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Multicast Routing
Algorithms
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Multicast Routing Algorithms 2
Outline
Introduction to Multi-point Communication
Three approaches to Multi-cast Routing
Steiner Tree Heuristics The MZQ algo
The SCTF algo
The Virtual Trunk algo for dynamic routing
The BSMA algo.
The KPP algo
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Multicast Routing Algorithms 3
Multipoint Communication
Concept: Single Source, Multiple Destinations,
Duplication only at branch points.
Present Day Support: Communication satellites.
e-mail lists, internet news distribution.
Tomorrow's multimedia applications require:
efficient use of bandwidth.
near simultaneous delivery.
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Multicast Routing Algorithms 4
Applications: Multicast & Multi-point
One to Many
Video Distribution
Wide scale Information
dissemination.
Many to Many
Video Conferencing
Computer Supported
Common Work.
Distributed interactive
simulation.
Large scale distributed
(super)computing.
Distributed Games
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Multicast Routing Algorithms 5
Semantics of Multi-point Communication
Reliability
are different reliability models required for different classes of
applications?
Allowing dynamic join and leave session has to be receiver controlled.
Addressing
how to address groups at each level?
Whether and how to identify groups in layers above the IP layer?
Directionality
one-to-many or many-to-many.
are the transmitters a subset of the receivers?
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Multicast Routing Algorithms 7
Multipoint Routing algos
Performance Metrics
Quality of a tree is judged according to the following three
dimensions
Low Delay:
End to end delay between source and receiver relative to theshortest unicast path delay.
Low Cost :
Cost of total bandwidth consumption
Cost of tree state info
Light Traffic Concentration :
Maximum number of flows on a unidirectional link.
How evenly the routes are distributed.
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Multicast Routing Algorithms 8
Routing Algorithms
All multi-point services use some kind of a distribution tree.
Multicast trees can be
Shared across sources. (shared trees)
Only one tree needs to be established for each group, which is
shared by all the sources within that group.
Source specific. (shortest path trees).
A shortest path tree rooted at each sending node needs to be
established
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Multicast Routing Algorithms 9
SOURCE BASED MULTIPOINT ROUTING
The Technique.
A Source Rooted Shortest Path Tree (SRSPT) algo:
Computes the shortest paths between the source and each of the
receivers within the group.
Eliminates duplicate data copies on common links.
Maintains one SRSPT per sender.
Concept:All receiving nodes compute path towards the source
independently.
Used by:current day IP multicast protocols as applications are still
small scale.
local area.
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Multicast Routing Algorithms 11
SHARED TREE APPROACH OF MULTIPOINT ROUTING
Characteristics of Steiner Tree based algorithms.
The Minimum Steiner Tree: The minimal cost subgraph
spanning a given subset of nodes in a graph.
The Steiner Tree problem is NP-complete. finding the minimum steiner tree in a graph has exponential cost.
The tree designed is undirected.
solution feasible only for symmetric links.
Monolithic algorithm.
has to be run each time group membership changes.
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Multicast Routing Algorithms 12
SHARED TREE APPROACH OF MULTIPOINT ROUTING
Characteristics of Steiner Tree based algorithms.
The SMT defines an absolute limit on the minimum tree
cost to serve as a reference for gauging the cost-optimality
of heuristic alternatives.
The SMT for all members of a multicast group is the same
irrespective of the role of sender or receiver.
only one state entry needs to be maintained per group.
it scales well for larger groups.
The SMT may have unbounded delay.
Worst case maximum end-to-end path length of a SMT can be the
longest acyclic path within the graph.
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Multicast Routing Algorithms 13
An example of a minimum Steiner Tree
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Mcast group members
Relay Nodes
AG = 6AE = 5AD = 3AC = 3AI = 8
Shortest Paths from Atotal distribution cost = 16
*
AG = 8AE = 5AD = 3AC = 3AI = 8
KMB Tree paths from Atotal distribution cost=13
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Multicast Routing Algorithms 14
SHARED TREE APPROACH OF MULTIPOINT ROUTING
Characteristics of Core Based Tree algorithms.
Concept:
Use the shortest Path Tree rooted at a node in the center of the
network
Steps: Choose an optimal center for the group. Multiple cores can be used
for better fault tolerance & delay characteristics.
Group members send a join message to the center.
Intermediate nodes mark interface from which the multicast info is
received and forward it to the center.
Choose the center to:
minimize max/avg delay for all members on the tree.
Minimize the sum of tree-link costs.
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Multicast Routing Algorithms 15
An example of a core based tree.
{A, B, C} = multi-cast group members
A
B
C
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Multicast Routing Algorithms 16
SHARED TREE APPROACH OF MULTIPOINT ROUTING
Advantages of Core Based Tree algorithms
Work well with multiple senders/receivers
state information is stored per group, therefore scalable.
Receiver based approach.
Supports dynamic group membership with relativeease.
Suitable for sparsely distributed receivers.
SPTs will not have many common links.
Do not have the unbounded delay problems of SMTs.
Simple to implement
used as the basis of PIM and of The CBT interdomain
Routing Protocol.
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Multicast Routing Algorithms 17
SHARED TREE APPROACH OF MULTIPOINT ROUTING
Disadvantages of Core Based Tree algorithms
Incur extra delay as compared to the RPF approach.
Suffer from traffic concentration on links converging
towards the center.
Choosing the optimal center is an NP complete problem.
Locating the center requires complete knowledge of the
network topology.
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Multicast Routing Algorithms 18
MULTIPOINT ROUTING
TradeOffs between algos
Any single tree cannot achieve Minimal Cost and Minimal
Delay both.
Shortest Path TreesMinimize delay at expense of Cost.
Steiner Minimal TreesMinimize cost at expense of Delay.
Between thesespectrum of different types of trees offering
different tradeoffs.
Different strategies to place the routes results in different
degrees of traffic concentration.
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Multicast Routing Algorithms 19
MULTIPOINT ROUTING
Ideals
Ideally multicast routing algorithms should
Compute trees with the desired cost and delay
characteristics.
Adapt to dynamic group behavior. Algorithm should be incremental (like CBT) instead of
monolithic (like SMT).
Maintain properties of the original route.
Not perturb ongoing data transfers.Be receiver driven.
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Multicast Routing Algorithms 20
STEINER TREE HEURISTICS
Problem Formulation
Given graph G= (V, E, c)
V= Set of vertices
E= Set of edges.
c= Cost function c: EZ0+ ( EdgesNon Negative Integers)
Z-Vertices: Set of Terminals (sometimes referred to as M)
S-Vertices: Set of non-terminals
TO: Initial tree = {s}.
Q : Priority Queue of vertices in the tree.
Vt: Vertices in the tree.
At: Edges in the tree.
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Multicast Routing Algorithms 21
STEINER TREE HEURISTICS
Pruned Dijkstra Heuristic (PDH)---Networks v17 `92
Take an arbitrary node as source.
Find the single source shortest path tree T for graph G
using Dijkstras algo. Delete from T, all S-vertices of degree 1.
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Multicast Routing Algorithms 22
Dijkstras Shortest Path Algorithm.
Begin.
vV,
add v to set U,
initialize Distance(v) = cost(s, v) Distance(s) = 0; Remove s from U.
while U is not empty do
vany member of G with minimum distance.
Remove v from U. For each neighbor w of v, do
if member(w, U)
distance(w) = min(distance(w), cost(w, v) + distance(v) );
Stop.
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Multicast Routing Algorithms 23
STEINER TREE HEURISTICS
Matsuyamas Minimum Cost Path Heuristic (MPH)
---Math Japonica v24
Begin
T1: subtree of G containing one arbitrarily chosen Z vertexi .
k = 1;
Zk={i}.
Determine a Z-vertexiZ - Zkclosest to Tk
Construct a tree Tk+1by adding the minimum cost path from Tk toi
k = k+1.
If k < p go-to step2.
If k = p, output Tpas the solution
Stop.
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Multicast Routing Algorithms 24
Matsuyamas Minimum Cost Path Heuristic
B
A
D E
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C G
Tree built so far
Next Terminal
to be added to the tree
Idea:For each iteration while M is not empty--Pick up that node from M which closest to the tree built so far.
Data Structure Needed: All pair shortest paths (Floydd Warshalls algo O(n3)a)
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Multicast Routing Algorithms 25
STEINER TREE HEURISTICS
KMB - A Fast Algo for Steiner Trees.---Acta Informatica 1981
Output A Stiener Tree Thfor G and the Z-vertices.
Step 1: Construct a complete directed distance graph G1=(V1,E1,c1) fromG and Z.
Step 2: Find the minimum spanning tree T1of G1. (pick any to break ties)
Step3: Construct a subgraph GSof G by replacing each edge in T1by its
corresponding shortest path in G. (break ties arbitrarily).
Step 4: Find the minimum spanning tree TSof GS (break ties arbitrarily).
Step 5: Construct a Steiner tree THfrom TSby deleting edges in TSif
necessary, so that all the leaves in TH are Steiner points.
Worst case time complexityO(|S||V|2).
Costno more than 2(1 - 1/l) *optimal cost
where l= number of leaves in the steiner tree.
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Multicast Routing Algorithms 26
Working of KMB
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Multicast Routing Algorithms 27
Working of KMB
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Destination Nodes
Intermediate Nodes
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Multicast Routing Algorithms 28
Multicast Tree Generation AlgorithmsThe MZQ Algorithm for multicasting in all optical networks
-- Malli, Zhang, Qiao
Limited wavelength conversion : every node is capable of converting
an input wavelength to only a subset of output wavelengths.
Sparse wavelength conversion : an input wavelength can be converted
to any output wavelength, but only a few nodes posses this capability.
Sparse Splitting : only a fraction of nodes can forward as many copies
as needed, and the rest of the nodes have no splitting capability.
MZQ assumes there are always enough wavelengths on each link.
Constructs multi-cast trees based on splitting capability of the nodes.
Nodes without splitting capability can have at-most one child in the
tree.
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Multicast Routing Algorithms 29
Multicast Tree Generation AlgorithmsThe MZQ Algorithm: Routing
-- Malli, Zhang, Qiao
Algorithm maintains three sets of nodes
V:nodes in the tree through which the tree can grow .
(nodes with splitting capability).
V`:nodes in the tree through which the tree cannot grow.(nodes without splitting capability)
UV: set of Terminals not included in any tree so far.
Pick that node from UV which is nearest to the tree
Include as many destinations as possible in one multicastingtree.
For nodes not included in the preceding tree(s), algorithm called
recursively to construct another multicasting tree.
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Multicast Routing Algorithms 30
Multicast Tree Generation AlgorithmsThe MZQ Algorithm: Wavelength Assignment
-- Malli, Zhang, Qiao
PerformanceMetrics: Number of Wavelengths
Total amount of Bandwidth (Total number of Channels)
Two counters maintained on each link
Ihighest wavelength index being used.
Nnumber of wavelengths being used.
Unlimited wavelengths on each link.
First-Fit algorithm used for wavelength assignment.
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Multicast Routing Algorithms 31
The MZQ Algorithm
Multicasting forest in a NSF-NET like network
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Node with full Splitting Capability
Node with no splitting capability
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Multicast Routing Algorithms 32
Multicast Tree Generation AlgorithmsThe MZQ Algorithm for multicasting in all optical networks
-- Malli, Zhang, Qiao
Results:
The bandwidth savings from using multicasting saturate at
50%.
Multicasting reduces number of wavelengths required byas much as 60%.
Even when the network does not have any nodes that have
the splitting capability, multicasting reduces the
bandwidth consumed by 43% to 45%. No more than 75% of the nodes need to have the splitting
capability to obtain the same effect as having the splitting
capability in all the nodes.
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Multicast Routing Algorithms 33
Multicast Tree Generation AlgorithmsSCTF-Algo (Selective Closest Terminal First)
S. Ramanathan, ---IEEE Infocom 1996
Initially Tree T = {source}.
Repeat until M is empty
Extend one branch from T to a terminal in M.
remove that terminal from M. Stop
Vertices in Tree are maintained as a priority queue with
priority(source) > priority(terminals) > priority(non-terminals).
Bin B holds first vertices of the queue.
Choose the path of least cost from all vertices in B to all
non-terminals not in the Tree.
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Multicast Routing Algorithms 34
Multicast Tree Generation AlgorithmsSCTF-Algo: Formal Description.
---IEEE Infocom 1996
Init Q{s}, Vt{s}, At{ }.
While M not empty do
Bfirst min( , |Q| )vertices in Q.
Initialize PATH to any path from B to M.
For each vin B do
for each min M do
if cost( shortestPath( v, m) < cost (PATH) PATHP.
Branchsubpath ( w z ) : only wis in Vt.
Insert vertices in Branch into Q. VtVt{vertices in Branch} , AtAt{edges in Branch}.
MM { terminals in Branch}.
Return T.
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Multicast Routing Algorithms 35
Multicast Tree Generation AlgorithmsSCTF-Algo: Subsuming three other algorithms
---IEEE Infocom 1996
PDH, MPH, and KMB are special cases of the R-algo.
= 1. SCTF equivalent to PDH. = |M|+1 & Ignore non-terminals in B.
SCTF equivalent to KMB.
= |V|. SCTF equivalent to MPH.
As
from 1 to n, Tree_Cost
, and Running_Time
. Running_Time linearly.
Tree_cost very rapidly.
Low = good operating point.
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Multicast Routing Algorithms 36
Multicast Tree Generation AlgorithmsR-Algo. Performance Characteristics
---IEEE Infocom 1996
Running Time of R-Algo : O(m2+ e)
Assumption: Shortest Paths from every vertex to every
terminal are available. ( takes O( m. n. log(e) ) time )
Performance Guarantee
A= max{ A(I)/OPT(I)}
Tree cost 2 . m. Optimal_Cost
m= MAX[u, v] max(cost(u,v), cost(v, u))/ min(cost(u,v), cost(v, u))m = 1 for symmetric graphs.
A= O (m).
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Multicast Routing Algorithms 37
Multicast Tree Generation AlgorithmsFeatures of the SCTF-Algo
---IEEE Infocom 1996
Controlling knob enables use of the SCTF-algo for both
Delay Optimization. ( = 1) Cost optimization. ( = n)
Advantageous for MultiMedia applications
select the right tradeoff & operating point to accommodate the
differing requirements of voice, video and data.
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Multicast Routing Algorithms 38
Multicast Tree Generation AlgorithmsVTDM - A Dynamic Multicast Routing Algorithm
H.C.Lin & S.C. Lai ---IEEE Infocom 1998
Problem Formulation..
Source node s.
Sequence of requests R = {r1, r2, ... rm}
Each request rieither adds a destination node to
or removes a destination from the multicast group.
The DMRP: Find a sequence of multicast trees
{Ti, i = 1 .. m} such that certain overall costis minimum.
This does not allow re-routing of existing connections as
the sequence of requests proceeds.
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Multicast Routing Algorithms 39
Cost Modelling
w(e, i):Cost of using wavelength ion edge e.
Infinite if iis not available on edge e.
cv(p , q):Cost of wavelength conversion at node v, frompto q.
Infinite if p cannot be converted toq at node v.
If p = q, then cv(p , q)is zero.
C(T) = v Tw(p(v), v), (v)) +
v T-{s}C p(v)((p(v)), (v))
where p(v) means parent of v in the tree.
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Multicast Routing Algorithms 40
The VTDM algo
Concept of a Virtual Trunk
---Infocom 1998
A virtual trunk is a tree of the underlying graph.
Spans nodes which have the greatest probability of being a
part of the multi-casting tree. Used as a template for building the multicasting tree.
Nodes which have a greater number of shortest paths
passing through them, have a greater probability of being a
part of the multi-cast tree. Weight W(vi) of vertex vi= number of shortest paths
passing through vi.
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Multicast Routing Algorithms 41
The VTDM algo
Building the Virtual Trunk---Infocom 1998
Find the shortest paths for all pairs of nodes in G.
Assign weights to the vertices in G.
Find the set of trunk nodes F.
Construct a complete graph for the set of trunk nodes.
Find the minimum spanning tree for the complete graph.
Convert edges in min. span. Tree back to the corres
shortest paths in graph G.
Run the minimum spanning tree algo and remove
unnecessary nodes and links to obtain the virtual trunk.
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Multicast Routing Algorithms 42
The VTDM algo
The VTDM routing algorithm
---Infocom 1998
Build the virtual trunk.
Adding a node to the multicast group
establish shortest route from the node to the virtual trunk. is
established.
Route along virtual trunk to source node also established if not yet
there.
Add node to the multicast group.
Removing a node from the multi-cast group First remove the node from the multicast group.
If it is a leaf node, remove the node from the tree.
Prune the excess branch, if the node did not have any downstream
nodes.
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Multicast Routing Algorithms 43
The VTDM algo
Node Addition (adding node B)---Infocom 1998
Step1:
If node B on the virtual trunk, denote it as node A & go to step2.
Else, find the shortest route from node B to the virtual trunk.
Add portion of the shortest route not yet included in the multicast
tree to the multicast tree.
Let node A be the node on the virtual trunk which attaches node B
to the virtual trunk via the selected shortest route.
Step2:
If node A is already on the multicast tree go to step 3. Else add portion of route from node A to source node that has not
yet been included in the multicast tree to the multicast tree.
Step3:
Add node B to the multicast group.
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Multicast Routing Algorithms 44
The VTDM algo
Node Removal (removing node B)---Infocom 1998
Step1:
Remove node B from the multicast group.
Step2:
If node B has downstream nodes the procedure is done.
Else, if node B is a leaf, remove node B and its upstream link to
the multicast tree.
Step3: If the upstream node of node B is in the multicast group, the
procedure is done.
Else denote this node as node B and go to step 2.
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Multicast Routing Algorithms 45
The VTDM algo
Simulation Results---Infocom 1998
Mean Inefficiency = TreeCost using AlgoA/ TreeCost using AlgoB.
KMBis taken as the reference algorithm.
VTDM compared against dynamic greedy(DG), Shortest Path (SP).
Mean Inefficiency versus Number of nodes
significant improvement over SP, better than DG.
Mean Inefficiency versus Size of multicast group.
significant improvement over SP, better than DGfor large grps.
Max Degree of nodes in the multicast trees. (no. of data copies).
Much lesser degree than SP, less than DGalgorithm.
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Multicast Routing Algorithms 46
Multicast Tree Generation AlgorithmsBSMA - Bounded Shortest Multicast Algorithm
Zhu, Parsa & Aceves---IEEE Infocom 1995
Problem :
Minimize the tree cost.
Guarantee all delays are less than predetermined bounds.
Feasible region :
the set of all delay bounded Steiner trees.
Steps:
Construct minimum delay steiner tree T0using
Dijkstras shortest path algorithm
Refine T0iteratively for lower cost while staying within
feasible region.
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Multicast Routing Algorithms 47
Multicast Tree Generation AlgorithmsBSMA - Definition of the cost function.
---IEEE Infocom 1995
Utilization Driven Cost
Minimizes sum of link costs along the path.
Congestion Driven Cost
Minimizes maximal link cost requirement along paths.
Link cost function
Cost of the link associated with the utilization of the link.
Link delay function
Queuing, Transmission, and Propagation delays on the link.
Destination Delay Bound Function (DDF)
Upper bound to the delay along path from the source to each of the
destinations.
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Multicast Routing Algorithms 48
Multicast Tree Generation AlgorithmsBSMA - Refinement of the tree for lower costs.
---IEEE Infocom 1995
Path Switching:
refinement of Tj to Tj+1.
Choosing a path p to be taken out of Tj.
Tj = Tj1+ Tj
2p
Choosing the new path ps in G not in Tjthat replaces the
path to be deleted from Tj.
Tj+1 = Tj1+ Tj
2ps. Tj+1is delay bounded.
l G l h
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Multicast Routing Algorithms 49
Multicast Tree Generation AlgorithmsBSMA - Refinement of the tree for lower costs.
---IEEE Infocom 1995
From Tjget Tj/.
Tj/has the source, all destination nodes, and all relay nodes of
degree more than 2.
Tj/. Edges of Tj
/are called super_edges.
All nodes between the two end_nodes of a super_edge are relay
nodes of degree 1.
Every super_edge represents a candidate path in Tj for switching.
**
M lti t T G ti Al ith
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Multicast Routing Algorithms 50
Multicast Tree Generation AlgorithmsBSMA - Algorithm Details
---IEEE Infocom 1995
Initially allsuper_edgesare unmarked.
Step1:Among all unmarkedsuper_edges, BSMA selects the
super_edgePhwith the highest path cost.
Exchange it with anothersuper_edgeof lesser cost, such that resulting
tree is delay bounded.. One of the two cases must happen:
The delay bounded shortest path is the same as Ph.
Mark that super edge. Go to Step1.
The delay bounded shortest path is a path other than Ph.
Do The replacement.
Unmark all super_edges.
Go to Step 1.
Stops when all super edges are marked.
M lti t T G ti Al ith
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Multicast Routing Algorithms 51
Multicast Tree Generation AlgorithmsBSMA - Algorithm Details
---IEEE Infocom 1995
BSMA incrementally calculates k shortest paths between subtrees Tj1
and Tj2.
K is determined only after a shortest path is found which has resulted
in a delay bounded tree. So the shortest path incremental construction
stops when one of the following two conditions is satisfied. The shortest path found does not result in the new tree violating the
delay bound.
The shortest path found has equal path length to the one just
deleted.
Dijkstras algo is extended to construct shortest path between twosubtrees instead of two nodes
A pseudo source node s is connected to all nodes in Tj1 and a
pseudo destination node d is connected to all nodes in Tj2.
The shortest path algo starts from s and ends at d.
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Multicast Routing Algorithms 52
The BSMA algo
cont
6,2
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Delay Bound = 12,end-end delay = 12
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*
6,2
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3,3
End-end delay = 9
A B
CD
Th BSMA l
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Multicast Routing Algorithms 53
The BSMA algo
Greedy Path switching
---IEEE Infocom 95
Gain = cost reduction after a round of path switching
if c = cost of Tj and c_prime = cost of Tj+1, then
gain = c - c_prime.
BSMA computes gains of all pairs of possible path
switchings in Tj and then selects one with the maximum
gain.
BSMA continues the greedy switching and terminateswhen the maximum gain is zero.
The time complexity of this greedy approach is more.
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Multicast Routing Algorithms 54
The BSMA algo
Simulation Results
Time complexity of BSMA O(kn3log(n)).
Using the tightest possible delay bound, as determined by the
min. delay tree, the cost of the BSMA tree is substantially better
than the cost of the min. delay tree.
A Range of min. cost solutions can be obtained between the two
extremes of the KMB and the min. delay solution.
Compared to KMB the relative quality of results improve with
the number of destinations in the multicast group.
A tighter bound results in a larger value for k, and henceincreases the computation time required by the algo, a slightrelaxation of the bound often results in considerably fewercomputations.
M lti t T G ti Al ith
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Multicast Tree Generation AlgorithmsKPP algorithm for a delay constrained Steiner Tree
Kompella, Pasquale, Polyzos ---IEEE Infocom 1995
Problem :
Minimize the tree cost. is minimized.
Bounded end-to-end delay.
Features:
Edge_Cost and Delay are different functions.
Delay constraints are on individual path delay.
Assumption:
Source has all the info necessary for tree construction.
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Multicast Routing Algorithms 56
Multicast Tree Generation AlgorithmsKPP algorithm for a delay constrained Steiner Tree
---IEEE Infocom 1995
Constrained Cheapest Path between v and w
Least cost path between vand wthat has delay less than .
Cost of such a path is PC(v, w).
Delay on this path is PD(v, w). Closure graph G/
A complete graph on the nodes in N, with PC(v, w) as edge
costs PD(v, w) as the edge delay.
To compute Closure graph G/
Calculate all-pair-constrained-cheapest-paths using Floyds
algorithm. (is bounded, so possible in poly-time)
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Multicast Tree Generation AlgorithmsKPP algorithm details
---IEEE Infocom 1995
Cd(v, w) = minuV {Cd-D(u, w)(v, u)+ C(u, w)}
Cost of cheapest path from vto wwith delay exactly d.
PC(v, w) = min d
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Multicast Tree Generation AlgorithmsKPP algorithm details
---IEEE Infocom 1995
Compute the closure graph G/of G.
Construct a constrained spanning tree T/of G/, using one ofthe two selection functions fCor fCDas the selection
function.
Expand the edges in the constrained spanning tree T/ intothe constrained cheapest paths they represent, and remove
any loops that may be caused by this expansion.
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Multicast Tree Generation AlgorithmsKPP algorithm - Two Source Based Heuristics
---IEEE Infocom 1995
CSTCD
Tries to choose low cost edges but modulates the choice
to pick edges that maximize the residual delay.
This increases the chances of extending the path alongthis edge, and beyond to another destination.
Has a tendency to optimize on delay.
May find trees with delay far lower than at the
expense of added cost to the tree. Uses fCD= if P(v) + D(v, w) <
fCD= infinity otherwise
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Multicast Routing Algorithms 60
Multicast Tree Generation AlgorithmsKPP algorithm - Two Source Based Heuristics
---IEEE Infocom 1995
CSTC
minimizes fC.
Constructs cheapest tree possible, while remaining
within delay bounds. Minimizes cost, without unduly minimizing delay.
fc = C(v, w) if P(v) + D(v, w)