Multilevel Hypergraph Partitioning Daniel Salce Matthew Zobel

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.

Phases of Multilevel Graph 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

Multilevel Hypergraph Partitioning Example

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 1 st or 2 nd 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

Multi-phase Refinement showing v- and V-cycles

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 , 1997