Routing with Quality-of-Service Guarantees: Algorithm and Analysis

Jun Huang, Xiaohong Huang, Yan MaBeijing Univ. of Posts & Telecom.

AsiaFI 2011

Agenda

• Introduction• Problem Formulation & Notations• Related Work • Contributions• Main Algorithms and Analysis• Numerical result• Conclusion

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Introduction

• The problem of QoS routing is NP-hard• Design an efficient QoS routing algorithm is an

important open topic• Application of QoS routing

– Establishing label-switching paths in MPLS– Arranging service-delivering paths in IMS-enabled networks– Constructing wavelength-switching paths in fiber-optics

networks

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Problem Formulation

• MCP– Is there a path p from a to d such

that wK(p)<=WK?

• MCOP– Is there an optimal path p from a

to d such that wK(p)<=WK when K = 2?

• EMCOP– Is there an optimal path p from a

to d such that wK(p)<=WK when K > 2?

(W1, …, WK)

a

b

c

d

e

g

(w1, …, wK)

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Frequently Used Notations

• m number of links• n number of nodes • K number of QoS parameters• W1, …, WK K additive constraints

• w1, …, wK K QoS metrics on each link• p a path• popt an optimal path• epsilon approximation ratio

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Related Work

• MCOP– K=2– Ergun et al. [1] developed an improved “binary

searching” technique to approximate MCOP– The time complexity of Ergun’s method is

O(mn/epsilon) which is known as the best result– However, this algorithm is designed for acyclic

graph.[1] F. Ergun, R. Sinha, and L. Zhang, “An improved FPTAS for restricted shortest path,” Inf. Process. Lett., vol. 83, no. 5, pp. 287-291, Sept. 2002

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Related Work (cont)

• EMCOP– K>2– Xue et al. [2] proposed a FPTAS for EMCOP within time

O(m(n/epsilon)K-1)– However, such FPTAS do not guarantee any constraints to be

enforced.– Xue et al. [3] also proposed a FPTAS for EMCOP with time

complexity O(mnlog log log n + m(n/epsilon)K-1) which guarantees all constraints to be enforced.

[2] G. Xue, A. Sen, W. Zhang, J. Tang and K. Thulasiraman, “Finding a path subject to many additive QoS constraints,” IEEE/ACM Trans. Netw., vol. 15, no. 1, pp. 201-211, Feb. 2007.[3] G. Xue, W. Zhang, J. Tang and K. Thulasiraman, “Polynomial time approximation algorithms for multi-constrained QoS routing,” IEEE/ACM Trans. Netw., vol. 16, no. 3, pp. 656-669, Jun. 2008.

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Contributions

• A graph-extending dynamic programming process in our proposed FPTAS

• Extension for our proposed FPTAS to solve the problem of EMCOP

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Main Algorithms and Analysis

• MCOP

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Main Algorithms and Analysis○●○○○○○

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Main Algorithms and Analysis

• Theorem 1 The worst-case time complexity of proposed FPTAS is

• Theorem 2 FPTAS finds a (1+)-approximation for MCOP if

○○●○○○○

Moreover, both of the constraints are enforced.

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Main Algorithms and Analysis

• Proposed FTPAS– (1 + )-approximation with the same time complexity– Designed for a general undirected graph– asymptotically approximate both the cost and delay

• Ergun’s method– Designed for a specific acyclic graph– minimizes the cost under the delay constraint

• Conclusion– The proposed FPTAS outperforms Ergun’s method

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Main Algorithms and Analysis

• EMCOP

○○○○●○○

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Main Algorithms and Analysis

• Theorem 3 The worst-case time complexity of proposed EFPTAS is

• Theorem 4 EFPTAS finds a (1+)-approximation for EMCOP if

○○○○○●○

Moreover, all of the constraints are enforced.

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Main Algorithms and Analysis

• EFPTAS– Find a (1 + )-approximation for EMCOP– Runs much faster than Xue’s algorithm [3]– Find a (1 + )-approximation with the same complexity with

Xue’s algorithm [2]– The constraints of finding path to be enforced

• Conclusion– Together with the implications of Theorem 1 and

Theorem 2, we confirm that our proposed algorithm outperforms the previous best-known algorithms.

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Numerical Result

• NSFNet

1

2

3

4

5

6

8

7

9

10

11

12

13

14

(149.44, 6.65)

(88.

41, 6

.95)

(116

.72,

4.8

0)

(183.98, 6.05)

(188.24, 8.05)

(89.97, 3.33) (71.26, 3.74)

(225.25, 5.02)

(268.02, 6.33)

(74.59, 4.04)

(161.11, 4.15)

(67.8

8, 5.0

3)

(74.4

8, 5.3

6)

(100.18, 5.94)

(193.77, 5.84)

(225.8

2, 9.3

7)

(203.54, 5.97)

(106

.33,

4.5

1)(119.69, 8.57)

(115.41, 4.11)(163.03, 8.14)

●○○○○○

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Numerical Result

• Performance Metric– Average Running Time (ART) = Total running time

for each routing request / Number of runs– Average Returned Weight (ARW) = Total returned

weight for each routing request / Number of runs– ARTRQ = Total ART for all routing requests /

Number of routing requests– ARWRQ = Total ARW for all routing requests /

Number of routing requests

○●○○○○

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Numerical Result

• ART

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Numerical Result

• ARW

○○○●○○

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Numerical Result

• Random networks (ARTRQ)

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Numerical Result

• Random networks (ARWRQ)

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Conclusion

• This work addressed QoS routing related problems and proposed a Fully Polynomial Time Approximation Scheme (FPTAS) and an extended version for QoS routing.

• The theoretical analyses show that the proposed algorithms outperform the previous best-known studies. And the numerical results further confirm that FPTAS and its extended version are effective and efficient for QoS guarantees over different networks.

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Q&A

Thank you!