Multilevel Hypergraph Partitioning - PowerPoint PPT Presentation

Multilevel Hypergraph Partitioning

Daniel Salce

Matthew Zobel

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

• 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.

• 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)

• 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

• 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.

• 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

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

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

• 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.

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

• 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

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

• 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

• 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

Initial partitioning 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

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

• 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

• 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

• 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

• 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

• 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

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

• 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

• 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

• 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

• 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

• 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