F. Thomson Leighton

Efficient embeddings of trees in hypercubes

THE BOOLEAN HYPERCUBE IS A PARTICULARLY VERSATILE NETWORK FOR PARALLEL COMPUTING. IT IS WELL KNOWN THAT MULTI-DIMENSIONAL GRID MACHINES CAN BE SIMULATED ON A HYPERCUBE WITH NO COMMUNICATIONS OVERHEAD. IN THIS PAPER WE SHOW THAT EVERY BOUNDED-DEGREE TREE CAN BE SIMULATED ON THE HYPERCUBE WITH CONSTANT COMMUNICATIONS OVERHEAD. OUR PROOF IN FACT SHOWS THAT EVERY BOUNDED-DEGREE GRAPH WITH AN 0(1)-SEPARATOR CAN BE EMBEDDED IN A HYPERCUBE OF THE SAME SIZE WITH DILATION AND CONGESTION BOTH 0(1). WE PROVE ALSO THAT NOT ALL BOUNDED-DEGREE GRAPHS CAN BE EFFICIENTLY EMBEDDED WITHIN THE HYPERCUBE.

On the Fault Tolerance of Some Popular Bounded-Degree Networks

In this paper, we analyze the fault tolerance of several bounded-degree networks that are commonly used for parallel computation. Among other things, we show that an N-node butterfly network containing

On the fault tolerance of some popular bounded-degree networks

N^{1-\epsilon}

Optimal emulations by butterfly-like networks

worst-case faults (for any constant

On the max-flow min-cut ratio for directed multicommodity flows

\epsilon > 0

The challenges of delivering content on the Internet

) can emulate a fault-free butterfly of the same size with only constant slowdown. The same result is proved for the shuffle-exchange network. Hence, these networks become the first connected bounded-degree networks known to be able to sustain more than a constant number of worst-case faults without suffering more than a constant-factor slowdown in performance. We also show that an N-node butterfly whose nodes fail with some constant probability p can emulate a fault-free network of the same type and size with a slowdown of 2O(log* N). These emulation schemes combine the technique of redundant computation with new algorithms for routing packets around faults in hypercubic networks. We also present techniques for tolerating faults that do not rely on redundant computation. These techniques tolerate fewer faults but are more widely applicable because they can be used with other networks such as binary trees and meshes of trees.

Global hosting system

The authors analyze the fault-tolerance properties of several bounded-degree networks that are commonly used for parallel computation. Among other things, they show that an N-node butterfly containing N/sup 1- epsilon / worst-case faults (for any constant epsilon >0) can emulate a fault-free butterfly of the same size with only constant slowdown. Similar results are proved for the shuffle-exchange graph. Hence, these networks become the first connected bounded-degree networks known to be able to sustain more than a constant number of worst-case faults without suffering more than a constant-factor slowdown in performance. They also show that an N-node butterfly whose nodes fail with some constant probability p can emulate a fault-free version of itself with a slowdown of 2/sup O(log* N)/, which is a very slowly increasing function of N. The proofs of these results combine the technique of redundant computation with new algorithms for routing packets around faults in hypercubic networks. Techniques for reconfiguring hypercubic networks around faults that do not rely on redundant computation are also presented. These techniques tolerate fewer faults but are more widely applicable since they can be used with other networks such as binary trees and meshes of trees.<<ETX>>

Content delivery network using edge-of-network servers for providing content delivery to a set of participating content providers

The power of butterfly-like networks as multicomputer interconnection networks is studied, by considering how efficiently the butterfly can emulate other networks. Emulations are studied formally via graph embeddings, so the topic here becomes : How efficiently can one embed the graph underlying a given interconnection network in the graph underlying the butterfly network ? Within this framework, the slowdown incurred by an emulation is measured by the sum of the dilation and the congestion of the corresponding embedding (respectively, the maximum amount that the embedding stretches an edge of the guest graph, and the maximum traffic across any edge of the host graph) ; the efficiency of resource utilization in an emulation is measured by the expansion of the corresponding embedding (the ratio of the sizes of the host to guest graph). Three main results expose a number of optimal emulations by butterfly networks. Call a family of graphs balanced if complete binary trees can be embedded in the family with simultaneous dilation, congestion, and expansion 0(1). (1) The family of butterfly graphs is balanced. (2) (a) Any graph < from a family of maxdegree-d graphs having a recursive separator of size S(x) can be embedded in any balanced graph family with simultaneous dilation O(log(d Σ i S(2 -i |G|))) and expansion O(1). (b) Any dilation-D embedding of a maxdegree-d graph in a butterfly graph can be converted to an embedding having simultaneous dilation O(D) and congestion O(dD). (3) Any embedding of a planar graph G in a butterfly graph must have dilation Ω(log Σ (G)/Φ(G), where : Σ(G) is the size of the smallest (1/3, 2/3)-node-separator of G, and Φ(G) is the size of Gs largest interior face. Applications of these results include : (1) The n-node X-tree network can be emulated by the butterfly network with slowdown O(log log n) and expansion 0(1) ; no embedding has dilation smaller than Ω(log log n), independent of expansion. (2) Every embedding of the n x n mesh in the butterfly graph has dilation Ω(log n) ; any expansion-O(1) embedding in the butterfly graph achieves dilation O(log n). These applications provide the first examples of networks that can be embedded more efficiently in hypercubes than in butterflies. We also show that analogues of these results hold for networks that are structurally related to the butterfly network. The upper bounds hold for the hypercube and the de Bruijn networks, possibly with altered constants. The lower bounds hold-at least in weakened form-for the de Bruijn network.

HTML delivery from edge-of-network servers in a content delivery network (CDN)

We present a pure combinatorial problem whose solution determines max-flow min-cut ratio for directed multicommodity flows. In addition, this combinatorial problem has applications in improving the approximation factor of the greedy algorithm for the maximum edge disjoint path problem. More precisely, our upper bound improves the approximation factor for this problem to O(n3/4).

Global load balancing across mirrored data centers

In this talk, we will give an overview of how content is distributed on the internet, with an emphasis on the approach being used by Akamai. We will describe some of the technical challenges involved in operating a network of thousands of content servers across multiple geographies on behalf of thousands of customers. The talk will be introductory in nature and should be accessible to a broad audience.