing algorithms– Comparison of algorithms
4. Generalized greedy routing algorithm– New greedy-type algorithms
5. Sufficient condition for guaranteed packet delivery
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)
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.
S
V
T
Greedy Approach for Routing AlgorithmsGeometric 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.
S
V
T
Contents1. Ad-hoc network and geometric routing
2. Previous geometric routing algorithms
3. Desirable properties of routing algorithms– Comparison of algorithms
4. Generalized greedy routing algorithm– New greedy-type algorithms
5. Sufficient condition for guaranteed packet delivery
Greedy Routing
v t
smallest
Finn, 1987• The next neighbor w is the
node nearest to t
1w2w
3w
4w
Compass Routing
v t
smallest
Kranakis, Singh, Urrutia, 1999• Packet will be sent to w if the line vw
forms with vt the smallest angle.
1w2w
3w
4w
Midpoint Routing
mv t
smallest
Si, Zomaya, 2010• Choose next neighbor w that is
closest to midpoint m between v an t
1w2w
3w
4w
Modified Midpoint Routing
m pv t
smallest
Si, Zomaya, 2010• The next node w closest to po p = t : Greedy routingo p = m : Midpoint routing
1w2w
3w
4w
Contents1. Ad-hoc network and geometric routing
2. Previous geometric routing algorithms
3. Desirable properties of routing algorithms– Comparison of algorithms
4. Generalized greedy routing algorithm– New greedy-type algorithms
5. Sufficient condition for guaranteed packet delivery
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.
ST
Desirable Properties of Routing Algorithms
Power Consumption: Long edges should not be used as much as possible.
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 small large
Compass average average
Need appropriate routing algorithm satisfying desirable properties in response to the request of applications.
Contents1. Ad-hoc network and geometric routing
2. Previous geometric routing algorithms
3. Desirable properties of routing algorithms– Comparison of algorithms
4. Generalized greedy routing algorithm– New greedy-type algorithms
5. Sufficient condition for guaranteed packet delivery
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
}{: RTf
Generalized Greedy Routing: Example
v t
7
3
2
+∞
Choose next node w that minimize f (w, v, t)
Example:
1w2w
3w
4w
Congruence-Invariant Function
w’
tw t’
v’
v
• f is congruence-invariant if function value f (w ,v ,t) depends only on shape and size of .
)',','(),,( tvwftvwf
wvt
va
Congruence-Invariant Function
f is congruence-invariant function if there exists a function h such that:
t
w
v
),,,,,(),,( vwtvwwtvt aaadddhtvwf
vtd
wtdvwd
ta
wa
Greedy Routing: Min d(w, t) function wtG dh
tw
v
Compass Routing: Min function
wvtvc ah
Midpoint Routing: Min d(w, m) function
22 )2
cos()sin( vttwttwtMP
dadadh
M. Midpoint Routing: Min d(w, p) function
22 )cos()sin( vttwttwtMMP dadadh
tw
v
tw
vm
tw
v
p
)2
1(
New routing algorithms
New Greedy I max function
New Greedy II min
function
New Greedy III min
function
vwt
wah 1
tw
v
)cos(/),( tvwwvd )2/( tvw
)(cos 2/2 v
v
vw aa
dh
)cos(/),( wtvwtd )2/( wtv
)(cos 2/3 t
t
wt aa
dh
tw
v
tw
v
Contour Map
GREEDY - concentric circles about t COMPASS – rays with same endpoint v
MIDPOINT - concentric circles about m
MODIFIED MIDPOINT - concentric circles about p
Contour Mapof New
Routings New Greedy I – curves with same chord vt
New Greedy III – circles tangent at t
New Greedy II – circles tangent at v
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 Small Large
CompassNew Greedy I Average Average
New Greedy II Large Small
New Greedy III Small Large
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
New Greedy III – circles tangent at t
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 Small Large
CompassNew Greedy I Average Average
New Greedy II Large Small
New Greedy III Small Large
Contents1. Ad-hoc network and geometric routing
2. Previous geometric routing algorithms
3. Desirable properties of routing algorithms– Comparison of algorithms
4. Generalized greedy routing algorithm– New greedy-type algorithms
5. Sufficient condition for guaranteed packet delivery
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.
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)
vt
u
wC : open disk with diameter vtD : 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
C
D
DDG Condition
vt
u
wC : open disk with diameter vt
• Strong DDG Condition : For all and
• Strong DDG implies DDG
C
Strong DDG Condition
CwCu ),,(),,( tvwftvuf
Theorem.
f is a function satisfying (strong) DDG condition.
The algorithm with function f guarantees packet delivery on Delaunay triangulations.
Delivery Guarantee on Delaunay triangulations
t
u
w
v
Theorem. Greedy Routing, Midpoint Routing and Modified Midpoint Routing satisfy DDG condition
Routing Algorithms and DDG Condition
Theorem. New Greedy Routing I, II, III satisfy Strong DDG condition
Guarantee delivery of packet on Delaunay graphs
Example: New Greedy I on Delaunay triangulation
S
T
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.
ConclusionOur 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.
Future Worko Consider a metric space with the existence of obstacles and
other natural/social conditions in real ad hoc network design.
vt
uw
),,(),,( tuvftwvf
Thank You