A Survey on Topology Mapping for Large Scale

Interconnection NetworksSoheila Abrishami, Peyman Faizian

Overview• Background• Definition• Performance metrics• Mapping techniques

Background• Research in interconnect design can be classified as:• Communication infrastructure (topology)• Communication paradigm (routing, switching)• Evaluation framework (throughput, latency)• Topology/application mapping

Data Locality• The challenge deals with scalability and can be expressed in several

ways:• How to use the maximum of the available resources at their full potential?• How to do so with an energy consumption that remains acceptable?

• One global and practical answer to these questions is to improve the “Data Locality” of parallel applications

Data Locality…• Data locality: the way the data are placed, accessed and moved by the

multiple hardware processing units of the underlying target architecture• Improving data locality can cause:• Reduced Communication cost• Decrease in application’s execution time• Decrease in energy consumption

Mapping• One way to improve data locality is to dedicate physical processing

units to their specific software processing entities• This means that matching between the application virtual topology

and the target hardware architecture has to be determined

Mapping…• Virtual topology: Expresses the existing dependencies between

software processing entities• Static: number of processing entities and the dependencies between these

entities do not change• Dynamic: when one of the two above conditions (possibly both) is not

fulfilled.

• In addition, the maximum of details regarding the target hardware have to be gathered.

Mapping…• The matching between the virtual and the physical topologies is

achievable in both ways• the virtual topology can be mapped on to the physical one• the physical topology can also be mapped onto the virtual one

Topology mapping• The network is typically modeled by a weighted graph • : represent the execution units• : represent the weight of the edges between two vertices and • : represent the routing as a probability distribution

• The static application graph is often modeled as a weighted graph • : represents the set of communicating processes• represents some metric for the communication between two processes

Topology mapping…• The topology mapping considers mappings • Each concrete mapping has two metrics:• Dilation: is defined as either the maximum or the sum of the pairwise

distances of neighbors in mapped to (correlate with the dynamic energy consumption of the network)• Congestion: counts how many communication pairs use a certain link.

(correlates strongly with the execution time of bulk-synchronous parallel applications)

Mapping TechniquesFinding a perfect mapping (wrt to a specific metric) is NP-Complete.

• LP based algorithms• Constructive approaches (greedy)• Partitioning approaches• Transformative approaches• Graph similarity based approaches

LP formulation• Given a topology G (links and bandwidths)• Given a virtual topology H (communications graph)• Find a mapping from H -> G such that:• Maximum throughput• Minimum latency• Minimum congestion• Minimum dilation

• Software solutions are available to solve the linear program

Greedy Approaches• Select two starting vertices u, v from G and H respectively• Local

• Add next vertices from the neighborhood of initial vertices• Global

• Add next vertices based on a global property (i.e., node degrees)

• Continue until finding a full mapping

• The end result relies heavily on the choice of first vertices• Compute mappings based on different initial choices and select the best one• Define some kind of primary conditions to choose the initial vertices

Partitioning Approaches• Based on k-way graph partitioning (i.e., 2-way partitioning)• H and G graphs are recursively cut into k parts based on a property

(i.e., minimum weighted edge-cut)• The resulting graphs are mapped together using the same approach

• Several heuristics are available to perform the partitioning task which is NP-Complete

Transformative approaches• Start with an initial mapping• Iteratively transform it to better ones• Typically evolutionary techniques are used

• Genetic Algorithms• Ant Colony Optimization• …

• Fitness measure• Delay• Power consumption• …

Graph Similarity Based Approaches• Graph adjacency matrix can be modeled as a sparse matrix• Mapping problem would be transformed to bringing two matrices

into a similar shape

• One possible approach: • Reduce the bandwidth of two sparse matrices• Transform them to diagonal matrices• Do the mapping

Graph Similarity Based Approaches

Final Comments• None of the techniques give optimal results in all cases• Topology specific mapping approaches seem to work better• Some papers propose using a combination of above approaches to

achieve better results• Data locality is not always desirable (i.e., dragonfly)• So far a few papers have explored parallelized mapping techniques• Better outcomes if we consider the routing algorithms while mapping