Circuit Partitioning : KernighanLin and FiducciaMattheyses Algorithms

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Circuit Partitioning Kernighan-Lin and
Fiduccia-Mattheyses Algorithms

• EE652 class presentation by
• Bernadeta Srijanto
• instr. Dr. D. Bouldin

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Presentation Outline

• Introduction
• Defining Problem what
• Kernighan-Lin Algorithm example
• Fiduccia-Mattheyses Algorithm example
• Summary
• comparison between KL FM algorithms
• whats next ?

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Introduction

• Background
• Advance technology causes more logic used and
  increasing system size
• Partitioning is needed here because
• too large of a circuit subcircuits / ICs
• I/O pin limitation more pins more money
• IO t G r

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Defining Problem General

• Visualization transform circuit to graph
• vertices represent circuit elements
• edges represents interconnects

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• Defining problem from graph
• Given graph G(V,E), where each vertex v ÎV has a
  size of s(v) and each edge e Î E has a weight of
  w(e). We need to divide the set V into k subsets
  V1, V2, V3, , Vk that is, divides the graph
  G(V,E) into k subgraph
• Gi(Vi,Ei), i
  1,2,3, ,k

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• Partitioning Constraints
• placing an upper bound on the size of each
  subcircuit. Prefer to have roughly equal size.
• Vi V / k , where Vi size of set Vi
• V size of V
• thus,
• ni n / k , ni number of elements
  in Vi
• n number
  of elements in V

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• Partitioning Constraints
• minimizing external wiring wiring between
  subcircuits (see Figure 2.1)

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Defining Problem Specific

• TWPP (two way partitioning problem)
• G1(V1,E1) and G2(V2,E2)
• Lets call blocks of partitioning A and B
• Constraints
• balance if we have 2n elements, we want n
  elements for each partitioning
• minimum size of cutset (min. external wiring)

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Kernighan-Lin algorithm

• heuristic technique iterative
• idea in TWPP, optimizing partitioning from
  initial partition PA,B to PA,B by
  swapping a subset XÍA with YÍB such that
• X Y
• XAÇB
• YBÇA
• (see Figure 2.4)
• now, how to determine the X and the Y ?

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Kernighan-Lin algorithm

• Step1 Initialization
• Step2 Compute D-values for all vÎV
• Step3 Compute gains between 2 nodes in
  A and B
• Step4 Re-compute D-values for nodes connected
  to (ai, bi)
• Step5 Determine k

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KL Step1 Initialization

• Initial partition PA,B is done randomly, where
  A B n, and AÇBÆ
• use example in Fig. 2.2(b), initial partition
• A2,3,4
• B1,5,6

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KL Step2 compute D-values

• If Da is the benefit of moving cell a from
  block A to B so that cutset would decrease
• Da Ea - Ia
• Ea vÎBS Cav External cost of node a
  in block A
• the number of edges from aÎA
  and and end in B
• Ia vÎAS Cav Internal cost of node
  a in block A
• Update i 1, and queue Æ

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KL Step2 example Fig 2.2(b)

• D1 E1 - I1 1 - 0 1
• D2 E2 - I2 1 - 2 -1
• D3 E3 - I3 0 - 1 -1
• D4 E4 - I4 2 - 1 1
• D5 E5 - I5 1 - 1 0
• D6 E6 - I6 1 - 1 0
• i 1
• Q Æ

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KL Step3 compute gains between 2 nodes in A and
B

• gain gi Dai Dbi -2caibi, where aiÎA bi Î B
• find the largest gain
• add this pair to queue Q(ai , bi)
• update the blocks AA-ai
• BB-bi

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KL Step3 example fig 2.2(b)

• g21D2D1-2c21(-1)(1)-2(1)-2
• g25D2D5-2c25(-1)(0)-2(0)-1
• g26D2D6-2c26(-1)(0)-2(0)-1
• g31D3D1-2c31(-1)(1)-2(0)0
• g35D3D5-2c35(-1)(0)-2(0)-1
• g36D3D6-2c36(-1)(0)-2(0)-1
• g41D4D1-2c41(1)(1)-2(0)2
• g45D4D5-2c45(1)(0)-2(1)-1
• g46D4D6-2c46(1)(0)-2(1)-1
• largest gain 2 of (4,1)

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• KL Step3 example (contd)
• Q(4 , 1)
• update the blocks AA-42,3
• BB-15,6

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KL Step4 re-compute Ds for new updated block
A and B

• (if A and B are empty, go to step 5)
• re-compute Ds for vÎV connected to(ai, bi)
• update i 1 i
• go to step3

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KL Step4 example fig 2.2(b)

• re-compute Ds for v connected to(ai, bi)
• D2 -1 D5-2 D6 -2
• update i 2
• go to step3 g25g26g35g36-3 arbitrarily
  choose g36
• Q 3,6
• A2, B5 etc

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KL Step5 determine k which gives maximum of
total gains

• (if A and B are empty)
• find k so that Gi1to kSgi g1g2g3gk is
  max
• If Ggt0 swap Xa1,a2,,ak from A to B
• and Yb1,b2,,bk from B to A
• update new A and B as initial partition for the
  next loop
• go to step2

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KL Step5 determine k (contd)

• If Glt0 Stop
• (partitioning cannot be improved)

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KL Step5 example fig 2.2(b)

• as A and B are empty
• updated i 3
• g1 2 g1 g2 -1 g1 g2 g3 0
• we found Gmax g1
• thus k 1
• since Ggt0 swap Xa14 in A and Yb11
  in B
• new A 1,2,3 and B 4,5,6
• go to step2, with new A and B as the initial
  partition

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Fiduccia-Mattheyses algorithm

• heuristic technique iterative
• idea in TWPP, Partition of circuit consisting C
  cells into 2 blocks A and B by moving a cell
  (called base cell) at a time from one block of
  partition to the other in attempt to minimize the
  number of nets cut (instead of minimizing the
  number of edges cut)
• now, how to select the base cell, and what is the
  base sequence moves of base cells ?

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Fiduccia-Mattheyses algorithm

• Example of circuit for FM algorithm

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Fiduccia-Mattheyses algorithm

• Step1 Compute gains of all cells
• Step2 Select base cell
• Step3 Lock base cell
• Step4 Checking for more free cells
• Step5 Choose for best sequence of moves for
  k-base cells
• Step6 Make k moves permanent and free all cells

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FM Step1 Compute gains for all cells

• Gain computed gi FS(i) - TE(i)
• FS(i) number of nets connected to cell(i) and
  not connected to any other cell in From_block
  F(i) of cell (i)
• TE(i) number of nets that are connected to
  cell(i) and not crossing the cut
• F(i) From_block of cell i
• T(i) To_block of cell i

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FM Step1 example fig 2.8(a)

• If A(n) and B(n) are the numbers of cells of net
  n in block A and block B, from fig 2.8(a) we have
  
• A(j) 0, A(m) 3, A(q) 2, A(k) 1, A(p) 1
• B(j) 2, B(m) 0, B(q) 1, B(k) 1, B(p) 1

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FM Step1 example contd fig 2.8(a)

• Defining block A as From_block and block B as
  To_block,
• F(j) 0, F(m) 3, F(q) 2, F(k) 1, F(p) 1
• T(j) 2, T(m) 0, T(q) 1, T(k) 1, T(p) 1
• (vice versa if B is the From_block, and A is the
  To_block)

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FM Step1 example contd fig 2.8(a)

• Since only the critical nets affect the gain, we
  looking at F(n) 1 and T(n)0
• moving from block A to B, we look at F(k) 1,
  F(p) 1 and T(m) 0.
• For the cell number i
• i 1 net on cell 1 is m T(m) 0 g(1) 0 -
  1 -1
• i 2 net on cell 2 are m, q, k, p F(k) F(p)
  1 and T(m) 0 g(2) 2 - 1 -1
• i 3 net on cell 3 are m, q T(m) 0 g(1)
  0 - 1 -1

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FM Step2 Select base cells

• initial i 1
• base cell ci selected if
• has maximum gi
• satisfies balance criterion
• if no base cell found Exit

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FM Step3 Lock base cell

• lock cell ci
• update gain of cell affected critical nets

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FM Step4 Checking for more free cells

• If free cell ¹ Æ update i 1 i
• select next base cell ci
• If ci ¹ Æ go to step3

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FM Step5 Choose best sequence of moves

• (if no more free cell)
• find k so that Gi1to kSgi g1g2g3gk is
  max
• sequence of moves is then
• c1,c2,c3, , ck
• If Glt0 Stop

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FM Step6 Makes k moves permanent

• Move cells in sequence of c1, c2, c3, , ck
  permanently to To_block
• free all cells
• go back to step1

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Comparison between KL and FM algorithms
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Whats next ?

• Variation of KL-FM Algorithm
• for KL 1. Unequal sized blocks
• 2. Unequal sized elements

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The End