QDMR An E cient QoS Dep endent Multicast Routing Algorithm
Liang Guo Ibrahim Matta
College of Computer Science
Northeastern University
Boston MA
fguol mattag ccs neu edu
Abstract and QoS capabilities to e ciently supp ort these appli
cations
The routing of multicast tra c requires the con
Many real time applications such as video con
struction of a tree referred to as the multicast path
ferencing require the transmission of messages from
When a source sends a message this message is for
a sender to multiple receivers subject to Quality of
warded along the multicast path replicating the mes
Service QoS delivery constraints e g bounded de
sage only when the paths to di erent destinations di
lay This requires the underlying multicast protocol
verge thereby taking advantage of common path seg
to nd a QoS constrained minimum cost communica
ments
tion path tree However nding such a tree is known
to be computationally expensive In this paper we
There are many algorithms to construct multicast
present a fast heuristic called QDMR for generating
paths They can b e broadly categorized as QoS
delay constrained low cost multicast routing trees A
oblivious algorithms and QoS sensitive algorithms
salient feature of QDMR is that it dynamically adjusts
QoS oblivious algorithms e g build
its low cost tree construction policy based on how far
multicast trees in a b est e ort way without explic
the current on tree node is from violating the QoS de
itly accounting for the requested QoS Some minimize
lay bound This QoS dependent adaptive tree con
replication and bandwidth cost by building a so called
struction together with the capability to merge least
Steiner tree that spans all group members and
delay paths into the low cost tree in case of stringent
the sum of its link costs is minimum Others build a
delay requirements lead to the fol lowing properties
tree of shortest least delay paths ro oted at the source
QDMR guarantees to nd a feasible multicast tree
in order to minimize end to end delays to individual
if such tree exists this delay bounded multicast tree
destinations Shortest path trees can b e e ciently
is very rapidly generated and the tree has low cost
computed while Steiner trees are very exp ensive to
Through analysis and extensive simulations we con
compute i e NP complete For this reason some
rm the premise of QDMR by comparing it to many
QoS oblivious algorithms try heuristically to achieve
existing multicast algorithms
a balance b etween cost and delay by building a tree
of shortest paths ro oted at some heuristically chosen
no de in the center of the group
Introduction
These latter QoS oblivious algorithms may not
however provide feasible trees that satisfy the re
Distributed real time applications such as audio
quested QoS in terms of the end to end delay along the
and video conferencing collab orative environments
individual paths from the source to each of the destina
and distributed interactive simulation by and large
tion no des To overcome this limitation a number of
involve a source sending messages to a selected set of
QoS sensitive algorithms have b een prop osed These
destinations with varying Quality of Service QoS de
QoS sensitive algorithms e g
livery constraints This requires the underlying net
try to heuristically construct a low cost tree subject
work to provide multicasting group communication
to a given upp er b ound on end to end delay However
some of these heuristics may fail to provide a low cost
This work was supp orted in part by NSF grants CAREER
tree as they assume that network links are symmet
ANIR and MRI EIA and NEU grant RSDF
ric Furthermore the time required to construct such a
tree may b e prohibitive esp ecially for large networks tion de nes the delay constrained multicast routing
as they employ a brute force approach to search for problem Section describ es our QDMR algorithm
low cost delay bounded paths Section surveys some recently prop osed heuristics
against which we compare QDMR via simulations in
In this pap er we prop ose an e cient algorithm for
Section Section concludes the pap er with future
rapidly generating a low cost multicast tree sub ject to
work
end to end delay constraints in large networks with
asymmetric link costs and delays Our algorithm is
based on Shaikh and Shin s Destination Driven Multi
The Delay Constrained Steiner Tree
Casting DDMC algorithm DDMC is an e cient
Problem
algorithm for generating low cost unconstrained mul
ticast trees The basic idea of DDMC is to give the
low cost path going through a destination no de prior We represent the network by a directed graph G
V E where V is the set of all vertices no des repre
ity over other paths to b e extended in order to add a
senting routers or switches E is the set of edges links
new no de to the current tree This helps reduce the
tree cost as the cost incurred to go to one destina representing physical or logical connectivity b etween
no des Each link is bidirectional i e the existence of
tion no de can b e leveraged to reach other destination
a link e u v from no de u to no de v implies the
no des However this may result in a tree with long
0
existence of another link e v u for any u v V
paths connecting a chain of destination no des and
+
Any link e E has a cost C e E R and a de
hence the end to end delay QoS requirement could b e
+
violated We extend DDMC to overcome this problem lay D e E R asso ciated with it The function
C de nes the measure we want to optimize mini
by adjusting dynamically on the y the low cost tree
mize The function D de nes the measure we want
construction p olicy so as to take into account how far
the current destination no de is from violating the QoS to constrain b ound Due to the asymmetric nature
0
of computer networks it is p ossible that C e C e
delay requirement Furthermore in the case where
0
and D e D e
the requested QoS is still violated for some destina
tion no de we merge the least delay path into the tree
We denote a multicast group by the set ffsg R g
Thus the new algorithm guarantees to nd a feasible
V where s is a source no de which will send messages
path if there is one while keeping the tree cost low
to a set of receivers denoted by R A multicast tree
We call this algorithm QoS Dependent Multicast Rout
T s R E is a tree ro oted at s and spanning all mem
ing QDMR for short To summarize QDMR has the
b ers of R We denote by P r T the set of links
T i
following features compared to other algorithms a
in T that constitute the path from s to r R The
i
tree generation pro cess that is QoS delay dependent so
total path delay from s to r denoted by D el ay r
i i
as to minimize tree cost while at the same time keeping
is simply the sum of the delay of links along P r
T i
an eye on how far it is from violating the delay b ound
i e
X
a very fast algorithm for generating low cost delay
D e D el ay r
i
b ounded multicast trees b ecause of its QoS dep endent
e2P (r )
T i
adaptive tree generation pro cess it guarantees
that a feasible tree will b e constructed if one exists by
The total cost of the tree denoted by C ost T is
merging least delay paths into the low cost tree if nec
de ned as the sum of the cost of all links in the tree
essary The merging pro cess of least delay paths stops
i e
X
as so on as the low cost tree is encountered and the de
C ost T C e
lay b ound is satis ed thus keeping the tree cost as low
e2T
as p ossible
Through extensive simulations we show that
The application may assign an upp er b ound to
i
QDMR yields trees that have comparable cost to other
D el ay r and the upp er b ound can b e di erent for
i
existing heuristics Furthermore these low cost trees
each destination r For simplicity in our network
i
are very rapidly generated the execution time of
mo del we assume that the upp er b ound for all desti
QDMR is several orders of magnitude lower than that
nations is the same and is denoted by r R
i i
of other heuristics including the fastest heuristic re
We are currently extending this mo del to consider mul
cently prop osed in For lack of space pro ofs of
tiple delay b ounds
correctness and time complexity are omitted and can
Given these de nitions we can formally present the
b e found in
Delay Constrained Steiner Tree DCST Problem as
The rest of the pap er is organized as follows Sec follows
DCST Problem Given a network G a source
The Modifications to DDMC
node s destination node set R a link delay function
D a link cost function C and a delay bound
Because the DDMC algorithm treats each destina
the objective of the Delay Constrained Steiner
tion no de as a new source after that no de is added
Tree DCST Problem is to construct a multicast tree
to the tree the nally constructed tree can easily have
T s R such that the delay bound is satis ed i e
some very long branches which lo ok like a chain of
destination no des This enables DDMC to e ciently
D el ay r r R
i i
generate a low cost multicast tree However this may
and that the tree cost C ost T is minimized
also lead to some destination no des violating the delay
b ound We illustrate the ab ove observation by an ex
Prop osition The DCST problem is NP complete
ample in Figure For ease of presentation we assume
here that links are symmetric
Our QoS Dep endent Multicast Rout
Figure a shows the tree generated by the DDMC
ing Algorithm
algorithm This tree runs through no des S D D
1 2
D and D b ecause at each step the path ending at
3 4
destination no de will always b e selected since it has
3.1 Motivation and Mechanics a
a lower cost This gives a tree cost of as opp osed
Our QoS Dep endent Multicast Routing QDMR al
to a cost of if a tree of least delay paths LDT is
gorithm is based on the Destination Driven MultiCas
constructed as shown in Figure b However with a
ting DDMC algorithm recently prop osed by Shaikh
DDMC tree the path delays for no des D and D are
3 4
and Shin The idea of DDMC comes from the
and resp ectively and hence for b oth destinations
fact that the well known algorithms of Prim s mini
the delay b ound of is violated To satisfy the delay
mum spanning tree and Dijkstra s shortest path tree
b ound a solution is to then replace the paths to D
3
b oth use essentially the same greedy strategy In
and D by the least delay paths LDP as shown in
4
Prim s algorithm a minimum cost tree that spans all
Figure c We call such a tree DDMC LDP Now
no des is constructed by augmenting the current tree
the cost of the nally formed tree b ecomes which
with the minimum cost edge emanating from some on
is not the lowest cost p ossible as we show b elow
tree no de not necessarily the source In Dijkstra s
To e ciently obtain delay bounded low cost trees
algorithm at each step a new no de with the minimum
we mo di ed the original DDMC algorithm so as it dy
path cost from the source is added to the current tree
namically adjusts its tree construction p olicy accord
Thus the di erence b etween the two algorithms lies
ing to how far a destination no de is from the delay
in the choice of the cost function used at each step
b ound our QoS requirement the further the desti
In Prim s algorithm the cost of a new not on tree
nation no de is away from the delay b ound the more
no de to b e added to the current tree is simply the cost
priority it has in b eing selected as parent of the newly
of the minimum cost edge to it whereas in Dijkstra s
added no de In other words if we are still far from
this cost is the minimum total cost from the source to
reaching the delay b ound our QDMR algorithm would
the no de The DDMC algorithm de nes the cost of a
b ehave as DDMC i e it can pro duce less bushy trees
new no de v T via no de u T to b e
with p ossibly long paths connecting a chain of des
tination no des This is desirable so as to minimize the
C ost v I u C ost u C u v
D
cost of the tree However if the delay b ound is ab out
to b e violated our QDMR algorithm would give less
where the indicator function I u V f g
D
priority to destination no des and result in bushier
is de ned as
trees so as to reduce the likelihood of violating the delay
if u R
b ound A salient feature of QDMR is that it is simple
I u
D
otherwise
and hence computationally inexp ensive by adjusting
its tree construction p olicy dynamically as no des are
The algorithm thus tries to make the destination
added to the tree This is in sharp contrast to other al
no des app ear as new sources This gives preference
gorithms that exhibit higher computation cost b ecause
to the paths going through destination no des since the
they pre compute multiple paths b etween the source
path to a new no de via a destination no de is likely to
and each destination and then select those paths that
have a lower cost and thus b e added to the tree Since
are least costly and do not violate the delay b ound
we have to absorb the cost of reaching a destination
no de anyway the DDMC algorithm can multicast to Because it is easy to maintain the delay from the
all destination no des on a low cost tree source to each on tree no de our QDMR algorithm D2 D2 Legend: Source node 1/2 D4 D4 2/3 D3 D3 Non−Destination nodes D1 2/1 D1 2/1 2/1 2/1 2/1 Destination nodes 2/1 2/1 Network links N1 N2 N1 N2 2/2 Tree links 3/3 2/2 3/3 Removed links 3/1 3/1 Added least−delay links S S c/d cost/delay (a) (b)
D2 D2 1/2 D4 D4 2/3 D3 2/3 D3 D1 D1 2/1 2/1 2/1 2/1 Delay Bound = 6 N1 N2 N1 N2 2/2 2/2 3/1 S S (c) (d)
Figure 1. Example to show difference between DDMC and QDMR: (a) Example network and DDMC
tree; (b) Least-delay tree (LDT); (c) DDMC+LDP tree; (d) QDMR tree.
is lower than the cost of the DDMC LDP tree shown achieves its delay dep endent tree construction ob jec
in Figure c tive by replacing the simple indicator function I u
D
V f g used in DDMC with a new indicator func
We note that even after applying the new indicator
tion de ned as the ratio of the no de delay to the delay
function our QDMR algorithm may still fail to nd a
1
b ound i e
feasible path to a destination no de i e a path that
satis es the delay b ound even if such path do es ex
D el ay u if u R
ist The reason for this is that we are using a greedy
I u
D
otherwise
strategy to minimize tree cost when selecting the next
not on tree no de to b e added Thus we might miss
Using this new function when QDMR constructs
some feasible path b ecause of its high cost Therefore
the multicast tree cf Figure d no de D will have
2
we add an additional phase in QDMR to fall back on
a lower priority to b e added to the tree than no de D
1
using the least delay path if a feasible low cost path
b ecause D is farther away from the source or equiv
2
is not found for any destination no de In this case
alently its delay is closer to the delay b ound and so
2
we include the destination no de in the tree by merging
has a higher cost than D On the other hand D
1 2
the least delay path from the source to that no de into
still has a little higher priority than no des N and N
1 2
3
the partial QDMR tree as follows we start from each
b ecause it is a destination no de Since D has lower
3
4
such destination no de and trace back the least delay
cost via D than via D the nally constructed tree
1 2
path for it until some on tree no de another destina
would have a branch from no de D to D and D This
1 3 4
tion no de a relay no de or the source is encountered
tree satis es the delay b ound and has a cost of which
If the merged path satis es the delay b ound the merg
1
Other de nitions of I u are also p ossible but we found
D
ing pro cess terminates for that destination However
through simulations that this simple de nition provides su
if the merged path do es not satisfy the delay b ound
ciently go o d results
2
we continue tracing back the least delay path In the
C ost D I s C ost s C s D
1 1
D
2 8
C ost D I D C ost D C D D
2 1 1 1 2
D
worst case the merging succeeds at the source no de In
6 3
3
C ost N I s C ost s C s N
1 1
D
this case the least delay path in its entirety b ecomes
C ost N
2
4
part of the tree This merging pro cess is illustrated in
C ost D minfI D C ost D C D D I D C ost D
3 1 1 1 3 2 2
D D
5 8 8 29 2
g minf g C D D g minf Figure
2 3
6 6 3 3 9 Legend: Source Node
Destination Node
G V E s R D C D1 On−Tree Node QDMR
D2 QDMR Tree Branch
Initial Tree Construction Phase Added Least−Delay Tree Branch
D3 Least−Delay Path
Call Dijkstra s algorithm DJK G s R D to compute the
Deleted QDMR Branch
tree LDT to nd out the lowest p ossible
N1 least delay
delay b ound min fD el ay r g
min i
r 2R i
N3 N2
if
S min
Return F AI LE D a feasible tree do es not exist
D4
C ost s D el ay s
C ost u D el ay u for u s
T Q V
Figure 2. Merging process.
while Q and R T do
u Extract Min Q pick next no de
T T fug
In this example no de D can not b e included in
for each vertex v Adj u for each adjacent no de
4
if D el ay u D u v and v T
the QDMR tree in the initial construction phase since
if C ost v I u C ost u C u v
D
QDMR fails to nd a feasible path to D The merging
4
C ost v I u C ost u C u v
D
phase then starts to merge the least delay path shown
v u u is the parent
as dashed line into the tree by tracing it back to the
Merging Phase
source until the on tree no de N is encountered How
2
if R T some destinations haven t b een reached
ever at this p oint it nds that the delay b ound still
for each u R T do
can not b e satis ed i e D el ay N D N D
2 2 4
d p u merge LDP
cumm
LD P
while u s
It then continues the merging pro cess and up
u p d d D p u
cumm cumm
dates the parent p ointer of no de N from no de N
2 3
T T fug
to no de N cutting o the branch N N from the
1 3 2
if D el ay p d
cumm
tree At no de N the merging pro cess nds that
go to
1
u p p u
D el ay N D N N D N D so the sub
LD P
1 1 2 2 4
path N D is added as a new branch in the tree and
1 4
Pruning Phase
now the destination no de D is included in the tree
4
return PruneLeaves T
Finally a pruning phase is necessary to prune s N
3
since no de N is a leaf no de that is not a destination
3
pseudo co de of our QDMR algorithm is given in
The Figure 3. Our QDMR algorithm.
Figure Steps show the initial tree construction
phase Steps show the merging phase Steps
show the pruning phase
converts the original network graph into a complete
graph connecting multicast group members where a
edge represents the least cost path b etween
3.2 Correctness and Time Complexity of QDMR logical
two members Then KMB constructs a minimum cost
spanning tree and nally replaces each logical edge
Theorem The path generated by QDMR is loop
in the tree with the corresp onding physical least cost
free
path
Theorem The QDMR algorithm always constructs
KPP extends KMB to compute delay bounded
a delay bounded multicast tree if such a tree exists
paths assuming the delay b ound and link delays are
all integers This assumption limits its applicability
Lemma The time complexity of QDMR is
although an approximate solution was prop osed that
O jE j log jV j
transforms real delay values into integer values This
approximation however may compromise the accu
Related Work
racy of the algorithm by failing to nd a feasible tree
even though one exists The time complexity of KPP
3
In this section we brie y discuss some recently pro is O jV j CKMB also extends KMB without as
p osed delay constrained Steiner tree heuristics Ref suming integer delay values However CKMB like
erence provides a go o d survey The KPP KPP builds a minimum cost tree of logical edges Thus
and the CKMB heuristics b oth extend KMB an they may fail to exploit common physical edges lead
unconstrained Steiner tree heuristic KMB rst ing to high tree cost The time complexity of CKMB
2
is O jR jjV j We consider randomly generated network top ologies
using a mo di ed version of the graph generation algo
The CDKS algorithm di ers from QDMR in
rithm describ ed in The average degree of a no de
that the cost of a no de is simply the total cost from
is set to and the capacity of each link is Mbps
the source i e unlike QDMR it do es not attempt
The link delay function D e is de ned as the propa
to reduce tree cost by using a dynamic cost function
gation delay of the link plus a xed switching delay of
Furthermore if the delay b ound is violated for some
6
sec
destination no de the least delay path is simply used
The link cost function C e is de ned as the current
in its entirety leading to further increase in tree cost
7
2
total bandwidth reserved on the link We conducted
The time complexity of CDKS is O jV j
two sets of exp eriments We present each next
BSMA is another multicast algorithm that is
considered the b est in terms of tree cost BSMA
replaces the edges in the tree until the tree
iteratively Experiments 1
cost can not b e further reduced This makes the al
gorithm computationally very exp ensive to b e used for
In this rst set of exp eriments we investigate the
large scale networks The average time complexity of
quality of the multicast tree generated by each heuristic
3
BSMA is O k jV j log jV j where k is the average num
and their computation requirement This was done by
b er of lower cost paths examined to replace a lower
measuring for an arriving group the cost of the tree and
delay path
the actual execution time to generate the tree under
each heuristic All simulations were running on a SUN
Another algorithm is CAO It incremen
SPARCstation and the co des for all the simulated
tally adds new tree branches by merging in delay
algorithms were not optimized for sp eed For each run
constrained low cost unicast paths leading to new desti
the cost of a link is the amount of bandwidth reserved
nation no des The time complexity of CAO is O t A
on the link for some background tra c This reserved
jR j where A is the algorithm used to nd unicast
bandwidth is a random variable uniformly distributed
paths The original version of CAO uses a constrained
b etween B and B We set B to Mbps and
breadth rst search algorithm thus its time complex
min max min
B to Mbps The di erence b etween B and
ity can b e exp onential in some cases However even
max min
B re ects the asymmetry of the link loads when
with an e cient unicast routing heuristic e g it
max
the multicast group arrives The group members are
may still need a much longer time than QDMR to form
also randomly selected We obtain results for di erent
a tree
network sizes multicast group sizes and di erent delay
In Section we compare the various heuristics in
b ounds For a sp eci c network group con guration
terms of actual execution time via extensive simula
each algorithm is executed This was rep eated until
tions Our results show that QDMR is the fastest
con dence intervals of less than using con
in generating multicast trees that are comp etitive in
dence level were achieved for all measured quantities
terms of tree cost This indicates that QDMR is an
Figure shows the tree cost and the execution time
attractive algorithm for real time multicasting in large
for varying network size For clarity execution times
networks
are shown in log scale We x the group size at
and we choose a stringent delay b ound of ms Fig
Simulation Mo del and Results
ure a shows that as the size of the network increases
from no des to no des the di erence b etween the
sub optimal BSMA heuristic and other heuristics b e
To evaluate the e ciency of our QDMR al
comes larger CAO always yields the closest tree cost
gorithm we use the multicast routing simulator
to that of BSMA The remaining three heuristics p er
MCRSIM developed at North Carolina State Uni
form very close to each other less than CKMB
versity MCRSIM allows the simulation of arbitrary
has relatively lower tree cost followed closely by KPP
networks and supp orts multiple tra c types as well
then QDMR Figure b shows that BSMA and KPP
as background tra c on each link MCRSIM already
are the most computationally exp ensive algorithms
implements many existing Steiner tree heuristics in
QDMR is the fastest algorithm followed by CKMB
cluding BSMA CAO KPP CDKS
and CKMB We implement our QDMR algorithm
6
It is assumed that queueing delays are negligible given
in the MCRSIM simulator and compare it to these
enough bandwidth is reserved on the link for each multicast
5
other heuristics
group and that transmission delays are very small
7
This de nition re ects the relationship b etween cost and link
5
In our simulation exp eriments for the KPP algorithm we utilization a highly loaded link usually costs more Other de
map real delay values in to integer values in nitions are also p ossible
then CAO QDMR is orders of magnitude faster than icantly outp erforms DDMC LDP in terms of tree cost
all other algorithms due to its use of a QoS dep endent tree construction
pro cess while maintaining its fast execution feature
Figure shows the p erformance measures versus
This indicates that QDMR is a promising QoS multi
group size for a no de network and a delay b ound
cast routing algorithm for large networks
of ms It can b e seen from Figure a that the p er
formance of CAO in terms of tree cost degrades as the
size increases Other heuristics have similar p er
group Experiments 2
formance in general QDMR gives slightly higher tree
In this second set of exp eriments we compare the
cost less than than KPP and CKMB
e ciency of di erent heuristics in terms of how well
Figure b shows that the execution times of BSMA
they distribute arriving multicast groups tra c over
and KPP are very large almost orders of magnitude
the network so as to increase network utilization or
higher than that of QDMR As the group size increases
revenue In each run we start with an empty network
the execution times of b oth CAO and CKMB grow On
and generate successive arrivals of multicast groups A
the contrary the execution time of QDMR remains al
multicast group is randomly generated The source of
most the same with increasing group size and it can b e
each arriving group is assumed to b e a variable bit rate
up to order of magnitude lower than that of CKMB
VBR video source The QoS requirement is guaran
Thus QDMR scales very well to large networks
teed by reserving some amount of bandwidth or the
and multicast groups It can pro duce low cost
equivalent bandwidth of the source on each link of
trees at a signi cantly higher sp eed
the multicast tree In these exp eriments the equiva
Figure shows the p erformance of di erent heuris
lent bandwidth of each multicast session is Mbps
tics for varying delay b ound for a no de network and
Each group requests a delay b ound of msec
a group size of Figure a shows that all heuristics
A group is not admitted into the network i e re
can nd a lower cost tree as the delay b ound increases
jected if the total bandwidth reserved on a link would
CKMB is slightly more sensitive to the delay b ound
exceed of the link s capacity The run is termi
than others When the delay b ound is not so stringent
nated when the total group rejection rate exceeds
i e large b oth CKMB and QDMR p erform b etter
We note that an arriving group could b e rejected either
than KPP Figure b shows that the execution time
b ecause of failure to reserve bandwidth or b ecause a
of all heuristics is not very much a ected by the delay
8
feasible tree could not b e found Figure shows the
b ound
number of admitted multicast groups on a no de net
Since CDKS was suggested to b e a suitable al
work for di erent group sizes Again con dence
gorithm for large networks Figure compares
intervals were computed for the results We observe
our QDMR algorithm to CDKS as well as CKMB
that QDMR p erforms as well as BSMA which was
for relatively large networks We also show re
suggested in to b e the most e cient in distribut
sults for DDMC LDP The only di erence b etween
ing sessions over the network and thus increasing the
DDMC LDP and QDMR is in the de nition of
likelihoo d of an arriving group b eing admitted into the
the indicator function used to give priority to paths
network We also note that an algorithm that simply
going through destination no des f g is used in
builds a tree of least delay paths LDT p erforms the
DDMC LDP whereas QDMR uses the linear delay
worst as it pro duces costly trees
dep endent function cf Section We vary the net
work size from no des to no des and set the
Conclusions and Future work
group size to b e of the network size and the de
lay b ound to b e ms Figure a shows that CKMB
We presented an e cient algorithm for obtaining
yields the lowest tree cost QDMR has a slightly higher
delay constrained low cost multicast trees Our algo
tree cost always less than QDMR always has
rithm extends a recently prop osed unconstrained al
a lower tree cost than CDKS Figure b shows that
gorithm so as to rapidly generate a low cost tree
QDMR is signi cantly faster Both QDMR and CKMB
while adapting on the y the generation pro cess to
have a lower rate of increase in execution time than
account for application s end to end delay demands
CDKS as the network size increases Note that even
We showed through extensive simulations that the pro
for a very large network no des the time required
p osed algorithm generates low cost trees with a com
for constructing a QDMR tree is less than second on
a SUN SPARCstation Recall that the co de for the
8
KPP is the only simulated algorithm which may not nd a
various algorithms under study was not optimized for
feasible tree even though one exists b ecause of its real to integer
conversion of delay values sp eed It is also imp ortant to note that QDMR signif
putation overhead that is several orders of magnitude M R Garey and D S Johnson Computers and
Intractability A Guide to the Theory of NP
lower than that of other algorithms
Completeness W H Freeman and Company New
Since the problem of nding a delay constrained
York
minimum cost path is NP complete we resorted to a
R Guerin H Ahmadi and M Naghshineh Equiva
lent Capacity and its Application to Bandwidth Allo
simple heuristic in order to keep the computational cost
cation in High Sp eed Networks IEEE J Select Areas
low Like other existing algorithms such as CKMB
Commun SAC September
and KPP our algorithm may fall back on using least
L Guo and I Matta QDMR An E cient QoS Dep en
delay paths or segments of them so as to nd a feasi
dent Multicast Routing Algorithm Technical Rep ort
ble tree A more sophisticated search may yield lower
NU CCS College of Computer Science North
eastern University August
cost paths that are also feasible To that end we will
L Guo and I Matta Search Space Reduction in QoS
investigate the use of our recently prop osed heuristic to
Routing Technical Rep ort NU CCS College of
nd low cost QoS constrained unicast paths An
Computer Science Northeastern University Octob er
interesting question here is whether the gain in tree
cost will justify the additional computational cost
F K Hwang and D S Richards Steiner tree prob
lems IEEE ACM Transactions on Networking
It is worth mentioning that our QDMR algorithm as
January
well as all the other algorithms presented in this pap er
V P Komp ella J C Pasquale and G C Polyzos Mul
assumes that information regarding the state of each
ticasting for Multimedia Applications In Pro c IEEE
link i e link cost and delay and the lo cation of group
INFOCOM pages
members is available at each no de This is the same
L Kou G Markowski and L Berman A Fast Algo
link state approach used in current Internet proto cols rithm for Steiner Trees Acta Informatica
April
like MOSPF If the network provides b est e ort
J Moy Multicast Extensions to OSPF Internet draft
service then the delay b ound satis ed by the computed
Network Working Group September
tree is not guaranteed i e may b e violated for some
G N Rouskas Multicast Routing with End to End
packets For all packets to satisfy the delay b ound
Delay and Delay Variation Constraints IEEE J Se
resources must b e reserved over the computed tree to
lect Areas Commun April
guarantee the asso ciated link delays
H F Salama D S Reeves and Y Viniotis Evaluation
of Multicast Routing Algorithms for Real Time Com
Other future work include investigating other QoS
munication on High Sp eed Networks IEEE J Select
constraints e g delay jitter and applications where
Areas Commun April
group members join and leave the multicast group at
A Shaikh and K Shin Destination Driven Routing for
will Such dynamic group membership makes it very
Low Cost Multicast IEEE J Select Areas Commun
April
hard if not imp ossible to maintain all the time an op
Q Sun and H Langendo erfer E cient Multicast
timal cost multicast tree that also satis es given p er
Routing for Delay Sensitive Applications In Pro c Sec
formance delay constraints One p ossible solution to
ond Workshop on Protocols for Multimedia Systems
this problem is to whenever a new group member joins
PROMS pages Octob er
or an existing member b ecomes out of b ound add or
Q Sun and H Langendo erfer An E cient Delay
Constrained Multicast Routing Algorithm Techni
replace the old path with a new low cost delay bounded
cal Rep ort Internal Rep ort Institute of Op erating
unicast path QDMR could b e applied regularly to
Systems and Computer Networks TU Braunschweig
restore optimality of the whole multicast tree We will
Bueltenweg Braunschweig Germany
January
investigate this approach in our future work
D Waitzman C Partridge and S Deering Distance
Vector Multicast Routing Proto col Request for Com
References
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D W Wall Mechanisms for Broadcast and Selective
A Ballardie P Francis and J Crowcroft Core Based
Broadcast PhD thesis Stanford University Depart
Trees In Pro c SIGCOMM San Francisco Cali
ment of Computer Science
fornia September
B M Waxman Routing of multipoint connections
T H Cormen C E Leiserson and R L Rivest Intro
IEEE J Select Areas Commun pages De
duction to Algorithms The MIT Press and McGraw
cember
Hill Bo ok Company
R Widyono The Design and Evaluation of Routing
S Deering D Estrin D Farrinacci V Jacobson
Algorithms for Real Time Channels Technical Rep ort
C Liu and L Wei Proto col Indep endent Multicast
ICSI TR International Computer Science In
PIM Proto col Sp eci cation Internet Draft
stitute U C Berkeley June
H F Salama et al MCRSIM simulator source co de
Q Zhu M Parsa and J J Garcia Luna Aceves
and Users Manual Center for Advanced Comput
A Source Based Algorithm for Delay Constrained
ing and Communication North Carolina State Uni
Minimum Cost Multicasting In Pro c IEEE INFO
versity Raleigh Available by anonymous ftp at
COM pages ftp csc ncsu edu pub rtcomm Tree Cost vs Network Size Execution Time vs Network Size 50 1000 CAO 45 CAO KPP KPP 100 QDMR 40 QDMR CKMB CKMB 35 BSMA 10 30 25 1 20 0.1 15 Execution Time (sec) 10 0.01 % excess cost relative to BSMA 5 0 0.001 20 30 40 50 60 70 80 90 100 20 30 40 50 60 70 80 90 100
# of network nodes # of network nodes
a b
Figure 4. Tree cost and execution time versus network size: group size=10, delay bound=30 ms.
Tree Cost vs Group Size Execution Time vs Group Size 50 1000 CAO 45 CAO KPP KPP 100 QDMR 40 QDMR CKMB CKMB 35 BSMA 10 30 25 1 20 0.1 15 Execution Time (sec)
% excess cost over BSMA 10 0.01 5 0 0.001 5 10 15 20 25 30 35 40 45 5 10 15 20 25 30 35 40 45
# of group members # of group members
a b
Figure 5. Tree cost and execution time versus group size: network size=50, delay bound=30 ms.
Tree Cost vs Delay Bound Execution Time vs Delay Bound 50 1000 CAO 45 CAO KPP KPP 100 QDMR 40 QDMR CKMB CKMB 35 BSMA 10 30 25 1 20 0.1 15 Execution Time (sec) 10 0.01 % excess cost relative to BSMA 5 0 0.001 25 30 35 40 45 50 55 60 25 30 35 40 45 50 55 60
Delay bound (ms) Delay bound (ms)
a b
Figure 6. Tree cost and execution time versus delay bound: network size=50, group size=15. Tree Cost vs Network Size Execution Time vs Network Size 10 1000 CDKS CDKS DDMC+LDP DDMC+LDP 8 QDMR QDMR 100 CKMB
6 10 4
Execution Time (sec) 1 2 % excess cost relative to CKMB
0 0.1 200 250 300 350 400 450 500 550 600 200 250 300 350 400 450 500 550 600
# of network nodes # of network nodes
a b
Figure 7. Tree cost and execution time vs network size: group size=10% of network size, delay
bound=30 ms.
Accepted Sessions vs Group Size 2000 BSMA 1800 CAO DDMC+LDP 1600 QDMR CKMB 1400 CDKS LDT 1200
1000
800
600 # of successfully established sessions 400 4 6 8 10 12 14 16 18 20 # of group members
Figure 8. Number of admitted sessions versus group size: network size=20, delay bound=30 ms.