Kernighan and Lin partition algorithm Fideccia and Mattheyses partition algorithm

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Kernighan and Lin partition algorithmFideccia
and Mattheyses partition algorithm

• Presented by Bhanu Rekapalli.

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Papers

• An Efficient Heuristic procedure for Partitioning
  Graphs.
• By B.W.KERNIGHAN and S.LIN
• ( Bell Systems Technical Journal, 49(2)
  291-307, 1970.)
• A Linear-Time Heuristic for Improving Network
  Partitions.
• By C.M.FIDUCCIA and R.M. MATTHEYSES
• (Proceedings of the Design Automation
  Conference, pp
• 175-181, 1982.)

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TERMS TO UNDERSTAND

• Network-circuit modules
• Terminals-connectors or pins
• Graph
• Vertices-nodes or points
• Edges-net or signal
• Cutset

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Algorithms

• Minimize the costs.
• Effective in finding the optimal partitions.
• Fast enough to be practical in solving large
  problems.

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Definition of the problem

• Connection between the FPGAs or modules
  (external) have high cost compared to connections
  within them(internal).
• The object is to minimize the number of external
  connections.

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Mathematically

• G is a graph of n nodes of weights wi gt0,i1.n
• Let p be a positive number,such that 0lt wi ltp
  for all i.
• Partition the nodes of G into k subsets
• Weighted Connectivity matrix C(cij),i,j1,.n
  describing the edges of G.
• A k-way partition of G is a set of nonempty
  pairwise disjoint subsets of G,v1,..,vk such
  that Uk i1 viG.
• A partition is admissible if mod vi ltp for all
  i.
• The partitioning problem, is to find the
  minimal-cost admissible partition of G.

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Exact Solutions

• Graph G has n nodes of size 1 to be partitioned
  into k subsets of size p.
• kpn.
• ways to to chose the first subset
• ways to chose the second subset
• The number of cases is 1/k!
• This expressions yields a very large number
• Examplen40, p10 (k4), it is gt1020

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False Starts

• Random Solutions
• Simply generate random solutions, keeping the
  best seen to date, and terminating after some
  predetermined time or value is reached.
• Fast,although it is an n2-procedure
• This approach is unsatisfactory for moderate
  size,since few optimal solutions, which appear
  randomly with very low probabilities.
• Egmatrix of 32X32 there are 3 to 5 optimal
  solutions out of partitions giving
  probability lt10-7

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Max Flow-Min Cut

• Theorem The maximum flow between any pair of
  nodes the minimum cut capacity of all cuts which
  separate the two nodes
• Cut is a 2-way partition and the cut capacity is
  the cost of the partition.
• The most severe difficulty of the algorithm is
  that the algorithm has no provision for
  constraining the sizes of the resultant subsets.
• notefor 2-way unconstrained partition,the
  minimal cost it produces is a lower bound for
  solutions produced by any method.

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Clustering

• Identifying the natural cluster in the given
  cost matrix.
• Again satisfying constraints on the sizes of the
  subsets is a problem
• Systematic assignment of the stragglers is not
  provided.

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Two-Way Uniform Partitions

• Consider the set S of 2n points,with an
  associated cost matrix C(cij), i,j1,.,2n.
• Assume C is symmetric and that cii 0 for all i.
• We want to partition S into two subsets A and B,
  each with n points, such that the external cost
  is minimized.
• Start with any arbitrary partition A,B of S and
  try to
• decrease the initial cost T by a series of
  interchanges of subsets of A and B
• When no further improvement is possible, the
  resulting partition A,B is a local minimum
  and has a fairly high probability of being a
  global minimum with this scheme
• This process is repeated with another arbitrary
  partition,to obtain many locally minimum
  partitions we desire.

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KL-algorithm

• Let A ,B be any arbitrary 2-way partition
• Subsets with XY
• such that interchanging X and Y produces A and
  B as shown below
• ?

X
Y
Y
X
B
A
B
A
AX-XY BB-YX
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KL-algorithm
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Phase 1 optimization Algorithm

• Compute the D values for all elements of S
• Choose ai and bi such that gain
  giDaiDbi-2caibi is maximum.
• Set ai and bi aside, and call them ai and bi.
• Recalculate the D values for the elements of

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• Choose a pair a2 and b2 from A-(a1) and
  B-(b1) such that the gain is maximum.
• Continue until all the nodes have been
  exhausted,identifying (a3,b3)..(an,bn) and
  the corresponding gains g3,..gn.
• If Xa1,.,ak, Yb1,bk, then the decrease in
  cost when the sets X and Y are interchanged is
  precisely g1g2 gk
• Choose k to maximize the partial sum
• If Ggt0,a reduction in cost of value G can be made
  by interchanging X and Y, and the resulting
  partition is treated as initial partition and the
  procedure is repeated.
• G0?locally optimum partition is reached called
  phase 1 optimal partition.

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Effectiveness of the Procedure

• The procedure sequentially finds an approximation
  to the best exchange of lambda pairs.
• This technique sacrifices a certain amount of
  power for a considerable gain in speed.
• The procedure finds the maximum partial sum, the
  process does not terminate immediately when gain
  is negative.
• The probability that it finds an optimal solution
  in a single trial p(n)2-n/30,eg30X30 matrix p
  is 0.5, 0.2 to 0.3 for 60X60 and .05 to .1 for
  120X120.

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Running time of the Procedure

• The upper bound is n2logn.
• The lower bound with fast scan method is
  proportional to
• This function grows as

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Plot of Running times
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Drawbacks

• It minimizes the number of edges cut not the
  number of nets cut.
• It does not allow logic cells to be of different
  sizes.
• It is expensive in computation time.
• It does not allow partition to be unequal.
• It does not allow for selected logic cells to be
  fixed in place.
• The results are random.
• It does not directly allow more than two
  partitions.

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Fiduccia and Mattheyses algorithm

• To deal with cells of various sizes , the
  algorithm progresses by moving one cell at a time
  between the blocks of the partition while
  maintaining a desired balance based on the size
  of the blocks rather than the number of cells per
  block.
• Two cells which share atleast one net are said to
  be neighbors.
• Each cell is assumed to have a size s(i).Number
  of pins p(i).
• Given a fraction 0ltrlt1 partition the network in
  to two blocks A and B such that A/(AB)r,
  and such that the size of the resulting cutset is
  minimized.

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The Basic Idea

• The main idea of the algorithm is to move a cell
  at a time from one block of the partition to the
  other in an attempt to minimize the cut set of
  the final partition.
• Base cellthe cell to be moved.
• Gain g(i)no. of nets by which the cutset would
  decrease if cell i were moved from partition A to
  partition B (may be negative).integer range
  p(i)ltg(i)ltp(i).
• Keep in mind the balance criterion to prevent all
  cells from migrating to one block of the
  partition.
• To prevent thrashing, once a cell is moved it is
  locked for an entire pass
• Only O(N) passes are possible since the cutset is
  bound by the number of nets

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steps

• Choose a base cell having the largest gain in its
  block.
• Move it.
• Adjust the gains of its free neighbors.
• The total work required to perform one pass is
  O(P).

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Cell Gain

• Each cell has its gain in the range pmax to
  pmax, pmaxmax(p(i)cell(i) is initially free).
• The total amount of work required to maintain
  each Bucket array is O(P)/Pass.

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Establishing Balance

• Move the Cell
• Find the first cell of highest gain that is not
  locked and such that moving it would not cause an
  imbalance
• Break tie by choosing the one that gives the best
  balance
• Choose this as the base cell. Remove it from the
  bucket list and place it on the LOCKED list.
  Update it to the other partition.

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Computing and Maintaining Cell Gains

• Critical net
• Given a partition (A,B), define the distribution
  of net n, relative to this partition, as an
  ordered pair of integers (A(n),B(n)) which
  represents the number of cells the net n has in
  blocks A and B respectively. These can be
  computed in O(P) time for all nets.
• Net is critical if there exists a cell on it such
  that if it were moved it would change the nets
  cut state (whether it is cut or not).
• Gain of cell depends only on its critical nets
• If a net is not critical, its cut state cannot be
  affected by the move
• A net which is not critical either before or
  after a move cannot influence the gains of its
  cells
• This observation, and previous locked base
  cell,leads to the basis of linear-time claim.

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Net is critical if either A(n) or B(n) is equal
to 0 or 1
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• Let F be the from partition of cell i and T the
  to partition i.e. its complimentary block, FA
  and TB or vice versa.
• The gain of the cell(i) is g(i)FS(i)-TS(i)
• FS(i) no. of nets which have cell i as their
  only F cell
• TE(i) no. of nets which contain i and have an
  empty T side
• Requires O(P) work to initialize
• net is critical before the move if F(n)1 or
  T(n)0 or T(n)1
• F(n) 0 does not occur because base cell on F
  side before
• net is critical after the move if T(n)1 or
  F(n)0 or F(n)1
• T(n) 0 does not occur because base cell on T
  side before

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Propositions

• 1)The total amount of work required to maintain
  each bucket array is O(P) per pass.
• 2)Initialization of all cell gains requires O(P)
  work.
• 3)No more than four update operations per net are
  performed during one pass of the algorithm.
• 4)The total work required to initialize and
  maintain cell gains is O(P) per pass.
• Theoremthe minimization algorithm requires O(P)
  time to complete one pass.

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The interactive, visual bi-partitioning
environment

• Developed and tested in Java
• Netlists are loaded in two formats-.net and
  .mon(my own netlist)
• The .mon format retains partition and clustering
  information not available in .net files
• The control panel provides several methods by
  which to view the netlist and the current state
  of the partitioning
• controls for aborting, suspending, and resume
  algorithm execution

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Control panel
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Cutsize and the Partition size over time
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The number of nets connected to each node. The
number of nodes connected to each net.
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References

• Lecture notes from eecs244 course of Prof.Kurt
  Keutzer from berkley.
• Interactive Visual Environment for Bipartitioning
  Algorithm Development . CS294-7 Project Report,by
  James S. Young
• ASIC by smith.