Routing Algorithms for Rapidly Fluctuating Networks
Arka Bhattacharya
Computer Science Division
University of California
Berkeley,CA 94720
arka@cs.berkeley.edu
Shaunak Chatterjee
Computer Science Division
University of California
Berkeley,CA 94720
shaunakc@cs.berkeley.edu
Abstract
Routing in rapidly uctuating networks present
new challenges.Conventional algorithms will fail
in this setting because there will be too many up
dates generated and the network state has very
short temporal validity.Also,we argue that a
single path metric is not robust enough to the
rapid uctuations and will often result in sub
optimal routing decisions.In this work,we pro
pose a multipath metric  an approximate esti
mate of stochastic connectivity.We show why
this is always a better metric albeit more dicult
to compute than a single path metric.We are
currently setting up experiments to validate our
algorithms.
1 Introduction
Conventional routing algorithms depend very heavily on
stability of link costs.Commonly used algorithms such
as OSPF (Ordinary Shortest Path First ) for wired net
works and OLSR ( Ordinary Link State Routing protocol
) for wireless,are examples of link state algorithms which
broadcast updates throughout the network whenever a link
state changes.In most such algorithms,the state of the
link is assumed to be binary ( UP or DOWN ),or may
have an associated cost ( such as amount of congestion ).
In both cases,the state of a link is assumed to persist for
some time and hence not expected to change by the time
the update message reaches other nodes.Also,most com
mon routing algorithms such as AODV,DVR,etc carry
out bestpath routing,i.e they maintain the best path to
each of the other nodes in the network and route packets
along that path.
Wireless networks have nowadays become ubiquitous.
While various applications and usefulness of wireless net
works have been well documented,the fact that they use
electromagnetic waves over air for transmission of data
leave them open to external interference.Anything from
switching on a microwave oven to a packet being trans
mitted by another wireless channel can cause interference.
Hence the guarantee of a packet being delivered success
fully to the next hop through a wireless link ( also known
as PRR  Packet Reception Ratio ) uctuates rapidly.
Among others,there are two major problems which such
uctuating networks face.(1) Any linkstate ( or distance
vector ) algorithm will fail as too many updates will be
sent throughout the network.(2) Any best path metric
will fail because the current best path might temporarily
become extremely bad;and (3) Updates about the change
in state of a link might not be temporally valid,i.e by
the time the control message describing the state of the
link reaches a particular node,the linkstate may have
changed again.
There exist many routing algorithms for wireless networks
with capabilities of scoping link update messages.There
exist routing protocols which impose a hierarchy on the
topology,such as BVR [2] which directs packets to par
ticular beacon nodes,Landmark routing [8] which directs
packets to certain nodes marked as landmarks and S4 [5]
which modies landmark routing in certain ways to provide
mathematical guarantees of stretch.There exist back
pressure algorithms [6] where congestion and path failure
are indicated by the packet queues getting full and CTP [3]
where which uses adaptive beaconing and nondecreasing
path costs from source to destination to route packets.
Each of the abovementioned algorithms have its own ad
vantages and shortcomings,but none of them will per
form eciently under the scenario of rapidly uctuating
links.XL [4] is a routing algorithm which incorporates
update suppression mechanisms,but is designed primar
ily for wired networks.Hence,certain features in rapidly
uctuating wireless networks renders it useless.However,
our paper uses a lot of the techniques used in XL,the
details of which are provided in section 4.1.
Most of the problems mentioned could be solved if one had
knowledge about the model of uctuation of a wireless link
.However,despite a lot of studies extensively document
ing link churn [7],no concrete model has been found.The
Betafactor paper [9] reports that 802.15.4 links are bursty
and provide a metric to capture the burstiness but most
long distance communication takes place over 802.11.Also
[1] mentions that most 802.11 links are stochastic and do
not follow any well known model.All this leads to a
deeper scientic question  how does one maintain routes
and routing state when there is a prohibitive amount of
churn in the network?
In this paper,we present an insight into a\connectiv
ity"or multipath metric,which maintains multiple routes
from a node to its close neighbors,and thereby takes ad
vantage of the redundancy in the graph.Each collection
of paths to a particular nearby node is termed as a hyper
link.Hyperlinks are only maintained to nodes in a Chop
neighborhood,since there is an overhead required in main
taining the alternate paths in a hyperlink (the same short
comings to single bestpath routing applies ).Maintaining
short alternate paths is key to successful routing.Also
,a multipath metric has an added advantage of having
lower sensitivity to a linkstate change,thus enabling the
network to throttle most updates.The multipath metric
along with an XL style routing over hyperlinks results in
throttling of a large number of unwanted and irrelevant
link updates.
The paper is organized as follows:Section 2 describes the
problem formulation in details.Section 3 compares the
multipath\connectivity"metric against a single path met
ric.Section 4 enumerates the overall algorithm and com
bines it with XL.Section 5 describes other multi path met
rics.Section 6 presents some issues with XL which makes
our algorithm performance bad.We present our results in
Section 7 and outline some future work in Section 8.
2 Problem Formulation
We consider a network represented by a graph.Let (G;P)
represent a graph where G is the set of nodes and P is a
formof adjacency matrix.P
ij
is the probability of a packet
being successfully sent directly from node i to node j.In
a rapidly uctuating network,P
ij
can potentially change
very frequently.We also assume that nodes i and j are
always aware of the current value of P
ij
.
In this scenario,the task is to devise a routing algorithm
which is robust to frequent changes in the network state.
The assumptions we make are the following:
A link's P
ij
value could uctuate too frequently for
the entire network to be updated about its current
state at any point of time.
A lot of links could be uctuating,so the control mes
sage overhead for a conventional algorithm could be
prohibitive.
Thus,in this setting,all nodes in the network can never
have a consistent view of the network,since the overhead
is too high,and some updates will be invalid by the time
they reach a node.
3 Connectivity vs single path metric
In deterministic cost based networks,the optimal route
is the shortest path between the source node S and des
tination node D.In a stochastic network,the analogous
measure could be the maximumlikelihood path.If we con
sider the negative log likelihood values,then the maximum
likelihood path becomes the minimum cost path (shortest
path).However,in a stochastic network what we really
care about is not the maximum likelihood path but the
probability of any path existing between s and d.The de
cision of which path to use can be made locally based on
current information (also sampled locally).
This point can be illustrated through an example (see Fig
ure 1).When a packet needs to be sent from node S to
node D,the two choices available are via nodes XY or via
PQ.The maximum likelihood metric suggest the path
through X and Y since that path is individually is better
than all the paths available through P and Q.However,it
is more likely at any given point of time,that a path will
exist via the PQ route than via the XY route (where
there is only one possible path).Thus,the probability of
successfully delivering a packet is increased by choosing to
send the packet to P rather than sending it to X.
Figure 1:An example where the maximumlikelihood path
is dierent from the maximum connectivity path.
Hence,an intuitive metric could be the one mentioned be
fore  the probability of a path existing between a source
node S and a destination node D.However,in the gen
eral case,dening this value as a function of constituent
edge availability values can be semantically infeasible.This
is because it is dicult to account for paths which share
edges.It is easier to compute this probability value if the
paths are disjoint.Also,if all the paths between S and
D pass through a common point X,then the probabil
ity of a path existing between S and D (say P(SD)) can
be decomposed into P(SX) and P(XD),where P(SD) =
P(SX) P(XD).
Since the exact computation is often infeasible,we choose
to use a proxy function which we term the\connectivity"
metric.This metric is dened by a set of edgedisjoint
paths between a pair of nodes.The formal denition is:
conn(S;D) = 1
k
Y
i=1
(1 path
v
al(S;D;i)) (1)
where S and D are nodes and path
v
al(S;D;i) is the like
lihood of the i
th
most likely path between S and D.This
metric is a lower bound for the actual probability of a path
existing between S and D.It can be shown that if we chose
any K random edgedisjoint paths between S and D,then
the connectivity metric dened by them would be lesser
than or equal to the one dened by the K shortest (most
likely) paths.Thus,this is the best approximation to the
actual value using K edgedisjoint paths.
Computing the new metric requires information about
edges on all the K most likely paths.If we want to main
tain this metric between all pairs of nodes (i.e.globally),
then the path lengths could be arbitrarily long and updates
(of frequent uctuations) would need to be sent across the
entire network.This is a very expensive proposition since
the control overhead would be too large.Also,since the
uctuations are frequent,the time taken for the update
about a particular link to reach a far away node might be
more than the time of the next uctuation of that link.
In this case,the information that reaches the node is al
ready invalid.We need to reduce the amount of updates
being sent around,and at the same time,ensure that a
large portion of the updates reach the relevant nodes be
fore the information becomes invalid.This can be done by
considering local paths.
Let us add a constraint that any path that we consider be
tween a pair of nodes S and D have to be within C hops.
This automatically ensures that the updates for computing
the connectivity metric are only locally scoped.This could
even be done through local ooding for small values of C.
Once we use these\local"paths to compute the connec
tivity metric,node pairs which do not have short paths (of
C hops or less) between them are considered to not have
a direct'hyperlink'between them.After these hyperlinks
are created,we use Dijkstra's algorithm to perform nor
mal linkstate routing.Thus,the main idea here is to not
maintain global redundant paths since the updates will not
be received soon enough to make wellinformed decisions.
Saving on the control overhead is also an important con
sideration for this decision.
4 Using Hyperlinks with XL
Now that we have a basic setup of forming hyperlinks and
then performing a socalled\hyperlinkstate"routing,we
can use existing update suppression schemes.We chose to
work with the XL (an approximate link state) algorithm
since it proposes an update suppression scheme while still
providing performance guarantees.
4.1 XL:the algorithm
XL or Approximate Link State algorithm [ CITATION ] is
a linkstate algorithm which aims at lowering the number
of control messages sent by the network by suppressing
updates fromcertain parts of the network,and maintaining
slightly suboptimal paths as a tradeo.XL sends out link
updates only
1.when the update is a cost increase ( i.e bad news )
2.when the link is used in the node's shortestpath route
to another node in the network.This update is only
sent to the next hop node in that route.
3.when propagating the new link cost improves cost to
any destination from the a node by a factor of (1+)
,where is a design parameter.Lower values of
tends to maintain more optimal paths and suppress
less updates,while higher values of maintain more
suboptimal paths while suppressing more updates.
XL proves that the three constraints ensure soundness ( i.e
if XL reports a path from node uto node v in the network,
the path exists ) and completeness ( i.e if there is a path
between node u to node v in the network,XL will nd it ).
Also,the paper provides guarantees that any path main
tained by a node as a result of the XL routing algorithm
will have a stretch within a nite bound ( parameterized
by ).The authors simulate the XL algorithm on vari
ous types of graphs and show the improvement in number
of control messages sent through the network.Conditions
(1) and (2) maintain soundness and completeness,while
condition (3) ensures a bounded stretch.
4.2 Putting together the overall routing
algorithm
Although we have primarily discussed the\connectivity"
metric to form hyperlinks,we can potentially use any mul
tipath metric to form our hyperlinks and the overall algo
rithm can operate with such dierent choices of metrics.
Hence in the general version of the algorithm,we only refer
to a general multipath metric,instead of the\connectiv
ity"metric specically.
Link Estimation We assume that every node gets the
following information about every link (edge) attached to
it.
1.Any attached link's availability probability value
2.Current state of an attached link
Topology Formation I:Hyperlinks Parameters
C:maximum length of a path (in hops) K:number of al
ternate paths considered
For each node pair P,Q,
1.Find the K edgedisjoint shortest paths from P to Q
of length C hops or less.
2.Create a hyperlink from P to Q using a multipath
metric.
3.Upon receiving updates about links,recompute the K
shortest paths and update the hyperlink value.
No hyperlink exists between a pair of nodes if there does
not exist any path between them of length C hops of less.
Topology Formation II:Route creation
1.Replace original links with hyperlinks (as described in
the previous segment) to form the hyperlink graph H.
2.Use Dijkstra's algorithm to nd the shortest path be
tween every pair of nodes in H.
Update Dissemination
1.Flood the local C hop neighborhood about the change
in a link's value.
2.Recompute the hyperlink values for every relevant
node pair (aected by the changes propagated as
above).
3.Send out updates about hyperlink changes following
XL rules (S1,S2,C1).
Routing
1.When a packet arrives,identify the immediate hyper
link destination.
2.Send it to the current available neighbor (including
itself,i.e.waiting and not sending) which presents
the best chance of reaching the hyperlink destination.
The next subsection how to decide which is the best avail
able neighbor.
4.3 Finding the best available neighbor
We explored a few options for making this decision.Let
the source of the hyperlink be S and the destination of
the hyperlink be D.X
1
,:::,X
K
are the K immediate
neighbors along the K shortest paths from S to D and X
1
is the neighbor along the best path.
Firstly,we checked the rst edge along the K shortest
paths (from best to worst) and sent the packet to the rst
such node which was reachable via an available edge.This
approach tries to minimize waiting time and will work well
when the K alternate paths are of similar cost.However,
the problem with this approach is that if the second best
path is a lot worse than the best path,then it is often
wiser to wait for the rst link along the best path to be
come available rather than sending it along an alternate
path.
This insight led us to our second (improved) criterion.This
works as follows.Sort the K paths in increasing order
based on their X
i
(i=1;:::;K) to D path cost.Sample
in this sorted order,the S X
i
link until you nd one
available.Let this X
i
node be denoted as X
bestavail
.If
S X1 D path cost is lower than X
bestavail
D,then
wait else send it to X
bestavail
.
5 Various multi path metrics
5.1 More about the connectivity metric
5.1.1 The good
The\connectivity"metric is more'resilient'to changes.
The amount of change in a single path metric when a
link on that path changes is much more than the amount
of change on the\connectivity"metric.The tradeo is
that the\connectivity"metric changes (albeit by small
amounts) much more frequently since many more links are
constituents of the hyperlink.However,we have empiri
cally observed that the benet is more prominent.We set
up a simulation for a hyperlink similar in topology to the
one shown in Figure 3 but with more redundant paths be
tween A and D.We varied each edge value randomly (at
random time intervals) and compared the maximum like
lihood (ML) path metric against the\connectivity"met
ric.The\connectivity"metric had a higher mean and a
lower variance as shown in Figure 2.This lower variance
also implies longer temporal validity when we are ready
to tolerate small errors in our routing decisions (and this
is inevitable since exact information cannot be propagated
quickly enough).
5.1.2 The bad
Although the\connectivity"metric seems appealing intu
itively,it suers from a basic aw.It can lead to loops.An
Figure 2:The connectivity metric has a lower variance and
higher mean than the ML path
example demonstrating this is shown in Figure 3.Consider
the hyperlink AD.Since all the 3 paths are equivalent,
a packet arriving at A and going to D is routed to B
1
.
Once at B
1
,the optimal route to D is now through the
hyperlinks B
1
A and AD.Analytically the reason for
this is that a hyperlink cost (AD) can be lower than the
cost of a constituent hyperlink (B
1
D).This is not the
case with normal link costs.
However,analysing the semantic reason behind the loop
formation is more interesting.The\connectivity"metric is
semantically analogous to sampling a path atomically.At
A,if we could sample the availability of AB
i
C
i
D
a priori,and then send it down a selected path (using
sourcebased routing) to D,where the edges would retain
their state (up or down) until we reach D,then this met
ric would have been appropriate.However,we assume
that atomiticity lasts one time step or one hop.Edges can
change state after every time step.Hence,we allow the
routing decision to potentially change at every time step
as well (hence no source based routing).
Figure 3:An example demonstrating how the connectivity
metric can lead to loops
Using the\connectivity"metric,we detected a lot of loops
in our experiments.Loops will be formed whenever the
best among the K shortest (lowest in cost) paths consti
tuting a hyperlink have comparable cost.To remove loops,
the approach we adopted was we dened a modied version
of\connectivity".
5.2 Expected Connectivity
Let us use the same example (see Figure 3) with rede
ned edge costs to distinguish between paths.c(AB
1
) =
x
1
;c(AB
2
) = y
1
;c(AB
3
) = z
1
;c(B
1
C
1
) = x
2
;c(B
2
C
2
) = y
2
;c(B
3
C
3
) = z
2
;c(C
1
A) = x
3
;c(C
2
A) =
y
3
;c(C
3
A) = z
3
.Also assume x
2
x
3
y
2
y
3
z
2
z
3
.
The new multi path metric dening the cost of the hyper
link AD is:x
1
x
2
x
3
+(1 x
1
)(y
1
y
2
y
3
+(1 y
1
)z
1
z
2
z
3
)
This metric re ects the expected cost of a path from A to
D.If AB
1
,which is the rst link along the best path,is
available (probability x
1
) then the cost is x
2
x
3
.Otherwise
if it is unavailable (probability 1 x
1
),and if AB
2
,the
rst link along the next best path,is available (probability
y
1
),then the cost is y
2
y
3
and so on.
This metric semantically captures the single time step
atomicity that we assume and hence removes loops.How
ever,we still encountered a few loops in our experiments
because of the uctuations.All guarantees of loop avoid
ance only work when the network has reached steady state.
Our network is often not in steady state (since dierent
nodes have not yet received information which has been
sent their way by the update dissemination scheme).The
results that we have reported are based on this metric.
5.3 Expected Time metric
An alternate multi path metric that we have devised is
based on the expected number of time steps to go from
one node to another since (often) the most important met
ric is average transmission time.It can be shown that
the expected number of time steps required to transmit a
packet across a link whose availability is x is 1=x.Hence,
the expected number of time steps to send a packet along
AB
1
C
1
D is 1=x
1
+1=x
2
+1=x
3
.This also means
that this cost is additive along a path.
The routing scheme is required to compute the expected
transmission time across a hyperlink.Let the routing
scheme be that we forward the packet along an alternate
path only when the expected time along the remaining
path is less than the expected time along the best path.If
that is not the case,then we wait for a time step.The deci
sion at the next time step is taken using the same criterion.
It is straightforward to derive an expression for computing
the expected time considering K edgedisjoint paths.The
calculation is very similar to the previous metric and hence
is omitted.
6 Issues with XL
While XL,with its provable correctness,gives an ideal
framework to implement the concept of multipath or hy
perlink routing,there are a few points which make XL
highly unsuitable for use in a rapidly uctuating stochastic
network.First,at each node XL maintains the lowestever
cost of a link,and sends out an update when the current
link cost is better than the lowest cost by a factor of .
In stochastic networks,where links may assume extremely
low costs momentarily,one has to set a very high value of
to enable the nodes to suppress link updates.Very high
tends to yield highly suboptimal paths.Second,condi
tion (1) of XL,which sends out every bit of bad news is also
not suited to stochastic networks,where a link value might
uctuate and the cost may increase often.This problem is
compounded by the fact that one link may be part of mul
tiple hyperlinks,and the cost of a link increasing slightly
will adversely aect the cost of all the hyperlinks of which
it is a constituent  resulting in the generation of a large
number of updates.
7 Experiments
In this section,we evaluate the performance of the\con
nectivity"metric from two aspects  its role in lowering
the average packet delivery time,and its role in lessening
sensitivity to rapid link uctuation.The experiments are
performed on graphs of size 10,20,30 and 40 nodes.Five
instances of each size were generated  each having dier
ent amounts of redundancy.Also,we wanted to capture
the performance of the\connectivity"metric under low,
moderate and extremely high update suppression scenar
ios  and hence each instance was run for three values of
( 0.1,0.5 and 0.9 ).
7.1 Simulator
To simulate the algorithms developed,we built a discrete
time simulator.The initial graph topologies were gener
ated by random sampling.The change of the probability
value on an edge is termed henceforth was termed as an
\event".A list of events were described in an events le,
which was fed into the simulator at runtime,along with
the le containing the initial graph topology.The simula
tor does not perform the initial formation of the link and
hyperlink routing tables in a distributed manner,since we
are not bothered about the onetime overhead required to
freshly build up a routing table.Instead,the simulator
takes a centralized view of the system,and informs all
nodes about the its routing tables and the accurate Chop
view at the start time of the simulation.At every time
instant,a global function samples all the links of the graph
to assign the value'UP'or'DOWN'to each of them.A
packet can only be sent through an'UP'link.A packet
le containing the source,destination and time of genera
tion of each packet is given to the simulator as an input.
The simulator also takes as input values of C ( the number
of hops for which each node will attempt to maintain an
accurate view and a path diversity metric ),K ( the max
imum number of edgedisjoint alternate paths to maintain
to each node in a node's Chop neighborhood ) and (
the suppression factor,as mentioned in XL ).The simula
tor outputs the average packet delivery time,the number
of hyperlink and link control update messages which are
propagated through the network,and the redundancy of
the graph.
7.2 Results
This subsection reports and analyzes the various plots gen
erated to evaluate the main objectives behind coming up
with the above algorithm,viz.whether the connectivity
metric is able to suppress updates due to its low sensi
tivity to link value changes,whether the overhead due to
maintaining multiple paths is reduced when there is more
redundancy in the graph,whether the multipath metric
can take advantage of alternate paths in times of link fail
ure to route packets faster,and whether redundancy has
any eect on packet transmission times.In all of the re
sults discusses hence,the overhead refers to the number
of hyperlink updates.The number of link updates are not
taken into account,as they remain constant across K for a
certain value of C,and hence provide uninteresting conclu
sions.Also,note that C=1,K=1 boils down to ordinary
link state algorithm (akin to OSPF).
7.2.1 Increase of Overhead with multipaths
Figure 4 shows the percentage dierence in the overhead
of maintaining various alternate paths within a 2 and a
3hop neighborhood from the simple C=1 case ( ordinary
link state with same suppression algorithms).The average
message overhead in a graph is taken to be the number
of hyperlink update messages per node per event.As ex
pected,the average number of hyperlink updates sent by
nodes when C=2 and C=3 for K=1,2 and 3 are all higher
than an ordinary linkstate algorithm equipped with the
same suppression techniques.As the suppression rate gets
higher,the number of control updates sent by nodes when
C=2 and 3 decreases,until at =0.9,the nodes actually
send out lesser updates than an ordinary linkstate algo
rithm.Since the sensitivity of hyperlinks to link value uc
tuations is really small,a suppression factor of 0.9 requires
that a large number of constituent links of a hyperlink have
to change for the hyperlink to send out updates.Also,for
cases where =0.5,for C=2 and C=3,one nds that main
taining 2 edgedisjoint paths actually help in sending out
lesser updates.While this is a little counterintuitive an
explanation maybe as follows:a lot more links become
critical links ( constituent of one of the alternate paths
maintained by a node ) when K=3.Hence,the uctuation
of a single link aects the values of multiple hyperlinks,and
hence more updates are sent out.
Figure 4:Analysis of control overhead compared to
C=1,K=1
7.2.2 Change of Average Packet Delivery Time
with multipaths
Figure 5 shows the Average Packet Delivery Time analogue
to Fig 1.The values shown on the graph are the percent
age dierences from C=1( ordinary link state algorithm
with the suppression algorithms ).Surprisingly,the avg.
time required by packets to complete their trip seems to in
crease,albeit slightly ( mostly within 10%).In most cases,
it is only when K=2 that there are improvements in aver
age packet delivery time.The better performance of K=2
was seen in section 7.2.1 as well.This seems to suggest
that having alternate paths instead of a single path within
a hyperlink is better,but having more than one alternate
path involves a larger overhead and does not guarantee
better packet delivery times.
Figure 5:Analysis of average packet delivery time com
pared to C=1,K=1
7.2.3 Change of Overhead with redundancy
An aspect which had not been considered in the graphs
plotted so far was the amount of inherent redundancy in
the graph.From Figure 6,it becomes evident that more
the redundancy in the graph,lower is the overhead when
maintaining a multipath metric as compared to a single
path metric.The data represented in the graph is averaged
over all C values.The percentage dierence from K=1 is
reported.Each redundancy value on the Xaxis has three
pairs of columns,accounting for =0,1,0.5 and 0.9 from
left to right respectively.This graph conrms the objec
tive that we set out to achieve  which was to harness the
power of redundancy in graphs.For lower values of redun
dancy the overheads incurred by maintaining multipaths
are higher as expected.In such cases,the benets of low
sensitivity of hyperlinks is obscured by the fact that there
are fewer number of redundant paths in the graph.At
higher values of redundancy,the eect of hyperlinks in sup
pressing link value uctuations is more pronounced.Also,
the gains become larger as the redundancy is increased,
proving the value for having a hyperlink metric.
Figure 6:variation of control overhead with increase in
redundancy
7.2.4 Change of Average Packet Delivery Time
with redundancy
Figure 7 is the analogue of Figure 6,only that it reports
the Average Packet Delivery Time.The graphs point out
that in more cases than not,keeping alternate paths pay
o,even at low redundancy.In almost all cases,when
maintaining redundant paths have proved to be worse,it
is when three disjoint paths are maintained ( K=3).This
seems to indicate that keeping one primary path and one
alternate path may be the most optimal solution.Main
taining more than 3 edgedisjoint paths may often lead to
suboptimal decisions.
Figure 7:variation of average packet delivery time with
increase in redundancy
7.3 Conclusions and Caveat
All of the above results are to be taken with a pinch of salt.
The improved performance under greater redundancy has
been shown as a ratio to the case K=1 for the correspond
ing values of C.In other words the base case of K=1 for
higher values of C implies that one is essentially maintain
ing a larger local view,and choosing only one path to each
node in its Chop neighborhood.Figures 6 and 7 prove
that if a node maintains a larger view,it essentially needs
to maintain alternate paths to make use of the extra in
formation overhead.Also Figures 4 and 5 seem to suggest
that in the average case the control message overhead is
higher and path improvements are not signicant.How
ever,when we throw in redundancy into the graph  the
scenario changes,and there is a greater scope for suppress
ing updates and route along multipaths.
Figure 8:comparison of average control overhead for all
values of C and K with increase in redundancy
Figure 8 and Figure 9 show comparison for overhead and
Figure 9:comparison of average packet delivery time for
all values of C and K with increase in redundancy
packet delivery time respectively for values of C and K
for graphs of dierent redundancy ( the values being aver
aged across dierent values of ).The set of blue columns
denote the case of C=1 ( corresponding to ordinary link
state algorithm ),the set of brown columns denote the
C=2 case,and the set of green columns denote the C=3
case.These graphs are meant to clear up the dierences
in the inferences from Figures 47.Figures 8 and 9 clearly
show,that the overhead and the average time required for
delivery of a packet for a particular value of redundancy is
often worse in the case of higher C and K,for any value
of redundancy.It is only that when one is maintaining a
higher value of C ( i.e a node is maintaining extra infor
mation,maybe for orthogonal purposes ),higher values of
K gives better allround results.
8 Future work
There are several directions in which this work can be
taken forward.Modifying XL renders its correctness proof
null and void.It would be a challenge to come up with a
theoretical correctness proof for the algorithm described.
Also,one need not maintain a Chop information for ev
ery node.Having a criteria whereby nodes may select the
amount of state information to maintain ( based on local
redundancy ) is key to reducing the number of updates.
The expected time metric ( or an improved version of it
along similar lines ) captures the semantics of fast packet
delivery  and hence might be expected to give better re
sults.Also,various facets of percolation theory might be
applicable to such stochastic networks  it is a direction
worth exploring.
Acknowledgements
The author would like to thank Scott Shenker for sharing
with us his vision of rapidly uctuating networks and also
for his invaluable help in formulating this problem.The
authors are also grateful to David Culler for helping us
focus on the right bits and to Ion Stoica for his valuable
advice.
References
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