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 multi-path 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 best-path 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

electro-magnetic 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 link-state ( 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 link-state 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 non-decreasing

path costs from source to destination to route packets.

Each of the above-mentioned 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

Beta-factor 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 multi-path 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 C-hop

neighborhood,since there is an overhead required in main-

taining the alternate paths in a hyperlink (the same short-

comings to single best-path routing applies ).Maintaining

short alternate paths is key to successful routing.Also

,a multi-path metric has an added advantage of having

lower sensitivity to a link-state change,thus enabling the

network to throttle most updates.The multi-path 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 X-Y or via

P-Q.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 P-Q route than via the X-Y 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 edge-disjoint

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 edge-disjoint 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 edge-disjoint 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 link-state routing.Thus,the main idea here is to not

maintain global redundant paths since the updates will not

be received soon enough to make well-informed 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 so-called\hyperlink-state"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 link-state 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 sub-optimal paths as a trade-o.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 shortest-path 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

sub-optimal 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 edge-disjoint shortest paths from P to Q

of length C hops or less.

2.Create a hyperlink from P to Q using a multi-path

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

source-based 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 edge-disjoint 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 multi-path 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 lowest-ever

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 sub-optimal 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 run-time,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 one-time 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 C-hop

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 edge-disjoint alternate paths to maintain

to each node in a node's C-hop 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 multi-path 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 multi-paths

Figure 4 shows the percentage dierence in the overhead

of maintaining various alternate paths within a 2 and a

3-hop 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 link-state 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 link-state 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 edge-disjoint paths actually help in sending out

lesser updates.While this is a little counter-intuitive 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 multi-paths

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 multi-path 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 X-axis 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 multi-paths

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 edge-disjoint 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 C-hop 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 multi-paths.

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 4-7.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 all-round 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 C-hop 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.

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