This document summarizes techniques for mapping application topologies to interconnect network topologies. It discusses how improving data locality through topology mapping can reduce communication costs, execution time, and energy consumption. Several common mapping techniques are described, including linear programming formulations, greedy approaches, partitioning approaches, transformative approaches, and those based on graph similarity. The document notes that finding an optimal mapping is NP-complete and different techniques may work better depending on the topology.

1. A Survey on Topology Mapping
for Large Scale Interconnection
Networks
Soheila Abrishami, Peyman Faizian

2. Overview
• Background
• Definition
• Performance metrics
• Mapping techniques

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

4. 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

5. 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

6. 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

7. 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.

8. 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

9. Topology mapping
• The network is typically modeled by a weighted graph
𝐻 = 𝑉𝐻, 𝑊𝐻, 𝑅 𝐻
• 𝑉𝐻: represent the execution units
• 𝑊𝐻(𝑢, 𝑣): represent the weight of the edges between two vertices 𝑢 and 𝑣
• 𝑅 𝐻: represent the routing as a probability distribution
• The static application graph is often modeled as a weighted graph
𝐴 = 𝑉𝐴, 𝑤 𝐴
• 𝑉𝐴: represents the set of communicating processes
• 𝑤 𝐴 𝑢, 𝑣 : represents some metric for the communication between two
processes 𝑢, 𝑣

10. Topology mapping…
• The topology mapping considers mappings 𝜎 ∶ 𝑉𝐴 → 𝑉𝐻
• Each concrete mapping 𝜎 has two metrics:
• Dilation: is defined as either the maximum or the sum of the pairwise
distances of neighbors in 𝐴 mapped to 𝐻. (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)

11. Mapping Techniques
Finding 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

12. 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

13. 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

14. 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

15. 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
• …

16. 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

17. Graph Similarity Based Approaches

18. 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