Multilevel Hypergraph Partitioning Daniel Salce Matthew Zobel Overview

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 • v. V-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 v. V-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 hyperedgecut 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