CiteSeerX — On Algorithm for the Delay- and Delay Variation-Bounded Multicast Trees Based on Estimation ⋆

Variation-Bounded Multicast Trees Based on Estimation

Youngjin Ahn¹, Moonseong Kim¹, Young-Cheol Bang², and Hyunseung Choo¹

1 School of Information and Communication Engineering, Sungkyunkwan University, 440-746, Suwon, Korea

{watchman, moonseong, choo}@skku.edu

2 Department of Computer Engineering, Korea Polytechnic University, 429-793, Gyeonggi-Do, Korea

[email protected]

Abstract. With the multicast technology, demands for the real-time group applications through multicasting is getting more important. An essential factor of these real-time strategy is to optimize the Delay- and delay Variation-Bounded Multicast Tree (DVBMT) problem. In this pa- per, we propose a new algorithm for the DVBMT solution. The proposed algorithm outperforms other algorithms up to 9%∼25% in terms of the delay variation.

1 Introduction

Not only the tree cost as a measure of bandwidth efficiency is one of the impor- tant factors for the QoS, but also networks supporting real-time transmission are required to receive messages from source node in a limited amount of time.

Therefore, we should consider the multicast end-to-end delay and delay variation problem [3]. In this paper, we study the delay variation problem under the upper bound on the multicast end-to-end delay. We propose an efficient algorithm in comparison with the Delay and Delay Variation Constraint Algorithm (DDVCA [4]), known as the best algorithm so far. Even if the time complexity of our al- gorithm is same as the one of DDVCA, the proposed algorithm would have the better performance than DDVCA has in terms of the multicast delay variation.

The rest of the paper is organized as follows. In Section 2, we state the network model for the multicast routing, and the problem formulation. Section 3 presents the details of the proposed algorithm. Finally, section 4 concludes this paper.

This work was supported in parts by Brain Korea 21 and the Ministry of Information and Communication in Republic of Korea. Dr. H. Choo is the corresponding author and Dr. Bang is the co-corresponding author.

L.T. Yang et al. (Eds.): HPCC 2005, LNCS 3726, pp. 277–282, 2005.

Springer-Verlag Berlin Heidelberg 2005c

2 Related Works

We consider that a computer network is represented by a directed graph G = (V, E) with n nodes and l links or arcs, where V is a set of nodes and E is a set of links, respectively. Each link e = (i, j)∈ E is associated with delay d(e) ≥ 0.

The delay of a link, d(e), is the sum of the perceived queueing delay, transmission delay, and propagation delay. We define a path as sequence of links such that (u, i), (i, j), . . ., (k, v), belongs to E.

Let P (u, v) ={(u, i), (i, j), . . . , (k, v)} denote the path from node u to node v. If all u, i, j, . . . , k, v are distinct, then we say that it is a simple directed path. For a given source node s∈ V and a destination node d ∈ V , (2^s→d,∞) is the set of all possible paths from s to d.

(2^s→d,∞) = { Pk(s, d)| all possible paths from s to d, ^∀s, d∈ V, ^∀k∈ Λ }, where Λ is an index set. The path-delay of P_k is given by φ_D(P_k) =

e∈Pkd(e),

∀Pk∈ (2^s→d,∞). (2^s→d, ∆) is the set of paths from s to d for which the end-to- end delay is bounded by ∆. Therefore (2^s→d, ∆)⊆ (2^s→d,∞).

For the multicast communications, messages need to be delivered to all re- ceivers in the set M ⊆ V \ {s} which is called the multicast group, where

|M| = m. The path traversed by messages from the source s to a multicast receiver, mi, is given by P (s, mi). Thus multicast routing tree can be defined as T (s, M ) = 

mi∈MP (s, mi) and the messages are sent from s to M through T (s, M ). The multicast end-to-end delay constraint, ∆, represents an upper bound on the acceptable end-to-end delay along any path from source node to a destination node. The multicast delay variation, δ, is the maximum differ- ence between the end-to-end delays along the paths from the source to any two destination nodes.

δ = max{ |φD(P (s, mi))− φD(P (s, mj))|, ^∀mi, mj∈ M, i = j } The issue defined and discussed in [3], initially, is to minimize multicast delay variation under multicast end-to-end delay constraint. The authors referred to this problem as Delay- and delay Variation-Bounded Multicast Tree (DVBMT) problem. The DVBMT problem is to find the tree that satisfies

min{ δα| ^∀P (s, mi)∈ (2^s→mi, ∆), ^∀P (s, mi)⊆ Tα, ^∀mi∈ M ^∀α∈ Λ }, where Tαdenotes any multicast tree spanning M∪ {s}.

3 Proposed Algorithm

The algorithm gives an explicit solution about the DVBMT problem. We define the M ODE function, since the location of the core node influences the multicast delay variation. In addition to the M ODE function, we measure the delay varia- tion for each mode, using the CM P function. The M ODE and CM P functions are deployed as follows.

M ODE(c) =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

I if c = s

II if ^∃m in P (s, c), where^∀m∈ M III if ^∃s in P (m, c), where^∀m∈ M IV if II and III

V otherwise

,

where s is source node.

CM P (x) =

|(dscore+ max delay)− dsmk| if x = II or III or IV

dvcore otherwise ,

where core = M ODE⁻¹(x), max delay = max{min{φD(P (core, m))} |

∀m∈ M\{m^∗∈ M|^∃m^∗ in P (s, core) _∃

s in P (m^∗, core)} }.

As above functions M ODE and CM P , M ODE I is that the core node corresponds to the source node. In this case, CM P keeps dvcore. Otherwise, M ODEs II∼V are executed.

In the case of M ODE II [1], Fig. 1 (a) shows that there exists the destination node from the source node to the core node. CM P which can be computed stores the sum of the delay from the core node to the destination node and max delay.

The third case just satisfies M ODE III. In this case, the proposed algorithm initially adds the delay from the source node to the core node to max delay and subtracts the delay from the source node to the adjacent destination node, and stores the value with CM P . If the value is negative, CM P accepts the absolute value of it. The fourth case, MODE which satisfies II and III should find the factor to determine CM P , having situation that the destination nodes exist around the source node. The measure of the comparison is determined with the minimum delay from the source to the associated destinations. We can decide this situation because the shortest delay value from the source node affects the delay variation of the created tree. Therefore, the proposed algorithm selects either M ODE II or III. In M ODE V, if the proposed algorithm doesn’t satisfy M ODEs I∼IV, CMP stores dvcore. With respect to the time complexity, the proposed algorithm follows DDVCA so that it equals to O(mn²).

Fig. 2 shows the given network topology, and the link delays are presented on each link. It is supposed that the multicast end-to-end delay constraint ∆

s

c d₁

d₃

Core Node Source Node

Destination Nodes :

d₂

dsmk

dsci

max_delay

(a) M ODE II

s

c d₁

d₃

Core Node Source Node

d₂

dsci

max_delay ds_mk

(b) M ODE III Fig. 1. Main idea of M ODE function

3 2 1 9

3

4 1

3

2

2 5 ^{Core Node}

1 9

V₂ V₁

V3

V₈ V6

S

V7

V₅

V₄

(a) δDDV CA= 6

3 2

1 9

3

4 1

3

2

2 5

Core Node

1 9

V₂ V1

V3

V8

V6

S

V7

V₅

V4

(b) δP roposed= 1

Fig. 2. DDVCA and proposed algorithm

Table 1. The method by which proposed algorithm selects a core node

s v1 v2 v3 v4 v5 v6 v7 v8

source s 0 2 8 7 7 4 3 5 3

v1 2 0 9 6 5 6 5 4 1

destination v3 7 6 9 0 1 3 4 3 5

v8 3 1 10 5 4 6 5 3 0

maxi 7 6 10 6 5 6 5 4 5

mini 2 0 9 0 1 3 4 3 0

dvi 5 6 1 6 4 3 1 1 5

equals 10. Fig. 2 (a) and (b) give the multicast tree using DDVCA and the proposed algorithm, respectively. From Table 1, we can know that the nodes which have the minimum multicast delay variation dv_min are v₂, v₆, and v₇. However, node v2is excluded, because the upper delay bound ∆ is 10. And then, DDVCA randomly chooses node v6as the core node, but the proposed algorithm chooses node v7 as the core node. The propsed algorithm selects the less value of CM P (M ODE(v7)) = 4− 3 = 1 than the value of CMP (MODE(v6)) =

|(3 + 5) − 2|. In Fig. 3, when dvv₆ and dvv₇ have the same value, the proposed algorithm take the wise selection node with v7. As a result, the multicast delay variation of DDVCA is 6, but the one of proposed algorithm is 1.

4 Performance Evaluation

We compare our proposed algorithm with the DDVCA in terms of multicast delay variation. We describe the generation of random network topologies for the evaluation and the simulation results based on the network topology generated [2]. We now describe some numerical results, comparing the performance of the

3 3 3 2

2 Core Node 1

dv_v6= 1 = dv_v7 dvv6= 1 = dvv7 ds_v6

dsmk

max_delay S V1

V₆

V3 V8

V5

V₇

(a) M ODE III, CM P = 6

3

2

1 3 2

Core Node

1

max_v7= 4 min_v7= 3

S

V6

V₇

V₈ V1

V₃ V₄

(b) M ODE V, CM P = 1 Fig. 3. Core selection by CM P

proposed algorithm. Our algorithm is implemented in C⁺⁺. The 10 different network environments are generated for each size of given 100, 200, and 400 nodes. A source node is randomly selected and destination nodes are picked uniformly from the set of nodes except the source in the network topology.

Moreover, the destination nodes in the multicast group occupy 10% to 60%

of the all nodes on the network, respectively. An upper bound ∆ is randomly valued. We simulate 1000 times (10×100 = 1000) for each environment and P_e=0.3. For the performance comparison, the proposed algorithm and DDVCA are implemented in the same condition. Fig. 4 shows the simulation results for the multicast delay variations. It is apparent that the multicast delay variation of the proposed algorithm outperforms that of DDVCA. Fig. 4 (d) shows inefficiency in case of the number of nodes. We define inefficiency in the following equation.

δ = δDDV CA− δP roposed

δDDV CA

We present it as percentage (i.e, δ× 100%). The enhancement is up to about 9%∼25% in terms of the multicast delay variation.

5 Conclusion

In this paper, we consider the transmission of a message that guarantees cer- tain bounds of the end-to-end delays as well as the multicast delay variations computer network. The time complexity of our algorithm is O(mn²), which is the same as that of DDVCA. Furthermore, our algorithm results in the more efficient multicast delay variation than DDVCA.

References

1. M. Kim, Y.-C. Bang, and H. Choo, “Efficient Algorithm for Reducing Delay Vari- ation on Bounded Multicast Trees,” Springer-Verlag Lecture Notes in Computer Science, vol. 3090, pp. 440-450, September 2004.

2. A.S. Rodionov and H. Choo, “On Generating Random Network Structures: Con- nected Graphs,” Springer-Verlag Lecture Notes in Computer Science, vol. 3090, pp.

483-491, September 2004.

3. G. N. Rouskas and I. Baldine, “Multicast routing with end-to-end delay and delay variation constraints,” IEEE J-SAC, vol. 15, no. 3, pp. 346-356, April 1997.

4. P.-R. Sheu and S.-T. Chen, “A fast and efficient heuristic algorithm for the delay- and delay variation bound multicast tree problem,” Information Networking, Proc.

ICOIN-15, pp. 611-618, January 2001.

(a) V = 100 (b)|V | = 200

(c)|V | = 400 (d) δ× 100

Fig. 4. The multicast delay variations and inefficiency of three different networks, Pe= 0.3

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