### A Unified View to Greedy Routing Algorithms in Ad

```A Unified View to
Greedy Routing Algorithms
○Truong Minh Tien
Joint work with
Jinhee Chun, Akiyoshi Shioura, and Takeshi Tokuyama
Tohoku University
Japan
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.
Contents
1. 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




Self-organizing network without fixed pre-existing infrastructure
Communication between nodes are achieved by multi-hop links
Network topology can be represented by undirected graph G=(V, E)
Geometric Routing
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.
T
S
V
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.
T
S
V
Contents
1. 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
w2
w1
v
t
w3
Finn, 1987
• The next neighbor w is the
node nearest to t
w4
smallest (, )
Compass Routing
smallest ∠
w2
w1
v
t
w3
Kranakis, Singh, Urrutia, 1999
• Packet will be sent to w if the line vw
forms with vt the smallest angle.
w4
Midpoint Routing
w2
v
w1
smallest (, )
m
t
w3
Si, Zomaya, 2010
• Choose next neighbor w that is
closest to midpoint m between v
an t
w4
Modified Midpoint Routing
w2
v
w1
m
p
w3
Si, Zomaya, 2010
• The next node w closest to p
o p = t : Greedy routing
o p = m : Midpoint routing
t
smallest (, )
w4
Contents
1. 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.
S
T
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
very small
very large
Midpoint
small
large
small
large
average
average
Modified
Midpoint
Compass
guaranteed on
Delaunay graph
Need appropriate routing algorithm satisfying desirable properties in
response to the request of applications.
Contents
1. 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
f : T  R  {}
• Generalized greedy routing:
Choose a neighbor w of v that minimizes f (w, v, t) in each iteration
Generalized Greedy Routing: Example
Choose next node w that minimize f (w, v, t)
Example:
v
7
3
w2
w1
t
2
w3
+∞
w4
Congruence-Invariant Function
w
t’
t
v
f ( w, v, t )  f ( w' , v ' , t ' )
w’
v’
• f is congruence-invariant if function value f (w ,v ,t)
depends only on shape and size of wvt .
Congruence-Invariant Function
f is congruence-invariant function if there exists
a function h such that:
f ( w, v, t )  h( d vt , d wt , d vw , at , aw , av )
w
aw
d vw
v
d wt
at
av
d vt
t
Greedy Routing: Min d(w, t)
function hG  d wt
w
Compass Routing: Min
function hc  av
t
w
v
t
v
Midpoint Routing: Min d(w, m)
function
hMP  ( d wt sin at ) 2  ( d wt cos at 
w
t
d vt 2
)
2
M. Midpoint Routing: Min d(w, p)
function
hMMP  (dwt sin at )2  (dwt cosat  dvt )2
w
1
(  )
2
t
p
v
wvt
m
v
New routing algorithms
w
New Greedy I
max vwt
function
h1  aw
t
v
New Greedy II
w
t
w
t
min d (v, w) / cos(tvw) (tvw   / 2)
function
d vw
h2 
  / 2 (av )
cosav
v
New Greedy III
min d (t, w) / cos(wtv) (wtv   / 2)
function
d wt
h3 
  / 2 (at )
cosat
v
Contour Map
GREEDY - concentric circles about t
MIDPOINT - concentric
COMPASS – rays with same endpoint v
MODIFIED MIDPOINT concentric circles about p
Contour Map
of New
Routings
New Greedy I – curves with same
chord vt
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
Very small
Very large
Midpoint
Small
Large
Small
Large
Average
Average
New Greedy II
Large
Small
New Greedy III
Small
Large
Modified
Midpoint
Compass
New Greedy I
guaranteed on
Delaunay
graph
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
Very small
Very large
Midpoint
Small
Large
Small
Large
Average
Average
New Greedy II
Large
Small
New Greedy III
Small
Large
Modified
Midpoint
Compass
New Greedy I
guaranteed on
Delaunay
graph
Contents
1. 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)
DDG Condition
D
w
C : open disk with diameter vt
D : open disk with radius tv
A = max{f(u, v, t) | u ∈ C}
C
v
t
u
• DDG Condition :
For all w with f(w, v, t) ≤ A ;
w∈D
Strong DDG Condition
w
C : open disk with diameter vt
C
v
t
• Strong DDG Condition :
For all u  C and w C
f (u , v, t )  f ( w, v, t )
u
• Strong DDG implies DDG
Delivery Guarantee on
Delaunay triangulations
w
v
t
u
Theorem.
f is a function satisfying
(strong) DDG condition.
 The algorithm with
function f guarantees
packet delivery on
Delaunay triangulations.
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
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.
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.
Future Work
o Consider a metric space with the existence of obstacles and
other natural/social conditions in real ad hoc network design.
w
u
t
v
f (v, w, t )  f (v, u , t )  
Thank You
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