minimum-cost multicast routing for multi- layered multimedia distribution im phd forum, ntu...

IM PhD Forum, NTU

Minimum-Cost Multicast Routing for Multi-Minimum-Cost Multicast Routing for Multi-Layered Multimedia DistributionLayered Multimedia Distribution

Hsu-Chen Cheng and Frank Yeong-Sung LinDepartment of Information Management

National Taiwan University

[email protected]

Cheng, Hsu-Chen National Taiwan University 2

Outline Introduction

Multirate Multicasting Modified T-M Heuristic

Two Simple Enhanced Procedures Mathematical Formulation Lagrangean Relaxation

Dual Problem Getting Primal Feasible Solutions

Computational Experiments With Consideration of Link Capacity Constraint Conclusion and Future Research


Introduction With the popularity of the Internet, multimedia

multicasting applications such as e-learning and video conference based on network service are growing rapidly.

For Internet service provider, there is one way to achieve the goal of revenue maximization, namely: network planning or traffic engineering.

Traffic engineering is the process of controlling how traffic flows through a network in order to optimize resource utilization and network performance.


Introduction (cont.)

S 1N

1D

2N

2D

3D

4D

(a) Example NetworkRate: 2 Mbps

Rate: 0.5 Mbps

Rate: 0.5 Mbps

Rate: 0.5 Mbps

2D

3D

4D

1N 2N

1D

S

Rate: 0.5 Mbps

Rate: 0.5 Mbps

Rate: 0.5 Mbps

Rate: 2 Mbps

(b) Unicast video distribution using point-to-pointtransmission

S 1N

1D

2N

2D

3D

4DRate: 2 Mbps

Rate: 0.5 Mbps

Rate: 0.5 Mbps

Rate: 0.5 Mbps

(c) Multicast video distribution using point-to-multipoint transmission

Cost: 8.5

Cost: 6.5


Steiner Tree Problem The minimum cost multicast tree problem, which is the

Steiner tree problem, is known to be NP-complete. The Steiner tree problem is different to the minimum

spanning tree problem in that it permits us to construct, or select, intermediate connection points to reduce the cost of the tree.

Heuristics Shortest Path Tree Based (Minimum Depth Tree, MDT) Spanning Tree Based (MST) T-M Heuristic Others (Optimization based)


Steiner Tree Heuristics

MDT

MST

Cost: 8

Cost: 9

S(1) D(2,4)


T-M heuristic The T-M heuristic is known to have a solution that is

within a factor of two of the optimum.

Cost: 7


Multirate Multicasting Multi-layered Multicasting (Multirate Multicasting)

Taking advantage of recent advances in video encoding and transmission technologies the source only needs to transmit signals that are sufficient for the highest bandwidth destination into a single multicast tree.

A multi-layered encoder encodes video data into more than one video stream, including one base layer stream and several enhancement layer streams.


Multirate Multicasting (cont.)

S 1N

1D

2N

2D

3D

4D

(a) Example NetworkRate: 2 Mbps

Rate: 0.5 Mbps

Rate: 0.5 Mbps

Rate: 0.5 Mbps

2D

3D

4D

1N 2N

1D

S

Rate: 0.5 Mbps

Rate: 0.5 Mbps

Rate: 0.5 Mbps

Rate: 2 Mbps

(b) Unicast video distribution using point-to-pointtransmission

S 1N

1D

2N

2D

3D

4DRate: 2 Mbps

Rate: 0.5 Mbps

Rate: 0.5 Mbps

Rate: 0.5 Mbps

(c) Multicast video distribution using point-to-multipoint transmission

B+E

B+E

B

B

B

B

Base layer- 0.5 MbpsEnhancement layer- 1.5 Mbps

Cost: 8.5

Cost: 6.5 Cost: 6

S 1N

1D

2N

2D

3D

4D

Rate: 0.5 Mbps

Rate: 0.5 Mbps

Rate: 0.5 Mbps

(d) Multicast video distribution using multi-layeredMulticasting


Multirate Multicasting (cont.) For the conventional Steiner tree problem, the link costs i

n the network are fixed. However, for the minimum cost multi-layered video multicast tree, the link costs are dependent on the set of receivers sharing the link.

The heterogeneity of the networks and destinations makes it difficult to design an efficient and flexible mechanism for servicing all multicast group users.

For (i) a given network topology, (ii) the destinations of a multicast group and (iii) the bandwidth requirement of each destination, we attempt to find a feasible routing solution to minimize the cost of a multicast tree for multi-layered multimedia distribution.


Modified T-M Heuristic Takahashi and Matsuyama (1980) proposed an approxi

mate algorithm (T-M Heuristic) to deal with the Steiner Tree Problem.

Maxemchuk (1997) modifies the T-M Heuristic to deal with the min-cost multicast problem in multi-layered video distribution.

Charikar et al. (2004) prove that the problem we discuss here is NP-hard and the result of M-T-M heuristic is no more than 4.214 times of an optimal multicast tree.


1 Separates the receivers into subsets according to the receiving rate.

2 Constructs the multicast for the subset with highest rate by using T-M Heuristic.

3 Using the initial tree, the M-T-M heuristic is then applied to the subsets according the receiving rate from high to low until all destination are included in the tree.

Modified T-M Heuristic (cont.)


Example I

[0]

[0]

1

2

3 4

1

1

1

1

1[0][0]

Step 5

Rate: 2 Rate: 1

1

2

3 4

1

1

1

1

1

Graph

[1]

1

2

3 4

1

1

1

1

1

[0]

[1]

Step 1

[0]

[0]

1

2

3 4

1

1

1

1

1[1][0]

Step 4Step 3

1

2

3 4

1

1

1

1

1

[0]

[1]

[1][2]

[0]

[1]

1

2

3 4

1

1

1

1

1[1][2]

Step 2

1

2

3 4

Rate: 1

Optimal

Cost: 5 Cost: 4


Example II

1

2

3

4

1

3

2

4

3 Rate: 2Rate: 1

Graph

1

2

3

4

1

3

2

4

3

[0]

[3]

[1]

[4]

Step 2

1

2

3

4

1

3

2

4

3

[0]

[3]

[1]

[4]

Step 1

1

2

3

4

1

3

2

4

3

[0]

[3]

[1]

[4]

Step 3

1

2

3

4

1

3

2

4

3

[0]

[3]

[1]

[0]

Step 4

1

2

3

4

1

3

2

4

3

[0]

[3]

[1]

[0]

Step 5

1

2

3

4

1

3

2

4

3

[0]

[0]

[1]

[0]

Step 6

1

2

3

4

1

2 3 Rate: 2Rate: 1

OptimalCost: 11 Cost: 10


Two Enhanced Procedures Tie Breaking Procedure

When there is a tie, the node with the largest requirement should be selected as the next node to join the tree.

Drop and Add Procedure 1. Compute the number of hops from the source to the

destinations.

2. Sort the nodes in descending order according to {incoming traffic/its own traffic demand}.

3. In accordance with the order, drop the node and re-add it to the tree. Consider the following possible adding measures and set the best one to be the final tree. Either adds the dropping node to the source node, or to other nodes having the same hop count, or to the nodes having a hop count larger or smaller by one.


Example

[0]

[0]

1

2

3 4

1

1

1

1

1[0][0]

Step 5

[1]

1

2

3 4

1

1

1

1

1

[0]

[1]

Step 1

[0]

[0]

1

2

3 4

1

1

1

1

1[1][0]

Step 4Step 3

1

2

3 4

1

1

1

1

1

[0]

[1]

[1][2]

[0]

[1]

1

2

3 4

1

1

1

1

1[1][2]

Step 2

Without TBP

With TBP

[1]

1

2

3 4

1

1

1

1

1

[0]

[1]

Step 1

[0]

[1]

1

2

3 4

1

1

1

1

1[1][2]

Step 2

Rate: 2

1

2

3 4

Rate: 1

Optimal

Rate: 2 Rate: 1

1

2

3 4

1

1

1

1

1

Graph

[0]

[1]

1

2

3 4

1

1

1

1

1[1][2]

Step 3 Step 4

1

2

3 4

1

1

1

1

1

[0]

[1]

[0][0]


Mathematical Formulation

Decision Variables Routing Decision: and Traffic Requirement:

Objective Function

Capacity Constraint

min IP l glg G l L

Z a m

gd

gpd gd pl glp P

x m

, ,gg G d D l L (2)

gpdx gly

glm


Lagrangean Relaxation

Lagrangian Relaxation Problem

Primal Problem

sub-optimal

subproblem subproblem

sub-optimal

LB < Optimal solution < UB

LBMultiplier

Dual Problem

Adjust uUB


Dual Problem We transform the primal problem (IP) into the

Lagrangean Relaxation problem (LR) where Constraints (2) and (6) are relaxed.

LR:

Subproblem 1:

Subproblem 2:

Subproblem 3:

1( , ) min [ ( )]g gd

Sub pl gdl gd gl gpdg G d D p P l L

Z x

2 ( ) min ( )Sub gl g glg G l L

Z D y

3( ) min ( )g

Sub l gdl glg G l L d D

Z a m

( , ) min

g gd g

g gd

D l gl gdl gpd gd pl gdl glg G l L g G d D l L p P g G d D l L

gl gpd pl gl g glg G l L d D p P g G l L

Z a m x m

x D y


Getting Primal Feasible Solution

[Lagrangean based modified T-M heuristic]

Step 1 Use as link l’s arc weight and run the M-T-M heuristic.

Step 2 After getting a feasible solution, we apply the drop-and-add procedure described earlier to adjust the result.

gl dgld D

a


Computational Experiments Testing Networks:

Regular networks (Grid Network and Cellular Network) Random networks Scale-free networks

For each testing network, several distinct cases which have different pre-determined parameters such as the number of nodes, are considered.

The link cost, destination nodes and traffic demands for each destination are generated randomly.

(a) Grid Network (b) Cellular Network


Computational Results

Maximum improvement between M-T-M and LR: Grid networks: 16.18% Cellular networks: 23.23 % Random networks: 10.41% Scale-free networks: 11.02 %

Solution optimality: 60% of the regular and scale-free networks have a gap of less than 10%, but the result of random networks shown a larger gap.


Computational Results (cont.) The M-T-M heuristic performed well in many cases, such

as case D of grid network and case D of random network. The tie breaking procedure we proposed is not uniformly

better than random selection. The results of experiments shown that the drop and add

procedure does reduce the cost of the multicast tree.

H 50 1517 1572 1503 1357 1187.332 14.29% 11.79%

D 10 547 547 547 547 541.8165 0.96% 0.00%

Case Dest # M-T-M TB AD UB LB GAP Imp.


Computational Results (cont.) There are two main reasons of which the Lagrangean

based heuristic works better than the simple algorithm. First, the simple algorithm routes the group in accordance with

fixed link cost and residual capacity merely, whereas the Lagrangean based heuristic makes use of the related Lagrangean multipliers. The Lagrangean multipliers include the potential cost for routing on each link in the topology.

Second, the Lagrangean based heuristic is iteration-based and is guaranteed to improve the solution quality iteration by iteration.


With Consideration of Link Capacity Constraint Single commodity problem Multi-commodity

problem

gd

gpd gd pl glp P

x m

, ,gg G d D l L (1)

gl lg G

m C

l L (2)


Multiplier-based Adjustment Procedure (LAP) 1) Compute the aggregate flow of each link.2) Sort the links by the difference between aggregate flow of each link

and the link capacity in descending order.3) Choose the first link. If the difference value of the link is positive, go

to Step 4, otherwise Step 6.4) Choose the group, which have the minimal sensitivity value

on that link, to drop and use as link l’s arc weight and run the M-T-M heuristic to re-add it to the tree. Consider the following possible adding measures and set the best one to be the final tree. Either adds the dropping node to the source node, or to other nodes having the same hop count, or to the nodes having a hop count larger or smaller by one.

5) If a feasible solution is found, go to Step2, otherwise Step 6.6) Stop.

gl gdld D

a

g

l gdld Da


Computational Results Maximum improvement between M-T-M and LR-MTM:

Grid networks: 13.46 % Cellular networks: 8.83 % Random networks: 15.4 % Scale-free networks: 10.6 %

Solution optimality: 72% of the regular and scale-free networks have a gap of less than 10%, but the result of random networks shown a larger gap.


Conclusion The achievement of this paper can be expressed in term

s of mathematical formulation and experiment performance.

The model can also be easily extended to deal with the constrained multicast routing problem for multi-layered multimedia distribution by adding QoS constraints.

The min-cost model proposed in this paper can be modified as a max-revenue model, with the objective of maximizing total system revenues by totally, or partially, admitting destinations into the system.


References Hsu-Chen Cheng and Frank Yeong-Sung Lin, “Minimum-Cost Multicast Routing for

Multi-Layered Multimedia Distribution,” Proc. 7th IFIP/IEEE International Conference on Management of Multimedia and Network Services (MMNS ‘04), Lecture Notes In Computer Science 3271, pp.102-114, October 3- 6, 2004, San Diego, USA.

Hsu-Chen Cheng and Frank Yeong-Sung Lin, “A Capacitated Minimum-Cost Multicast Routing Algorithm for Multirate Multimedia Distribution,” Proc. 2004 International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS ‘04), November 18~19, 2004. Seoul, Korea.

Hsu-Chen Cheng and Frank Yeong-Sung Lin, “Maximum-Revenue Multicast Routing and Partial Admission Control for Multimedia Distribution,” Proc. International Computer Symposium, Dec. 15-17, 2004, Taipei, Taiwan.

Hsu-Chen Cheng and Frank Yeong-Sung Lin, “Maximum-Revenue Multicast Routing and Partial Admission Control for Multirate Multimedia Distribution,” accepted for publication, Proc. The 19th IEEE International Conference on Advanced Information Networking and Applications (IEEE AINA2005).

Frank Yeong-Sung Lin, Hsu-Chen Cheng and Jung-Yao Yeh, “A Minimum Cost Multicast Routing Algorithm with the Consideration of Dynamic User Membership,” accepted for publication, Proc. 2005 The International Conference on Information Networking (ICOIN 2005), Lecture Notes In Computer Science 3288,Jan. 31- Feb 2, Jeju, Korea.

Hsu-Chen Cheng, Chih-Chun Kuo and Frank Yeong-Sung Lin, A Multicast Tree Aggregation Algorithm in Wavelength-routed WDM Networks, Proc. 2004 SPIE Asia-Pacific Optical Communications Conference (APOC ‘04), November 7–11, 2004, Beijing, China.

Q&A

IM PhD Forum, NTU

Minimum-Cost Multicast Routing for Multi-Minimum-Cost Multicast Routing for Multi-Layered Multimedia DistributionLayered Multimedia Distribution

minimum-cost multicast routing for multi- layered multimedia distribution im phd forum, ntu...

Documents

mbps cost

minimum spanning tree

single multicast tree

minimum depth tree

mdt spanning tree

heuristics shortest

b e b b b b base layer

future research slide