Partitioning for minimizing both Cut and Maximum Subdomain Degree

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Partitioning for minimizing both Cut and Maximum
Subdomain Degree
Navaratnasothie Selvakkumaran (Selva) and George
Karypis Department of Computer Science
Engineering University of Minnesota selva,karypis
_at_cs.umn.edu
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Outline

• Motivation
• Applications
• Solution Methodology
• Description of Algorithms
• Results

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Background

• Research in the last 8 years has resulted in the
  development of sophisticated multilevel circuit
  partitioning algorithms
• hMetis, MLPart, etc
• Low stable cuts.
• Moderate computational requirements.
• Can scale to large designs.
• However, small cuts (or related objectives) is
  only one of the properties that a partitioning
  solution must satisfy in order to be an effective
  enabling technology for the overall EDA pipeline!
• Precise modeling of real problems requires a
  partitioning problem formulation that is
  multi-objective in nature.

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For Example

• Existing partitioning algorithms produce
  solutions with highly unbalanced interconnect
  distribution.
• The number of nets incident on different
  partitions can vary significantly!

• Ideally, we will like to
• minimize the cut, and
• balance the incident degree of the various
  partitions.

2.78
Our work addresses this problem
The situation worsens as the number of partitions
increases!
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Some Definitions

• A hypergraph G(V,E) is a set of vertices V and a
  set of hyperedges E. w(v) and w(e) denote weights
  of vertex v and hyperedge e respectively.
• A k-way partitioning of G is a decomposition of V
  into k disjoint subsets V1, V2, , Vk whose sum
  of vertex-weights is roughly balanced.
• Each of these subsets is called a subdomain or a
  partition.
• The cut of a k-way partitioning of V is equal to
  the sum of the weights of the hyperedges that
  contain vertices from different subdomains.
• The subdomain degree of subdomain Vi is equal to
  the sum of weights of the hyperedges that contain
  at least one vertex in Vi and at least one vertex
  in V-Vi.
• The maximum subdomain degree is the maximum
  subdomain degree over all k subdomains.

Our goal is to compute a k-way partitioning that
minimizes both the cut and the maximum subdomain
degree.
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Applications

• Congestion Minimization
• Congestion due to demand for routing resources
  exceeding the supply of routing resources
• Supply is almost the same for bins of equal size
• Bin degree is a lower bound for routing demand
• When wire-length driven placements and minimal
  congestion placements were compared, maximum bin
  degree and standard deviation of bin degree were
  lower by 14 and 15 respectively.

Source Perimeter-Degree A priori metric for
directly measuring and homogenizing
interconnection complexity in multilevel
placement, SLIP 03
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Equal pin density but varying routing demand
Clusters contain 10 pins but varying degree.
Degree is an essential element to control
congestion in multilevel placement algorithms.
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Applications (contd.)

• Multi FPGA prototyping/emulation systems
• Minimal maximum subdomain degree reduces the need
  for pin multiplexing.
• In communication systems, can be used to control
  maximum communication as well as overall
  communication.
• Communication in parallel computers.

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Challenges

• This is inherently a multi-objective problem
• Minimize cut minimize maximum cut
• In many cases the objectives are at odds with
  each other.

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Challenges (contd.)

• The maximum subdomain degree depends on the
  overall structure of the k-way partitioning
• Until the final k-way partitioning is obtained,
  the maximum subdomain degree is unknown.
• The maximum subdomain degree cannot be
  effectively minimized in the context of the
  popular recursive bisection framework.
• A subdomain with maximum degree may not produce
  the maximum subdomain degree for the subdomains
  of the subsequent level.

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Solution Methodology

• We solve this partitioning problem by treating it
  as a two-objective optimization problem.
• There are two key parameters
• How to couple the different objectives?
• How to optimize the coupled objectives?
• Our work focused on investigating different
  coupling and optimization alternatives and
  studying their cost-quality trade-offs.

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Coupling Methods

• The objectives can be numerically combined
• Or they can be prioritized.
• If a move results in gain in a higher priority
  objective, that move will always be accepted. The
  moves that result in gain of lower priority
  objectives are only accepted if they do not
  result in degradation of any of the higher
  priority objectives.

a,ß user parameters
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Optimization Challenges Methods

• Challenge
• The min-max subdomain degree objective contains
  many local minima.
• Insight
• Minimizing the cut and maximum subdomain degree
  are well correlated.
• Developed a framework that initially computes a
  solution that minimizes only the cut and then
  modifies it to optimize the coupled objective
  function.
• Modification methods
• Direct approach
• Solely focuses on the coupled objective function
  and optimizes it directly within the context of a
  multi-phase refinement framework.
• This approach is similar to the k-way V-cycle
  scheme used by khMetis.
• Aggressive approach
• Performs large-scale perturbations of the initial
  partitioning.
• Constructs more subdomains than necessary and
  then combines them together in order to optimize
  the multi-objective formulation.

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Obtaining Initial Partitioning

• There are two approaches for k-way partitioning.
• Recursive Bisection (RB)
• Myopic view.
• Can counter by terminal-propagation
• Minimizing maximum subdomain degree of
  intermediate solutions might over-constrain the
  solution.
• Native multi-way partitioning (Sanchis)
• Maintains concurrent view of partitions.
• Suitable for complex objectives.
• In practice, RB algorithms produce superior
  results for cut objective.
• Therefore we use RB to obtain initial K-way
  partition.

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Direct Approach

• V-cycle based optimization
• Consists of two phases
• Restricted coarsening
• greedy refinement (multi-objective)

• Shortcomings
• Greedy, non-hill climbing nature.
• Intermediate solutions do not violate balancing
  constraints.
• Not capable of large-scale perturbations.

• Advantages
• Fast approach
• Leverages the existing algorithm.

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Aggressive Approach

• We want to move large boulders between the
  partitions to enable good perturbation.
• One option is to use clustering to create
  boulders
• But we chose over-partitioning to create them.
• Balance constraints will be satisfied all the
  time.
• Each partition will have equal number of
  boulders.
• Pair wise swap of boulders to improve the
  multi-objective cost.

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Aggressive Approach in detail

• We compute 2Lk subdomains when we need k
  subdomains.
• Apply Direct Approach refinement to further
  reduce the multi-objective cost.
• Collapse these 2Lk subdomains into macro nodes.
  (boulders)
• For efficient computation.
• Good quality clusters
• in terms of cut
• Satisfy balance constraints automatically.

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Aggressive Approach in detail

• Combine macro nodes to constitute desired number
  of partitions.
• Cut-Focused
• To bias the initial solution in favor of lower
  cut.
• Inherit the recursive bisection solution. (i.e.
  place the macro-nodes in the parent partitions.)
• Max-Degree Focused (greedy approach)
• To bias the initial solution in favor of lower
  maximum subdomain degree.
• For L1, (i.e. 2k-gtk) sort all possible pairings
  of macro-nodes in increasing order of their
  resulting maximum subdomain degree, and traverse
  the list in that order to identify the pairs of
  unmatched macro-nodes.
• For Lgt1, we repeatedly apply the above scheme L
  times.

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Aggressive Approach in detail

• Randomized pair-wise swapping of macro nodes, to
  improve the multi-objective cost.
• For L1, (i.e. 2k-gtk), pre-compute the costs of
  all possible pairings and store them in a matrix.
• For Lgt1, evaluate the cost on the fly.
• Macro-nodes are almost similar in size.
• There is no need to check area balance
  requirements.
• In theory k-way Kernighan-Lin type of refinement
  is possible, but we used pair-wise swap for
  computational efficiency.

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Experimental setup

• Used ISPD98 benchmarks
• We used RB results of hMetis with default options
  (without any V-cycle) as the baseline to compare
  our new algorithms.
• Both quality and runtime are given as summary of
  ratios.

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4-way Partitioning Results
On average max-deg is lower by 4.5, cut is lower
by 2 Slower by 43
On average max-deg is lower by 6.2, cut higher
by 2 Slower by 179
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64-way Partitioning Results
On average max-deg is lower by 15, cut lower by
3 Slower by 53
Ibm18, has improved from 178 to 39
On average max-deg is lower by 35, cut higher by
4 Slower by 206
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Lessons Learned

• Method of combining objectives.
• Both prioritizing objectives and numerically
  combining objectives are equally effective.
• Direct Vs Aggressive
• Direct approach provides more favorable cut
  results, but if one requires substantial
  reduction in max then he/she needs to use
  aggressive.
• Max-Degree Focused Vs Cut-Focused
• Expected results were obtained.
• But the differences are marginal (few percentage
  points)
• Further improvements possible by relaxing lower
  bound constraint.

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THANK YOU
Navaratnasothie Selvakkumaran (Selva) and George
Karypis Department of Computer Science
Engineering University of Minnesota selva,karypis
_at_cs.umn.edu
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Pareto Curve

• Traversing the curve for finding trade-off
  between objectives.
• Start from one solution that is good in terms of
  one objective and move towards solutions that are
  good for both.

Objective 1
Objective 2