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Partitioning 2
• Outline
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Fiduccia-Mattheyses Algorithm
Approach
Algorithm
Example
Algorithm Properties
• Goal
– Understand Fidducia-Mattheyses
partitioning algorithm
Fiduccia-Mattheyses Algorithm
• Approach
– move one cell at a time between partitions A and B
» less restrictive than pair moves
» no longer need to maintain partition size balance
– use special data structures to minimize cell gain updates
– only move cells once per pass
• Complexity
– prove constant number of updates per cell per pass
– runtime per pass = O(P) - P pins/terminals
» vs. O(n2) for group migration
– small (3-5) number of passes to converge
– estimate total time is O(PlogP)
» more pins than nets
» usually more pins than cells
Definitions
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ni - number of cells on neti
si - size (area) of celli
smax - largest cell = max(si)
S - total size of cells = sum(si)
pi - number of pins on celli
pmax - most pins on a cell = max(pi)
P - total number of pins = sum(pi)
C - total number of cells
N - total number of nets
r - fraction of cell area in partition A
CELL - C-entry array, entry is linked list of nets on cell
NET - N-entry array, entry is linked list of cells on net
Cell Gain
• Cell Gain
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gi - reduction in cutset by moving celli
label each cell with its gain
-pi <= gi <= pi
-pmax <= gi <= pmax
+2
+1
0
-1
BUCKET
• Gain Data Structure
– BUCKET array of cell gains
» one per partition
– O(1) access to cells of a given gain
– MAXGAIN tracks cells of max gain
– remove cell once it has moved
» cells move once per pass
» gain does not matter after that
+pmax
MAX
GAIN
Cell #
Cell #
-pmax
CELL
1 2
C
Critical Nets
• Minimize cell gain updates
– if all cells are updated on each move - O(C2) algorithm
– only cells that share a net with a moved cell must be updated
» but big net implies many moves and many updates
– only cells on critical nets must be updated
• Critical nets
– moving a cell would change cutstate
» cutstate - whether net is cut or not
» critical only if net has 0 or 1 cells in A or B
A=0, B=3
critical
A=1, B=2
critical
A=2, B=2
not critical
Partition Balance
• Control size balance between partitions
– otherwise all cells move to one partition
– cutset = 0
• Balance criterion
– rS - smax <= |A| <= rS + smax
– permits some “wiggle room” for cells to move
no
yes
Algorithm
• Initially place cells randomly into A and B
• Compute cell gains
• Algorithm for each pass
– for all cells in A and B of maximum gain whose move would
not cause imbalance
» choose one with best balance result - the base cell
» if none qualify, quit pass
– move to opposite partition and lock (remove from BUCKET)
» unlocked cells are free cells
– update cell gains and MAXGAIN pointer
• Repeat passes until no moves occur
– unlock all cells at beginning of pass
Update Cell Gains
F = “from” partition of base cell
T = “to” partition of base cell
FT(n) = # free T cells of net n
FF(n) = # free F cells of net n
LT(n) = # locked T cells of net n
LF(n) = # locked F cells of net n
for each net n on base cell do
if LT(n) == 0
if FT(n) == 0 UpdateGains(NET(n))
else if FT(n) == 1 UpdateGains(NET(n))
FF(n)-LT(n)++
if LF(n) == 0
if FF(n) == 0 UpdateGains(NET(n))
else if FF(n) == 1 UpdateGains(NET(n))
Example
Initial partition
cutset = 3
si = 1, r = 0.5, 1 <= |A| <= 3
Final partition
cutset = 1
+3
+1
+1
-1
L
-3
L
+1
a
b
c
d
b
a
c
d
- b, c are highest-gain candidates
- choose b, move and lock
- recompute gains for a and b
- no candidates qualify
- quit pass
- second pass
- no candidates qualify
- quit
L
+1
+1
-1
b
a
c
d
- a, c are highest-gain candidates
- choose c, move and lock
- recompute gains for a, d
-1
-3
-1
+1
b
a
c
d
Properties of Algorithm
• Fast
– O(PlogP)
– constant factors are small
» arrays, pointer access
• Space Efficient
– 5 words/net overhead
– 4 words/cell overhead
– small constant overhead
• Suboptimal
– gets stuck in local minima
– any one move has negative gain
– need multiple moves for positive gain
• Example
– moving a or b results in -3 or -4 gain
– moving both a and b results in +1 gain
a
-3
-3
-4
-4
b