Abstract

This paper studies the multicast routing and admission control problem on unit-capacity tree and mesh topologies in the throughput model. The problem is a generalization of the edge-disjoint paths problem and is NP-hard both on trees and meshes. We study both the offline and the online version of the problem: In the offline setting, we give the first constant-factor approximation algorithm for trees, and an O((loglog n)²)-factor approximation algorithm for meshes. In the online setting, we give the first polylogarithmic competitive online algorithm for tree and mesh topologies. No polylogarithmic-competitive algorithm is possible on general network topologies (Lower bounds for on-line graph problems with application to on-line circuits and optical routing, in: Proceedings of the 28th ACM Symposium on Theory of Computing, 1996, pp. 531–540) and there exists a polylogarithmic lower bound on the competitive ratio of any online algorithm on tree topologies (Making commitments in the face of uncertainity: how to pick a winner almost every time, in: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 1996, pp. 519–530). We prove the same lower bound for meshes.

Article Outline

1. Introduction

1.1. Previous work on unit-capacity networks

1.2. Our offline results

1.3. Our online results

2. The offline algorithm for trees

2.1. Proof of the constant approximation ratio

2.2. The polynomial time implementation of the algorithm

3. The online algorithm for trees

3.1. The first stage of the algorithm

3.2. The second stage of the algorithm

4. The offline algorithm for a mesh

4.1. Long requests

4.1.1. Routing algorithm

4.1.2. Proof of correctness

4.1.3. Proof of the approximation ratio

4.2. Short requests

5. The online algorithm for meshes

5.1. The first stage of the algorithm

5.2. The second stage of the algorithm

5.2.1. Long requests

5.2.2. Short requests

5.2.3. The analysis

5.3. A lower bound for the online algorithm on meshes

Appendix A. The MC algorithm on tree networks

A.1. Definitions and algorithm

A.2. Proof of competitiveness

References