Unidirectional Links Prove Costly in Wireless Ad Hoc Networks
Ravi Prakash *
Depa,rtment of Computer Science
University of Texas at Dallas
Richardson, TX 75083-0688.
ema.il: ravip@utdallas.edu
Abstract
Most, of t,he routing algorithms for ad hoc net,works
assume t,hat all wireless links are bidirect,ional. In
realit,y, some links may be unidirect.ional. The pres-
ence of such links can jeopardize t,he performance of
t,he existing dist,ance vect.or rout.ing algorit.hms. In
this paper we show t,hat, dist.ance vector based rout,-
ing prot.ocols t,hat, account for unidirectional links
will require nodes t.o exchange O(n2) informat.ion
with each other, where n is t.he number of nodes
in t,he nebwork. We also present. modifications to
dist,ance vector based routing algorithms to make
t,hem work in ad hoc netsworks wit.h unidirectional
links.
1 Introduction
The mobility patt.ern of t.he nodes in an ad hoc net.-
work is often non-determinist.ic. Hence, t,he net,work
topology is always in a flux. There has been a sig-
nificant. amount. of effort, towards developing rout-
ing algorithms for such net,works. These algorithms
can be classified into
(a) cluster-based
algorit.hms,
and (b)
flat
algorit,hms. In cluster-based algorithms
[I, 2, 5, 61, t g 1
a re u ar intervals, a subset of nodes is
elected
as cluster-heads.
A node is eit,her a cluster-
head or one wireless hop away from a cluster-head.
Nodes t,hat, are not clust,er-heads will, hencefort.h,
be referred t.o
as ordinary
nodes. When an ordinary
node has to send a packet, t,he node can send t,he
packet. to the clust,er-head which routes that. packet,
towards t.he destination. In flat routing algorit,hms
[7, 9, 12, 14, 151
each node maint*ains routing infor-
mat.ion.
*The author work
is
supported
in part by the National Science
Foundation grants CCR-9796331 and ANI-9805133.
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15
These rout,ing algorithms have contribut.ed sig-
nificantly towards t.he underst,anding of 6he problem
and t,he feasible solution approaches. However, t.o
successfully deploy ad hoc net.works we need to un-
derst.and the various ways in which RF-propagation
charact,erist,ics can impact, the rout.ing problem. Mod-
els based on t,he IEEE 802.11 physical and medium-
access cont.rol layer probocol [8] consider the prop-
agat.ion issues.
We will not. go
int.o these issues in
det,ail. Inst.ead, we will c0ncentrat.e on a manifes-
t,at.ion of t,he realist.ic propagat,ion models, namely
presence of some
unidirectional links in the nebwork.
Some links may be unidirect.ional due to the
hid-
den terminal problem
[17] or due t,o disparit.y be-
t,ween t,he t,ransmission power levels of t,he nodes at.
either ends of t,he link. Node
A
may be able t.o re-
ceive messages from node B as t.here may very litt,le
interference in
A’s
vicinity. However, B may be in
t.he vicinity of an interfering node and, therefore, be
unable t,o receive
A’s
messages. So, the link bet.ween
A
and B is direct.ed from B to
A.
Link unidirection-
alit,y may be
a persistent
phenomenon, especially
if some nodes experience a significant, depletion of
t,heir energy supply or a persist,ent, and strong in-
t.erferer. Alternatively, unidirect.ionality may be a
transient
phenomenon where a link quickly transi-
t.ions from unidirectional to bidirectional st.ate. The
frequency of such transitions, and the duration of
st,ay in each state would be a function of offered
t,raffic, terrain, mobi1it.y pat,tern, and energy avail-
ability.
Almost. all exist.ing rout.ing algorithms tend t.o
assume t,hat all links are bidirect,ional. In this pa-
per we intend to evaluate t,he impact. of unidirec-
tional links on some of the existing distance vector
rout,ing algorithms for ad hoc networks. Based on
t,he understanding of the impact of such links, we
propose a st.rategy to modify existing algorithms
so t,hat. t.hey can work correct,ly in an ad hoc net-
work t,hat has a combinabion of unidirectional and
bidirectional links. EvaiuaGon of the impact of uni-
directional links on hierarchical clust,er-based rout-
ing algorit.hrns and link-at,at,e roiit.ing algorit~hma is
slat.ed for future research.
Sect,ion 2 presents a brief description of some of
t.he exist,ing flat. rout,ing algorit.hrns. As t,he focus
of t,his paper is on such algorit,hms, we do not. de-
scribe t,he hierarchical algorit,hms. In Section 3 we
discuss t,he impact. of unidirect,ional links on some of
t,he exist,ing algorit,hms for ad hoc net.works. In Sec-
t,ion 4 we prove that, O(nZ) size messages need t,o be
exchanged bet.ween nodes t.o account, for unidirec-
t.ional links if dist.ance vect,or based routing is em-
ployed. This is significant,ly great,er t.han t,he O(n)
size messages exchanged in exist,ing rout,ing algo-
rithms t,hat. assume all links t,o be bidirectional. We
also propose an extension to disbance-vector based
rout.ing algorit,hms. Finally, we present, t,he conclu-
sions in Sect,ion 5.
2 Previous Work
The
De&nation Sequenced Dtstance Vector (DSDV)
[153 approach is a modificat,ion of the dist,ance vec-
t,or rout,ing algorit.hm used earlier in ARPANET. In
DSDV, each node maintains a dist.ance vect,or that.
cont.ains ent.ries for each dest,inat,ion. The ent.ry in-
dicat.es t.he dist.ance estimat,e and t.he next hop t.o be
taken by a packet. to reach a destination. Each entry
has a sequence number associated with it, indicat.-
ing it,s
freshness.
If a dest,inat,ion is unreachable,
t,he dist,ance met.ric is set, to infinit.y. Periodically a
node’s dist,ance estimat.es are diffused t,o neighbors.
When a node
p
loses a link that, it was using to
forward packets meant for destination y,
p
set.s it,s
dist.ance met.ric for q t.o infinit,y and propagat,es this
information wit,h a higher sequence number. Such
updates are diffused immediat,ely, wit,hout. wait.ing
for the next update t,ime. Similarly, when a pat,h
is found to a hit,herto unreachable node the finite
distance metric t,o t.hat dest,inat,ion is propagated
immediat,ely through the net.work.
Dynumic Source Routing (DSR)
[9] uses a diffu-
sion based mechanism to find a route to the desti-
nat,ion. Inst,ead of periodically exchanging routing
informat,ion bebween nodes, rout,e(s) are discovered
when a node has bo send packets t,o some dest,ina-
tion node. During t.his process int,ermediate nodes
can use t,he discovered rout,es to updat,e t,heir own
routing informat.ion. Caching of recently discov-
ered routing information. is employed bo speed up
the routing process. The rout,e maintenance mech-
anism (i) sends a
route error
packet. to the source if
it detect,s t,hat. t,he route t,o the destinat,ion is broken,
and (ii) eit,her tries t,o use any other cached rout,e
to the dest.ination or invokes route discovery once
again. In order to route packet.s, the source com-
pletely specifies the path the packet should take.
Figure 1: ad hoc network wit.h unidirectional and
bidirect.ional links.
In t.he
ud hoc On-De,mand Distance Vector (AODV)
scheme [14], route discovery and maint,enance are
performed on demand, as in DSR, along wit,h hop-
based rout.ing as in DSDV. In order t.o reduce com-
municat,ion overheads, as compared t.o DSDV, up-
dat,es are propagat.ed only along
act,ive
rout,es, i.e.,
rout,es bhat. have seen some t,raffic in t,he recent. past,.
The
Temporally Ordered Routing Algorith,m (TORA)
[12] is based on t,he not.ion of edge-reversal [7]. One
instance of t,he algorit.hm is executed for each desti-
nat.ion and a direct,ed graph is maintained wit.h re-
spect t,o each de&nation. Only bidirect,ional links
- are considered, and a direction is associat,e wit.h
each link. Direct,ed paths between every pair of
nodes are initially det.ermined t.hrough a sequence of
edge reversals. When any node det.ect.s t.hat. it. has
lost. the path to a dest,ination (all edges incident, on
bhe node are direct.ed t.owards it., in the graph for
t.hat, dest.inat.ion) it performs full edge reversal so
t.hat. it. has only out,going links t.o all its neighbors,
and initiat,es route rediscovery for that, destinat,ion.
If a network part.it,ion is detect.ed, t.he source is in-
formed about. t,he same.
3 Problem Description
Several flat, routing prot.ocols [la, 14, 151 and hier-
archical routing prot.ocols [l, 2, 5, 61 assume t,hat, all
wireless links are bidirectional.’ In t.he presence of
unidirectional links several problems arise for dis-
t,ance vect.or based algorithms. For t,he purpose of
illust.ration, let. us consider DSDV [15]. AODV [14]
has similar behavior. Ot.her rout.ing prot.ocols may
also exhibit, similar problems.
Let, us consider three interest,ing phenomena, il-
lustrated wit.h t,he help of the network configurat,ion
shown in Figure 1.
1. Kwwledge Asym,metry:
There is a t,wo-hop path
from j t.o a: jia. However, due to link J< be-
ing unidirectional, i cannot direct.ly inform j
about, t,he path.
Just because i knows that j is
‘DSR [9] does not explicitly assume the presence of only bidi-
rectional links.
16
that i Is ifs
neighbor.
Sinlple diffusion st.rat,egy
may not, be sufFicient, t,o propagat,e informat,ion
about. network t,opology.
2. Routing Asymmetry:
In AODV, during t.he pat.h
discovery phase, let. an int.errnediat,e node, vi,
get. t,o know that, t,he short.est pat,h from CC Do
:I/ is ZU~‘U~ .IJ~-~~uu;‘u,+~ . .y. Then, vi con-
cludes t,hat, t,he short,est, pat.h from it,self t.o 2
is vizli- 1 . ~12: t,he lexicographical reversal of
t.he pat.h prefix ending at, vi. However, if t.here
exist.s a unidirect,ional link on t.he pat,h from z
t.o Vi, t.hen ui’s conclusion would be wrong. In
Figure 1, as t,he link J< is unidirect.ional, t,he
short.esb pat.h from i t.o j consists of seven hops
and t,he pat,h from j t,o i consist,s of one hop:
a routing asymmetry.
3. Sink Unreuchubility:
In DSDV path updat,es
are init,iat.ed by the dest,inat,ion node. In AODV
a source node finds a rout,e to t,he dest.inat,ion
only when a sequence of
route replies flows
back on t.he path from t.he dest,inat,ion t.o the
source. In Figure 1, t,here exist.s a path t.o
node 1. So, it could be t,he destinat,ion of pack-
et,s. However, t.here is no way node 1 can in-
form
k
that, t,he lat.ter can reach t,he former in
one hop. So, reachabilit,y information about.
1
cannot, propagate t,o ot,her nodes. Node 1 is a
sink
node as all its incident. links are directed
t.owards it,.
The network topology muy indi-
cute that a sink is reachable from other nodes.
But due to the li.mitations of the routing al-
gorithm no node knows
of
the existence
of
the
sink, making it effectively unreachable.
In fact, bhe problem wit.h DSDV and AODV in t,he
scenario shown in Figure 1 is quite serious. As t,hey
can only use bidirect,ional links for roubing purposes,
they will ignore links c>,
f>,
J?, and 6. As a result,
even t.hough nodes a and e are reachable from each
obher, DSDV and AODV will perceive a and c t.o
be in different, net,work part,it,ions.
In DSR, let i receive a pat,h discovery message
from j along ~2. When i has t.o send an acknowl-
edgment, t,o j it may need t.o initiate a new path
discovery to find a route t,o j. The acknowledg-
ment. should t.hen be sent, along this rout.e. Thus,
while DSR does not, ignore the possibilit,y of uni-
direct,ional links, it. makes an implicit assumpt.ion
t,hat, routes in bobh directions always exist between
a pair of nodes. Such an assumption
may not al-
ways be valid
in a network with a combination of
bidirectional and unidirectional links.
4 Solution Approach
Each node needs to rnaint.ain enough informat.ion t,o
distinguish bet.ween bidirectional and unidirect.ional
links t,o its neighbors. A node may not, be able t.o
direct,ly send informat.ion t.o a neighbor if t,here is no
link from bhe node t.o t.he neighbor. Once knowledge
of link orienbat.ions is available, appropriabe rout,ing
decisions can be made.
First., let, us det.ermine t,he minimum amount, of
informat,ion participat.ing nodes need bo maint.ain
to ensure correct,ness of bhe rout.ing probocol. We
will concenbrat,e on modifications t.o protocols like
DSDV and AODV t.o cope with t.he presence of uni-
direct,ional links.
4.1 Assumptions
We rnodel the network as a graph G = (V, E),
where V is t.he set of vert.ices and E is t,he set. of
edges. Some of t,he edges are assumed to be di-
rect,ed. Every vert,ex (also referred t,o as a node)
is reachable from every other vertex. Thus, every
node in t.he net.work can send packets t.o every ot,her
node in t,he network.
A node, on receiving a packet from some ot,her
node, can det,ermine t.he length of the pat,h t.aken
by t,hat, packet..
Let each packet, start. from t.he
source
x
wit.h its
Time-To-Live (TTL)
field initial-
ized t.o TTLmax. All nodes have agreed
a priori
on
the value of TTLmax. Each intermediate node z,
and t,he destination y on receiving t,he packet. decre-
ment,s the TTL field by one. Let us refer t.o t.he
result,ant. value as TTLreceive. When t,he packet
arrives at the destination node t.he length of the
path traversed by t.he packet t.hus far is equal to
TTLmax - TTL-recv.
Definitions:
path(ab):
the shorbest. pat.h frorn node a t.o
node
b.
As some links are unidirectional,
path(ab)
may be different, from
puth(ba).
path(avl,uz
.
.vkb):
the short,est pat.h from CL
t,o
b
that passes t,hrough vertices w; : 1 <
i 5 k
such that v; precedes vj if i < j.
length(path(x)):
number of wireless links in
path(x),
where I is a sequence of vertices.
directed pathcab): path(ab)
is said t,o be a di-
rect.ed path if it. has at least, one directed link.
Lemma 1 O(n) size distance vector exchange, as
performed in protocols like DSDV, is not sufficient
to determine routing information
for
distance vector
based algorithms in the presence
of
unidirectional
links.
17
Proof: The lemma is proved by corlt,raclict.io~l. Let,
us consider t.he graph G’ shown in Figure 2. In the
figure:
1. d?e is a direct.ed edge
2. length(path,(cd)) >_ 0.
3. length(path(ef)) > 0.
>D
s
c
d e f
Figure 2: Represent,at,ion of direct,ed and undirect,ed
pat,hs.
Let. each node i maintain a vect,or Vi of length n
such that Vi b] .dist. is node i’s knowledge of it,s path-
lengt,h t,o node j. Let t,he shortest. path from c t.o D
be t,he direct.ed path puth(cclefD) and let. puth(fD)
be an undirect,ed pat,h. Also: let. path(fpc) be a
pat.h of 1engt.h great,er t.han zero bet,ween f and c.
There are t,wo possibilities regarding t.his path:
1. It. is a direct,ed pat,h from f t,o c,
2. It, is an undirect,ed pat.h, or directed from c t.o
Possibility 1: As the dist.ance vect.ors are exchanged
bebween neighboring nodes, t.he reachabilit,y infor-
mation about,
D
reaches c alongputh(Dfpc). There-
fore, node c’s estimat.e bf t.he dist.ance to D is:
length(path(Dfpc)), which may be different. from
length(puth(cdefD)).
Possibi1it.y 2: If path(fpc) is directed from c to f,
node c cannot ‘learn about. it.s dist.ance to D as no
pat,h exists from
D
to c This is a violat,ion of the as-
sumption t.hat every pair of nodes can communicate
along a path.
Also, if path(fpc) is undirecbed or directed from
c to f, length(path(cpf)) 2 length(path(cdef)).
Otherwise, the short,est pabh from c to D would
have been
path(cpfD).
If
length(puth(fpc)) > length(puth(cdef)), V,[D].dist
-
length(path(fpc))
+
length(puth(fD)),
which is
greater t,han
length(puth(cdefD)).
Hence, main-
taining only a dist,ance vector will lead to erroneous
calculation of pat,h-lengths.
I
Lemma 2 It is necessary to exchange O(n’) size
m&ices of pair-wise distance estimates to correctly
construct path-length estimates for distance vector
bused ulgorithms.
Proof: Let us once again refer
t,o Figure 2.
We
assllrne t,hat. puth.(cdef) is t.he shortest. pat.h from c
t.0 j. Let:
X = {e: x is a node on
path.(cd)},
and
Y = {y: y is a node on
puth.(ef)}
As
path(cdef)
is t,he shorbest. pat,h frorn c to f,
for all x: and y,
path(xy)
goes t,hrough verbices cl
and e. As edge
d>
is directed, inforrnat,ion about,
length(puth(xy))
cannot, pr0pagat.e from y t.0 z along
t.he pat,h t,hat, goes along z. Therefore, every node
p
on
puth(yfcx)
has t.o propagat,e
length(path(xy)), Vx E
X, y E Y. As set,s X and Y can be as large as
V,
1 X I= O(n) and ( Y I= O(n), where n =I
V I.
Therefore, node
p
needs to store and forward
O(n”) unit.s of 1engt.h informabion. I
4.2 Data Structures and Algorithm
It is assumed that each node emits a beacon at regu-
lar intervals. A node can hear beacons transmit.ted
by a neighboring node provided the link bet.ween
t.hem is bidirect,ional, or direct.ed from the neighbor
t,o it,self. The t,ransmission of beacons by different.
nodes is not, synchronized as there is no global clock
in the system.
4.2.1 Data Structures
Each node
p
maintains t,he following data struc-
t.ures:
l
Nodesheard,:
set of nodes whose beacons have
been heard by node
p
within t,he last t time
units. If
q E Nodesheurdp
and
p E Nodesheard,,
then there exist.s a bidirectional link between
p
and
q.
However, if
q E NodesheurdP
and
p 6 Nodesheurdp,
then t,here is a unidirec-
tional link from
q
to
p.
This data structure is
modeled after the one by t,he same name used
in t.he Linked Cluster Algorithm [l, 21.
l
D:
an n x n mat,rix of 2-t.uples, where n is
the number of nodes in t,he net.work. D[i, j] =
(seq, dist)
means node
p
knows t.hat. the pat.h
from node i to node j if of 1engt.h
dist,
and the
sequence number associat.ed with t,his informa-
t.ion, pert,aining t.o node j, is
seq.
Due t,o the
possibility of unidirect.ional links,
D[i, j].dist
may not be equal t.o D[j,
i].dist.
The sequence
number associated with a destinat.ion is monobon-
ically increasing. Each t.ime a node sends up-
dates to its neighbors, t.he node increases it,s
sequence number by a constant. value. As in
AODV and DSDV, routing informat,ion with
a higher sequence number overrides the corre-
sponding information with a smaller sequence
18
nllnlber. As a result, st.ale rout.ing informa-
tion cannot suppress hew rout.ing inforn~at.ion.
Consequent.ly, knowledge about, link disruptions
propagates quickly and t,he count to
27ifi7~ity
problem (associat.ed wit.h distance vector al-
gorit,hms) is avoided.
l
To
and
From:
vect.ors of lengt,h
n,
where each
ent,ry is a 3-t,uple of t,he form (seg, c&t, next)
and (seq,
distT
prev), respect,ively. The
To
vector is similar to t,he dist,ance vect.or of DSDV
as it. mainbains informat,ion about. bhe pat.h
length from a node t,o all ot.her nodes, and bhe
next, hop on the pat.h t,o those nodes. From,,
vect.or cont,ains informat,ion about, pat,hs from
ohher nodes t.o p. Due to the presence of uni-
direct,ional links ,in t,he net.work, and t.he re-
sult,ant, routing asymmet.ry, the corresponding
dist
values in t,he
To
and
From
vect.ors may
be different. from each ot,her. When rout.ing in-
format,ion st,abilizes,
To,
should have t.he same
dist
and seq values as the corresponding en-
t.ries in bhe pth row of
DP.
There should be
a similar rnabch bet.ween
Fromp
and the pth
column of
DP.
Determination of Link Orientation:
We
employ t.he Nodesheard set, in a manner similar to
[l], to det,ermine net,work adjacency. Each node pe-
riodically transmit,s it.s Nodesheard set, with its bea-
con. Ib also cont,inuously listens for similar t.rans-
missions from other nodes. If node p hears t,hat,
p E Nodesheardp,
node
p
knows that. bhere exists a
bidirectional link bet.ween
p
and
q.
The next t.ime
p
broadcast,s its beacon it. includes Q in it,s Nodes-
heard set.. When q hears this beacon, it, t,oo, knows
of t.he presence of t.he bidirectional link.
If node
p
finds t,hat,
p I$ Nodesheardp, p
con-
cludes that t.here exists a unidirectional link from
q t,o p. However, how does
q
get to know of t.he
presence of t,his link? For this purpose we employ
the matrix
D.
4.2.2 Routing Algorithm
Let,
V
denot,e t,he set, of nodes in the net.work. Ini-
tially, the D matrix at. each node
p
only cont,ains
it.s adjacency informat,ion. Each node periodically
transmit,s it.s
D
matrix. The t.ime bet,ween succes-
sive transmissions of
D
is a mult,iple of t.he time be-
tween successive transmissions of t,he
Nodesheard
set. This is so for two reasons:
1. Transmission of
D
consumes much more band-
width t.han t,he t.ransmission of
Nodesheard.
2.
On
Transient. noise t.hat may int.erfere wit.h t.he rp-
cept.ion of a few sllccessive
Nodcsheard ~IIYS-
sages from a neighbor does not, lead a node ir1t.o
erroneously concluding t,hat. it.s pat.h t,o/from
t.hat. neighbor is broken.
link discovery:
If
p
discovers a bidirec-
t.ional link bet,ween
p
and q, t,hen
DP[y, q].dist =
Dp[q, p].dist
= 1. If p cliscovers t,hat. there exist,s a
unidirect.ional link from
q
t,o
p,
t.hen
D,[q,p].dist =
1. The sequence number associat,ed wit.h each en-
t,ry is analogous t,o t,he sequence nurnber associat,ed
wit.h roubing t.able ent.ries in DSDV and AODV. The
sequence numbers are init.ialized t,o zero.
On receiving D matrix from neighbor:
Let.
node p receive mat.rix
D,,,,
from node
q.
If m is a
bidirect,ional link or a unidirectional link from
q
t,o
p,
p modifies its
D
rnat,rix in t.he following manner
on receiving t,he matrix:
l
For all nodes
T
E
V,
different from
p
and
q:
- Er,.riiL;/;q < D[r,pl.w then per-
-
If
((Dreev[r, q].seq == D[r,p].seq)
OR
((D,,,, [r, q].seq > D[r, p].seq)
AND
(Fromp[r]! = q))):
* D[r,p].dist = min(D,,,, [r, q].dist +
1,
D[r,p].dist)
*
if
D[r, p].dist
has decreased as a result,
then
Fromp[r].prev = q
- If((Drec,[r,q].seq > D[r,p].seq) AND (From,[r]
== q)):
* D[r, p].dist = D,,,, [r, q].dist +
1
-
If
D[r,p].dist
has changed as a result,, in-
crement.
D[r, p].seq.
- If
Drecv[~, q].seq == D[r, q].seq
* D[r, q].dist =
min(D,,,,[r,
q].dist, D[r, q].dist)
- If Drecu[r, ql.seq > D[r, ql.seq
* D[r, u] = Drecv[~, q]
These operations enable node
p
to det.ermine
its distance from ot.her nodes.
l
For any arbit,rary pair of nodes
T
and s in
V,
different from
p
and
q:
If
((Drecv[r, s].seq > D[T, s].seq)
OR
((D,,,,
[T,
s].seq == D[T, s].seq)
AND
(D,,,, [r, s].dist < D[r, s].dist)))
- D[r, ~1 = Drecv[~, 4
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