1. Partitioning 2 Outline Goal Fiduccia-Mattheyses Algorithm Approach
Example
Algorithm Properties
Goal
Understand Fidducia-Mattheyses partitioning algorithm

2. 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

3. Definitions 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

4. Cell Gain Cell Gain Gain Data Structure
gi - reduction in cutset by moving celli
label each cell with its gain
-pi <= gi <= pi
-pmax <= gi <= pmax
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 +2 +1 -1
MAX
GAIN 1 2 C
Cell #
+pmax
-pmax
BUCKET
CELL

5. Critical Nets Minimize cell gain updates Critical nets
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

6. 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
yes no

7. 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

8. 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))

9. 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
- quit L +1 +1 -1 b a c d -1 -3 -1 +1
- a, c are highest-gain candidates
- choose c, move and lock
- recompute gains for a, d b a c d

10. 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