Simulated annealing is a general purpose Monte Carlo optimization technique that was applied to the problem of placing standard logic cells in a VLSI ship so that the total interconnection wire length is minimized. An improved standard cell placement algorithm that takes advantage of the performance enhancements that appear to come from parallelizing the uniprocessor simulated annealing algorithm is presented. An outline of this algorithm is given

Banerjee, Prithviraj

Jones, Mark

English

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AN IMPROVED SIMULATED ANNEALING ALGORITHM FOR
STANDARD CFT.T. PLACEMENT
Mark Jones and Prithviraj Bonerjee \
Computer Systems Group
Coordinated Science Laboratory
University of Illinois at Urbana-Champaign
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Acknowledgment: This research wu supported in part by the National Aeronautics and Space Administration under
Contract NASA NAO 1-613.
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1. Introduction
Simulated annealing is a general purpose Monte Carlo optimization technique that has been
applied extensively in recent years [l. 2. 3.4]. In VLSI design, this technique has been applied to
the problem of placing standard logic cells in a VLSI chip so that the total interconnection wire
length is Tn'"'""*ed [5.6. 7]. Recently, many researchers have started to investigate techniques for
speeding up simulated annealing algorithms by running them on parallel processor systems
[8.9.10.11.12.13]. Many of these parallel algorithms have been reported to be considerably fas-
ter, and to converge to a final placement which is more optimal than similar uniprocessor simulated
annealing algorithms. The better performance of these parallel algorithms appears to be caused by
the constraints that they place on the distances over which moves can occur, and the use of slightly
outdated cell placement information caused by multiple interacting moves being accepted at each
parallel iteration.
In this paper..we present an improved standard cell placement algorithm that takes advantage
of the performance enhancements that appear to come from parallelizing the uniprocessor simu-
lated annealing algorithm. An outline of this new algorithm is given in Figure 1. Two important
differences between this algorithm and previously reported uniprocessor algorithms are in Steps 6
and 9. First, a modified generate function is proposed in Step 6 to allow for nonuniformly distri-
buted distances over which exchange and displacement moves can take place: this technique will be
called multi-windowing. Secondly, multiple cell movements are considered before updating cell
placement data in Step 9. hence the cost calculations are based on slightly outdated placement data.
This technique is referred to as pseudo-parallel moves.
2. Use of Multi-windowing ,
In the parallel versions of the simulated annealing placement algorithm, it appears that the
average distance a cell moves in the course of the algorithm has a profound effect on the final place-
ment. Specifically, it appears that movement of cells should be biased so that they are restricted
more to their local vicinity. Also, parallel simulated annealing algorithms running on architectures
such as the Hypercube are constrained to certain probability distributions because of the way the
cells are mapped to the individual processors, i.e.. the movement of cells from one section to certain
other sections in the physical circuit space is not possible in a single move because processing nodes
controlling those sections of the circuit space are not directly connected. A uniprocessor version of
the simulated annealing algorithm is not constrained in this manner and can therefore incorporate
rather complex windowing techniques and distance probability distributions. For example, in Fig-
ure 2. if cell M was picked to perform a displacement, a simple triple windowing scheme could be
used to determine where the cell will be displaced to. In Figure 2, the outermost window (Wl) is
always equal to the physical work space of the circuit. The inner windows. W2 and W3, have sizes
proportional to 2/3 and 1/3 of the physical work space and are centered about cell M. In order to
favor local movement, the probability of being in the innermost window (W3) is made greater than
being in the outer windows. In Figure 2. cell M has a 50% probability that its proposed new posi-
tion will be within the innermost window W3. a 25% probability of being within window W2 but
not window W3, and a 25% probability of being in the physical work space but not within win-
dows W2 or W3.
3. Use of Psendoparallel Moves
In Figure 1. a conditional data update statement has been added which allows multiple
number of accepted moves to accumulate before new cell locations are updated. This amounts to
allowing all moves after the first successful move determine the cost function on the basis of out-
dated placement information. For example, in Figure 3(a). if module Ml is successful in perform-
ing a displacement from (x i. y i) to (x2. y 2) during the first iteration of the inner loop, then the cir-
cuit should be as shown in Figure 3(b). but because Mi's position is not updated, the remainder of
the cells still calculate cost functions which involve Ml as though it were still at position (x i. y
 t).
Because of this, if module M2. which is connected to Ml via a net connection, is chosen for an
attempted move during iteration two. then the half-perimeter wiring cost associated with the net
will be computed using the old position of Ml.
After each move acceptance, a counter is incremented to keep track of the number of success-
ful moves, since the last cell position update and the new positions of the cells are placed in tem-
porary storage for use later in updating the cell positions. Random cell selection for movement in
subsequent iterations is not able to select cells which have made successful moves, but whose posi-
tions have not yet been updated. This amounts to freezing the cells' positions until the required
number of moves has been accepted. After a specified number of successful moves, the conditional
if statement criterion will be satisfied, and all cell positions will be updated using the information
saved in temporary storage.
The effect of using slightly outdated information appears to give a higher probability of get-
ting out of local minima, since this technique will accept a higher percentage of moves with uphill
changes in the cost function. The accept function limits the magnitude of uphill moves but does
not affect the total number. The number of uphill moves is affected by the random cell selection
and the "observed" placement. By having the observed placement slightly different from the actual
placement, it appears more uphill moves are accepted. By having greater numbers of uphill moves
which are all limited in magnitude by the accept function, the probability of getting out of local
minima is increased.
4. Placement Results
The advantage of this algorithm over conventional, uniprocessor simulated annealing algo-
rithms such as Timber Wolf is that it converges to a better final placement in a given amount of
time. We have implemented the algorithm in the C programming language on a Gould 9050 com-
puter system, running under UNIX 4.2.
We first report the performance of this algorithm using a small 64-standard cell circuit,
which was randomly generated and has several known clusters of cells with high connectivity. At
each temperature, approximately 100 new state moves were attempted per cell. The first set of
tests was concerned with determining the optimal number of moves that should be accepted before
cell updating is performed without any windowing. Table 1 shows final placement costs generally
decreasing as the number of multiple moves is increased. The optimal solution occurs when 16
moves have to be accepted before placement update will occur.
Simple testing of a few windowing schemes using 16 multiple moves showed consistent
decreases in final placement cost over using windowing alone as seen in Table 2. Because of this,
the remainder of the windowing scheme testing was performed using 16 multiple moves. A few of
the more promising windowing schemes were applied to two larger industry standard circuits of
sizes 183 and 286 cells with promising results, as shown in Table 3.
A variety of windowing schemes and size of windows have been extensively tested, results of
which will be reported in the full paper. Several generalized results can be deduced from these
studies. For example, if the inner windows are not significantly smaller than the physical work
space, the final placements tend to be decidedly inferior. This is in agreement with an earlier obser-
vation that the movement of cells should be localized to the area immediately surrounding the cell.
This statement is reinforced by noting that in all the windowing schemes, better performance is
generally obtained as the percentage of localized moves is increased.
5. Conclusions
This paper proposed an improved uniprocessor simulated annealing placement algorithm
which is based on the results obtained from parallelization of the sirnvlatffd annealing algorithm
proposed earlier by us and other researchers. Two new techniques were incorporated: multi-
windowing and pseudo-parallel moves. The algorithm has already been implemented and detailed
results on the performance of the algorithm on various industry example circuits will be reported
in the full paper. The results show that the algorithm is faster and better (in terms of final place-
ment quality for a fixed number of moves) than conventional algorithms.
[I] S. Kirkpatrick. C. D. Gelatt. and M. P. Vecchi, "Optimization by Simulated Ambling," Sci-
ence, voL 220. pp. 671-680, May 1983.
[2] S. Nahar, S. Sahni. and E. Shragowitz. "Experiments with Simulated Annealing." Proc. 22nd
Design. Automation Con/., pp. 748-752. June 1985.
[3] S. R. White. "Concepts of Scale in Simulated Annealing." Proc. Int. Con/. Computer Design
(ICCDS4). pp. 646-650. Oct. 1984.
[4] J. W. Greene and K. J. Supowit. "Simulated Annealing without Rejected Moves." Proc. Int.
Con/. Computer Design (ICCD84). pp. 658-663. Oct. 1984.
[5] C. Sechen and A. S. Vincentelli. "The TimberWolf Placement and Routing Package." Proc.
Custom Integrated Circuits Con/., pp. 522-527. May 1984.
[6] C. Sechen and A. S. Vincentelli. "TimberWolf3.2: A New Standard Cell Placement and Glo-
bal Routing Package." Proc. 23rd Design Automation Con/., pp. 432-439. Jun. 1986.
[7] L. K. Grover. "A New Simulated Annealing Algorithm for Standard Cell Placement." Proc.
Int. Con/, on Computer-Aided Design, pp. 378-380. Nov. 1986.
[8] E. H. L. Aarts. F. M. J. de Bont. E. H. A. Habers. and P. J. M. van Laarhoven. "Parallel Im-
plementations of the Statistical Cooling Algorithm." Integration, the VLSI Journal, vol. 4.
pp. 209-238.1986.
[9] M. J. Chung and K. K. Rao. "Parallel Simulated Annealing for Partitioning and Routing."
Proc. IEEE Int. Con/, on Computer Design (ICCD-86). pp. 238-242. Oct. 1986.
[10] S. A. Kravitz and R. A. Rutenbar. "Multiprocesssor-Based Placement by Simulated Anneal-
ing." Proc. 23rd Design Automation Con/., pp. 567-573. Jun. 1986.
[II] R. A. Rutenbar and S. A. Kravitz. "Layout by Annealing in a Parallel Environment." Proc.
IEEE Int. Con/, on Computer Design (ICCD-86). pp. 434-437. Oct. 1986.
[12] P. Banerjee and M. Jones. "A Parallel Simulated Annealing for Standard Cell Placement on a
Hypercube Computer." Proc. IEEE Int. Con/. Computer-Aided Design (ICCAD-86), Nov.
1986.
[13] M. Jones and P. Banerjee. "Performance of a Parallel Algorithm for Standard Cell Placement
on the Intel Hypercube." Proc. 24th Design Automation Con/.. June 1987.
STEP 1. Perform initial random placement of N standard cells
STEP 2. Determine initial temperature.
STEP 3. While "Stopping criteria" : temperature < 0.1 not reached
STEP 4. Generate new temperature
STEP 5. For inner_loop_count - 1 to ( N X attempt_parameter )
STEP 6. Generate new move using multi-windowing technique
STEP 7. Evaluate change in cost for move
STEPS. Accept/reject move using exponential function
STEP 9. IF the number of accepted moves is equal to limit (maz_accepted)
THEN update all saved cell positions and zero number of accepted moves counter
ELSE increment accepted moves counter and save cell movements in temporary storage
STEP 10. ENDFOR:
STEP 11. ENDWHILE:
Figure 1. Improved simulated annealing algorithm.
r-iiii
te.
iVl
Window
Wl
W2
W3
Fraction of
Max Dimension
1
2/3
1/3
Example
Size
120 X 10
72X60
36X30
Probability
Within Window
25%
25%
50%
Figure 2. Example use of windowing in determining cell movement for cell M.
M3
Ml
Figure 3(a). Original net placement.
M2
Mr J
......4^ ,
Calculated
- bounding
box
Actual
• bounding
box
Figure 3(b). Placement after initial acceptance.
Table 1. Cost vs number of multiple moves for 64-cell circuit.
Number of
Multiple Moves
1
2
4
8
12
16
24
32
Final
Placement Cost
24125
24138
24003
23984
23924
23821
24173
24829
Percentage
Change
40.0%
+0.1%
-0.5%
-O.6%
-0.8%
-13%
+0.2%
+2.9%
Table 2. Comparison of final cost for using or not using 16 multiple moves.
Number
Windows
1
2
2
2
3
4
Window Sizes
as Fraction
of max
1
1:%
l : fe
1:%
1 : 2/3 : 1/3
!:«:%:*
Distribution of
Moves in Windows
100%
20%: 80%
13% : 87%
10%: 90%
10% : 30% : 60%
6% : 12% : 24% : 58%
Final Cost
multiple moves
23821
22893
23654
23823
22148
22813
Percent
Change
-1.3%
-4.1%
-2.7%
-1.7%
-7.3%
-3.1%
Table 3. Comparison of cost vs windowing scheme for industry standard circuits.
Number
Cells
183
183
183
183
286
286
286
286
Number
Windows
1
2
3
4
1
2
3
4
Window Sizes
as Fraction
of max
1
1 : 1/3
1 : 2/3 : 1/3
1 :%:%:*
1
1:1/3
1:2/3: 1/3
1 :•%:%: %
Distribution
of moves
in windows
100%
20%: 80%
8% : 24% : 68%
5%: 10%: 20%: 65%
100%
20%: 80%
8% : 24% : 68%
5% : 10% : 20% : 65%
Final
Cost
76498
67159
69010
650034
115359
102398
102478
98312
Percent
Change
+0.0%
-12.2%
-9.8%
-10%
+0.0%
-11.2%
-11.2%
-14.8%


An improved simulated annealing algorithm for standard cell placement

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