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