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## Multilevel Hypergraph Partitioning

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Presentation Transcript

Overview

- Introduction
- Multilevel Algorithm Description
- Multi-phase Algorithm Description
- Experimental Results
- Conclusions
- Summary

Introduction

- VLSI circuit design requires many steps from design to packaging. Partitioning seeks to find the minimal number of clusters of vertices inside of a design. This will allow a smaller amount of interconnections and cuts in a design, which will allow for a smaller area and/or fewer chips.

Previous Algorithms

- Iterative refinement partitioning algorithms
- An initial bisection is computed (often obtained randomly) and then the partition is refined by repeatedly moving vertices between the two parts to reduce the hyperedge-cut.
- Types (KLFM)
- Kernighan-Lin (KL)
- Fiduccia-Mattheyses (FM)

Disadvantages:Poor for Large Graphs

- Local information, not global
- It may be better to move a vertex with a small gain, because it will be more advantageous later
- Vertices with similar gain
- There is no insight on which vertex to move, and the choice is randomized
- Inexact gain computation
- Vertices across a hyperedge will not transfer gain value across the hyperedge

New Type: Multilevel

- In these algorithms, a sequence of successively smaller (coarser) graphs is constructed. A bisection of the smallest graph is computed. This bisection is now successively projected to the next level finer graph, and at each level an iterative refinement algorithm such a KLFM is used to further improve the bisection.

Why Does Multilevel Work?

- The refinement scheme becomes more powerful (small sets of KLFM)
- Movement of a single node across partition boundary in a coarse graph can lead to movement of a large number of related nodes in the original graph
- The refined partitioning projected to the next level serves as an excellent initial partitioning for the KL or FM refinement algorithms

Multilevel Hypergraph Partitioning

- Contributions
- Hypergraphs instead of graphs
- less information loss
- Development of new hypergraph coarsening and uncoarsening techniques
- New multiphase refinement schemes
- v- and V- cycles

Algorithm Overview

- Coarsening Phase
- Edge coarsening
- Hyperedge coarsening
- Modified hyperedge coarsening
- Initial Partitioning Phase
- Uncoarsening and Refinement Phase
- Single Refinement
- Multilevel Refinement

Purpose of Coarsening Phase

- To create a small hypergraph, such that a good bisection of the small hypergraph is not significantly worse than the bisection directly obtained for the original hypergraph
- Helps in successively reducing the sizes of the hyperedges; large hyperedges are contracted to hyperedges connecting just a few vertices.

Edge Coarsening (EC)

- Vertices are matched by edges of highest weight
- Decreases hyperedge weight by factor of 2

Hyperedge Coarsening (HEC)

- Vertices that belong to individual hyperedges are contracted together
- Preference is given to higher weight and smaller size
- Non grouped vertices are copied to next level

Modified Hyperedge Coarsening (MHEC)

- Same as HEC, except after contraction the remaining vertices are grouped together
- Provides the largest amount of data compaction

Initial Partitioning Phase

- Bisection of the coarsest hypergraph is computed, such that it has a small cut, and satisfies a user specified balance constraint
- Since this hypergraph has a very small number of vertices the time to find partitioning is relatively small
- Not useful to find an optimal set, because refinement phase will significantly alter hypergraph
- Random selection or region growing

Initial Partitioning Phase Details

- Different bisections of coarsest hypergraph will result in different quality selections
- Partition of a hypergraph with smallest cut does not always result in smallest cut in original
- Possible for a higher cut partition to lead to a better original hypergraph
- Select multiple initial partitions
- Will increase running time and data set but overall quality will be increased
- Limit partitions accepted at each level by a percentage

Multilevel Hypergraph Partitioning Example

Initial partitioning phase

Uncoarsening and Refinement Phase

- A partitioning of the coarser hypergraph is successively projected to the next level finer hypergraph, and a partitioning refinement algorithm is used to reduce the cut-set (and thus improve the quality of the partition) without violating the user specified balance constraints.
- Since the next level finer hypergraph has more degrees of freedom, such refinement algorithms tend to improve the quality

Refinement Techniques

- Modified Fidduccia-Mattheyses (FM)
- Hyperedge Refinement (HER)

Modified Fidduccia-Mattheyses (FM)

- Limit FM passes to 2
- Greatest reduction in cut produced in 1st or 2nd pass
- Early-Exit FM (FM-EE)
- Aborts FM before moving all vertices
- Only a small fraction of moved vertices lead to a reduction in cuts

Hyperedge Refinement (HER)

- Can move all vertices with respect to a hyperedge for hyperedges that straddle a bisection
- Lacks the ability to climb out of local minima
- Can be further refined by FM (HER-FM)
- HER forces movement for an entire set of vertices, whereas FM refinement allows single vertices to move across a boundary

Multi-Phase Refinement with Restricted Coarsening

- Multilevel is robust, but randomization is inherent especially in coarsening phase
- Given an initial partitioning of hypergraph, it can be potentially refined depending on how the coarsening was performed
- A partition can be further refined if it’s coarsed in a different manner

Restricted Coarsening

- Preserves initial partitioning
- Will only collapse vertices on either side of partition
- Do not want to drastically change partitions, just redefine for possible better solutions

Multi-phase Approaches

- V-cycle
- Taking the best solution obtained from the multilevel partitioning algorithm and improve it using multi-phase refinement repeatedly
- v-cycle
- Select the best partition at a point in the uncoarsening phase and further refine only this best partitioning
- Reduces the cost of refining multiple solutions
- vV-cycle
- Use v-cycle to partition the hypergraph followed by the V-cycles to further improve the partition quality

Experimental Results

- Coarsening Phase
- MHEC produces best quality results
- HEC is close
- A robust scheme would run both types and select the best cut

Experimental Results

- Refinement Schemes
- MHEC coupled with either FM or HER+FM performs very well
- Multi-phase Refinement Schemes
- EE-FM with vV-cycles is a very good choice when runtime is the major consideration

Conclusions

- The multilevel paradigm is very successful in producing high quality hypergraph partitioning in relatively small amount of time
- The coarsening phase is able to generate a sequence of hypergraphs that are good approximations of the original hypergraph.
- The initial partitioning algorithms is able to find a good partitioning by essentially exploiting global information of the original hypergraph
- The iterative refinement at each uncoarsening level is able to significantly improve the partitioning equality because it moves successively smaller subsets of vertices between the two partitions

Conclusions (continued)

- In the multilevel paradigm, a good coarsening scheme results in a coarse graph that provides a global view that permits computations of a good initial partitioning, and the iterative refinement performed during the uncoarsening phase provides a local view to further improve the quality of the partitioning

Conclusions (continued)

- Hypergraph-based coarsening cause much greater reduction of the exposed hyperedge-weight of the coarsest level hypergraph, and thus provides much better initial partitions that those obtained with edge-based coarsening
- The refinement in the hypergraph-based multilevel scheme directly minimized the size of hyperedge-cut rather than the edge-cut of the inaccurate graph approximation of the hypergraph

Summary

- Introduction
- Multilevel Algorithm Description
- Multi-phase Algorithm Description
- Experimental Results
- Conclusions
- Reference
- G. Karypis, R. Aggarwal, V. Kumar, and S. Shekhar, "Multilevel Hypergraph Partitioning: Application in VLSI Domain", Proceedings of the Design Automation Conference, pp 526-529, 1997

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