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# Minimizing Congestion in General Networks

FOCS, pp.43-52, (2002)

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Abstract

A principle task in parallel and distributed systems is to reduce the communication load in the interconnection network, as this is usually the major bottleneck for the performance of distributed applications. In this paper we introduce a framework for solving on-line problems that aim to minimize the congestion (i.e. the maximum load of ...More

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Introduction

- In large parallel and distributed systems, such as networks of workstations or the Internet, the bandwidth of the interconnection network usually is the major bottleneck for the performance of distributed applications.
- Speaking, the weight function w (X) counts for a subset X, the bandwidth of all edges that are adjacent to nodes in X and that do not connect nodes of the same cluster with respect to the sub-partition corresponding to Vt .
- Based on this weight function, the authors define for a level node vt of the decomposition tree, the bandwidth-ratio λvt as λvt :=

Highlights

- In large parallel and distributed systems, such as networks of workstations or the Internet, the bandwidth of the interconnection network usually is the major bottleneck for the performance of distributed applications
- This is due to the fact that it is often more expensive or more difficult to increase the bandwidth of the interconnection network than to increase processor speed and memory capacities at individual nodes
- The load has to be distributed evenly among all network resources. This corresponds to minimizing the congestion, i.e., the maximum taken over all network links of the amount of data transmitted by the link divided by the respective bandwidth
- In this paper we show the somehow surprising result that knowledge of the current load in the network is not needed in order to achieve a polylogarithmic competitive ratio, w.r.t. the congestion

Results

- Theorem 1 For any request sequence σ for an online problem on network G that can be processed with congestion C the straightforward simulation on TG yields congestion Ct ≤ C.
- In the first part the authors give a general, i.e., problem-independent, simulation result that relates the expected relative load of any edge of G to the congestion on the tree TG.
- In the second part the authors show for the data management and the online routing problem that this simulation can be adopted such that the relative load of any edge e in G does not deviate too much from its expectation.
- The simulation of such a tree edge induces load on e = (x, y) only if both endpoints x and y of e lie in the cluster Sut .
- This holds because the routing paths between the nodes simulating ut and vt do not leave the cluster Sut .
- Theorem 6 Given a graph G and an associated decomposition tree TG there exists an oblivious online routing algorithm that is O(log n · h · max{δ, λ})-competitive with respect to the congestion.
- Remark 7 By utilizing the results of Section 4 on the parameters of TG this theorem gives an O(log3 n)-competitive online routing algorithm for general networks.
- Theorem 8 There is an O(log3 n)-competitive online algorithm for the data management problem on general networks.
- A tree TG in which any cluster Svt fulfills both properties has bandwidth-ratio at most λ and weight-ratio at most δ.
- Lemma 10 Given a level cluster S ⊂ V that fulfills the bandwidth property, it is possible to partition S into subclusters Si with the following characteristics.

Conclusion

- The cluster V that contains all nodes in the network fulfills the bandwidth-property, because out(V ) = 0.
- Let for the running time of the algorithm Ru and Rv denote the set in the partition PR that contain the node u and v, respectively.
- Later the authors will show that if no subset violates this condition the cluster S fulfills the weight-property, i.e., Requirement 4 holds.

Reference

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