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Flattened Butterfly : A Cost-Efficient Topology for High-Radix Networks

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### Flattened Butterfly : A Cost-Efficient Topology for High-Radix Networks

John Kim, William J. Dally & Dennis Abts

Presented by

Ajithkumar Thamarakuzhi

Outline

- Introduction
- Flattened Butterfly Topology
- Routing algorithms and performance comparison
- Topology cost comparison
- conclusion

Introduction

- Interconnection networks are widely used to connect processors and memories in multiprocessors , as switching fabrics for high-end routers and switches , and for connecting I/O devices.
- The performance of the interconnection network plays a central role in determining the overall performance of the system
- Low-radix networks, such as k-ary n-cubes, are unable to take full advantage of the increased router bandwidth
- With modern technology, high-radix networks based on a folded-Clos topology provide lower latency and lower cost than a network built from conventional low-radix routers

Flattened Butterfly Topology

- The butterfly network can take advantage of high-radix routers to reduce latency and network cost. However, there is no path diversity in a butterfly network which results in poor throughput for adversarial traffic patterns.
- Flattened butterfly is a topology which provides better path diversity than a conventional butterfly
- The flattened butterfly can scale more effectively than a hypercube network and also exploit high radix routers.

Flattened Butterfly Topology

- A Clos network provides many paths between each pair of nodes.
- This path diversity enables the Clos to route arbitrary traffic patterns with no loss of throughput.
- A Clos or folded Clos network has a cost that is nearly double that of a butterfly with equal capacity and has greater latency than a butterfly.
- Flattened butterfly has approximately half the cost of a comparable performance Clos network on balanced traffic.
- Flattened butterfly is routed similar to a folded-Clos network

Butterfly to Flattened Butterfly

4-ary 2-fly butterfly

Butterfly to Flattened Butterfly

2-ary 4-fly butterfly

Butterfly to Flattened Butterfly

- Flattened butterfly can be constructed by combining or flattening the routers in each row of the conventional butterfly network a into a single router.
- As a row of routers is combined, channels entirely local to the row are eliminated.
- If N is the total number of nodes in k-ary n-flat flattened butterfly, then
- number of routers = N/k
- Radix k’ =n(k-1)+1
- The routers are connected by channels in n’ = n − 1 dimensions

Butterfly to Flattened Butterfly

- In each dimension d, from 1 to n’, router i is connected to each router j given by

for m from 0 to k-1, where the connection from I to itself is omitted.

- In Figure , R4’ is connected toR5’ in dimension 1, R6’ in dimension 2, and R0’ in dimension3

Network size scalability as the radix and dimension is varied

- The figure shows that this topology is suited only for high-radix routers
- Networks of very limited size can be built using low-radix routers (k0 < 16) and even with k0 = 32 many dimensions are needed to scale to large network sizes. However with k0 = 61, a network with just three dimensions scales to 64K nodes.

Routing and Path Diversity

- label each node with a n-digit radix-k node address
- In Figure there are two minimal routes between node 0 (00002) and node 10 (10102).
- In general, if two nodes a and b have addresses that differ in j digits, then there are j! minimal routes between a and b.
- This path diversity derives from the fact that a packet routing in a flattened butterfly is able to traverse the dimensions in any order.
- Routing non-minimally in a flattened butterfly provides additional path diversity and can achieve load-balanced routing for arbitrary traffic patterns.

Flattened Butterfly Vs Generalized Hypercube

- The flattened butterfly connects k terminals to each router while the GHC connects only a single terminal to each router
- Adding this k-way concentration gives the flattened butterfly the following advantage compared to GHC

A) Reduced cost by a factor of k

B) Improved scalability

C) more suitable for high-radix routers

- Use of non minimal globally-adaptive routing gives Flattened butterfly more load-balancing in adversarial traffic patterns compared to GHC

Routing algorithm comparisons on flattened butterfly

uniform random traffic worst case traffic pattern

VAL = Valiant’s non-minimal oblivious algorithm

MIN = minimal adaptive , UGAL = non-minimal adaptive algorithm

UGAL-S = UGAL using sequential allocation

CLOS AD = non-minimal adaptive routing in a flattened Clos

Routing algorithm comparisons on flattened butterfly

- Valiant’s algorithm operates by picking a random intermediate node b, routing minimally from s to b, and then routing minimally from b to d. So VAL achieves only half of network capacity regardless of the traffic pattern
- In an adversarial traffic pattern all of the nodes connected to a router will attempt to use the same inter-router channel. So MIN is limited to approximately 3% throughput
- In both the traffic conditions CLOS AD performs well so this algorithm is suitable for flattened butterfly topology

Comparison to Other Topologies

- To compare the performance, a network of node size 1024 is taken and is constructed using the following topology by maintaining a constant bisection bandwidth.

Topology comparisons

uniform random traffic worst case traffic

Topology comparisons

- By holding bisection bandwidth constant across the topologies, the folded Clos uses 1/2 of the bandwidth for load-balancing to the middle stages – thus, only achieves 50% throughput in uniform random traffic.
- On WC traffic the conventional butterfly throughput is severely limited due to the lack of path diversity.
- The folded Clos has slightly higher latency because of the extra middle stage and the hypercube also has much higher latency because of its higher diameter
- Flattened butterfly provides 2x increase in performance over the folded-Clos on benign traffic while providing the same performance on the worst-case traffic pattern when the cost is held constant

Topology Cost Comparison

- Network cost is determined by the cost of the routers, backplane and cable links

Technology and packaging assumptions used in the topology comparison is shown below

Topology Cost Comparison

- Since flattened butterfly have lesser number of links, it gives a 35-53% reduction in cost compared to the folded-Clos.
- for example, with N = 1K network, the folded Clos requires 2048 links while the flattened butterfly requires 31 x 32 = 992 links, not 1024 links.
- The conventional butterfly is a lower cost network with radix-64 routers, because it can scale to 4K nodes with only 2 stages. At the same time flattened butterfly shares the radix of its router across stages (dimensions), and so it has more stages for the same number of nodes
- However, when N > 4K, the cost of the flattened butterfly becomes very comparable to the conventional butterfly.

Conclusion

- This paper introduces the flattened butterfly topology that exploits recent developments in high-radix routers and global adaptive routing to give a cost-effective network
- The flattened butterfly gives lower hop count than a folded Clos and better path diversity than a conventional butterfly
- On adversarial traffic, the flattened butterfly exploits global adaptive routing to match the performance of the folded Clos, at the same time cost of the flattened network is approximately half of the close network.

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