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Multilevel Partitioning
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History The circuit partitioning problem: studied since 1970’s.
The Kernighan-Lin (KL): introduced in 1970. Extended to Fiduccia-Mattheyses (FM) in 1982. Little progress: in a period of 15 years Significant progress: during the mid-to-late-90’s in less than five years the best reported cutsize on commonly used benchmarks was reduced by almost 50%. مرتضي صاحب الزماني
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hMetis Algorithm “Multilevel Hypergraph Partitioning,” Karypis, Aggarwal, Kumar, Shekhar, DAC 97, pp (Univ of Minnesota) “Multilevel Circuit Partitioning,” Alpert, Huang, Kahng, DAC 97, pp (UCLA) مرتضي صاحب الزماني
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Clustering Initial graph Possible clusterings a d a d b c,e b c e d
a,b,c Initial graph e Possible clusterings مرتضي صاحب الزماني
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Multilevel Partitioning Schemes
Produce high-quality partitionings. Outperformed everything else (for large designs)! Very fast Eg: 1M-node graph takes 35s. Easily parallelized Eg: 1M-node graph takes 0.8s on 64 processors. مرتضي صاحب الزماني
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Ingredients of Multilevel Partitioning
Coarsening Initial Partitioning Refinement Successive coarse graphs must make it easier to find a good partition. Uniform vertex weights. Exposed edge-weight must decrease rapidly. The `how to coarsen’ computation must be fast. مرتضي صاحب الزماني
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Ingredients of Multilevel Partitioning
Coarsening Initial Partitioning Refinement Easiest of the three phases. Everything reasonable works fine. e.g. FM. It requires very little time Operates on small graphs (~100 vertices). مرتضي صاحب الزماني
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Ingredients of Multilevel Partitioning
Coarsening Initial Partitioning Refinement Needs a local partitioning refinement algorithm. Any vertex-swapping algorithm can be used KL, FM, etc. If coarsening is done correctly, simple refinement algorithms work extremely well and this phase requires very little time. مرتضي صاحب الزماني
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Metis: Multilevel Graph Partitioning
Coarsening Maximal independent set of edges (matching). Preference to high weight edges: heavy-edge. Effective in reducing the exposed edge-weight! Initial Partitioning FM Refinement A simplified version of FM Only up to 4 passes, Early exit Very fast refinement. 2 1 مرتضي صاحب الزماني
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Coarsening مرتضي صاحب الزماني
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Coarsening مرتضي صاحب الزماني
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Number of Levels No. of Levels: logm(|V| / v0) |V|: No. of nodes
v0: No. of nodes in the final (clustered) graph nodes (can be processed efficiently by FM) m: cluster ratio Average number of nodes per cluster (e.g., 3) 1.3 recommended ( how many levels for 1M nodes?) مرتضي صاحب الزماني
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Going from Graphs to Hypergraphs
Hypergraph partitioning is significantly more complicated than graph partitioning. Hypergraph Coarsening Schemes: Edge-based Hyperedge-based مرتضي صاحب الزماني
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Hypergraph Coarsening Schemes
Edge-based coarsening schemes Pairs of connected vertices are collapsed together, using the heavy-edge heuristic. Easy and fast to compute. Does not dramatically decrease the exposed hyperedge weight. Cannot easily remove moderate-size hyperedges. Requires a lot of refinement in order to obtain good partitionings Requires sophisticated refinement schemes Can lead to good partitionings but very slow! مرتضي صاحب الزماني
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Hypergraph Coarsening Schemes
Hyperedge-based coarsening schemes Collapses together all the vertices of an entire hyperedge. Preference is given to the heavier hyperedges. Easy and fast to compute. It dramatically decreases the exposed hyperedge weight. Leads to very good initial partitionings. 3600 as opposed to 6200 for golem3! Requires very little refinement time. High-quality partitionings can be obtained with simple refinement schemes. مرتضي صاحب الزماني
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Coarsening مرتضي صاحب الزماني
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hMetis: Multilevel Hypergraph Partitioning Algorithm
Uses hyperedge-based coarsening. Uses a simplified version of FM for refinement Limits the number of passes, Early-exit hMetis is an extremely fast, robust, high-quality hypergraph partitioning algorithm. مرتضي صاحب الزماني
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Cut During One Pass مرتضي صاحب الزماني
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V-Cycle Multi-Level Partitioning V-Cycle مرتضي صاحب الزماني
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Restricted Coarsening
preserves the initial partitioning that is input to the algorithm. will collapse vertices together that belong only to one of the two partitions. Uncoarsening is the same as before. مرتضي صاحب الزماني
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Multi-Phase Refinement
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Bisection Quality مرتضي صاحب الزماني
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Bisection Runtime مرتضي صاحب الزماني
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Conclusions The multilevel paradigm with the right coarsening and refinement scheme works extremely well for hypergraphs. The quality of the partitionings can be further improved by running the algorithm multiple times. hMetis: available (binaries for many OS’s) in: URL: MLPart: available in source code مرتضي صاحب الزماني
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Multilevel k-way Hypergraph Partitioning
“Multilevel k-way Hypergraph Partitioning”, Karypis, Kumar, DAC 99. مرتضي صاحب الزماني
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Why k-way Partitioner? A k-way partitioning of a hypergraph can be easily computed via recursive bisection. However, developing an algorithm that directly computes a k-way partitioning has a number of advantages: Recursive bisection cannot directly optimize global objectives: SOED (Sum of External Degrees: external degree of e = the number of partitions that is spanned by e (K-1) -metric as SOED but –1. A direct k-way partitioning algorithm can better enforce tighter balance constraints. A direct k-way partitioning can potentially produce better partitionings. Recursive bisection is by its nature sub-optimal. A direct k-way partitioning can potentially be faster. مرتضي صاحب الزماني
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Multilevel k-way Partitioning
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Partitioning with Fixed Vertices
FM passively accommodates such fixed constraints. By not moving fixed vertices. Caldwell, Kahng and Markov show: Faster partitioning possible for fixed vertices. Open question: heuristics that actively exploit fixed vertices. “Hypergraph Partitioning with Fixed Vertices,” Alpert, Caldwell, Kahng, Markov, TCAD 2000. مرتضي صاحب الزماني
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Optimality Optimality, Scalability and Stability Study of Partitioning and Placement Algorithms Jason Cong, Michail Romesis, Min Xie ISPD 2003 مرتضي صاحب الزماني
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Optimality State-of-the-art, multilevel two way partitioning algorithms find solutions very close to the optimal solutions of BEKU benchmarks. Existing circuit partitioning techniques are fairly mature Multiway: 18% worse مرتضي صاحب الزماني
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FPGA-Based System Partitioning
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Multi-FPGA System Motivation: Large systems
FPIC FPIC FPIC FPIC FPIC: field-programmable interconnect chips مرتضي صاحب الزماني
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FPGA-Based System Mapping of a typical system architecture onto multiple FPGAs FPGA FPGA RAM Logic Logic مرتضي صاحب الزماني
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Challenges Multi-way system partitioning onto FPGAs: Key challenges:
Low utilization of FPGA gate capacity Reason: hard I/O pin limits Low clock speeds Reason: long interconnect delays between multiple FPGAs Long runtimes for system partitioning Run-time reconfigurability: Another dimension added (evolve through time) مرتضي صاحب الزماني
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Multi-FPGA Partitioning
Primary objective: No. of FPGAs Constraints: Area (FPGA size) IO (no. of FPGA pins) Different from single chip partitioning: Small change in balance and cut size infeasible partitioning مرتضي صاحب الزماني
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Multi-FPGA Partitioning
Secondary objective: After no. of partitions determined, minimize amount of communication improve speed Different from single chip: Not distinguish the number of FPGAs a net spans to (k) minimize k مرتضي صاحب الزماني
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Summary Circuit netlists can be represented by graphs
Partitioning a graph means assigning nodes to disjoint partitions The total size of each partition (number/area of nodes) is limited Objective: minimize the number connections between partitions
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Summary Basic partitioning algorithms Multilevel partitioning
Move-based, move are organized into passes KL swaps pairs of nodes from different partitions FM re-assigns one node at a time FM is faster, usually more successful Multilevel partitioning Clustering FM partitioning Refinement (also uses FM partitioning)
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