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 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

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 526-529, 1997