The document describes a new placement migration technique called "placement migration based on diffusion process" to address post-layout design issues like timing optimization and routing congestion mitigation. The technique models cell movement as a diffusion process, where cells move from areas of high density to low density over multiple time steps. This achieves smooth spreading while preserving the original placement's structure. Experimental results show the diffusion-based legalization approach improves wirelength and timing compared to other legalization methods.

1. R.PRABHAKAR, Dr K E Sreenivasa Murthy, Dr K Soundara Rajan / International Journal of
Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.1181-1184
Placement Migration Based On Diffusion Process For Future
VLSI Design
R.PRABHAKAR Dr K E Sreenivasa Murthy Dr K Soundara Rajan
Asso.Prof, ECE, HITS Principal Professor, ECE dept
Bogaram, R.R Dist, A.P SVITS ,Anantapur, A.P JNTUA, Anantapur, A.P
ABSTRACT
The VLSI placement problem is to place to its local density gradient. The more time steps the
the objects into fixed die such that there are no process is run, the closer the placement gets to
overlaps among the objects and some cost metric achieving equilibrium.
such as wire length and routability is optimized. We also need to mention another technique
For this purpose we use new type of placement in the context of global placement. This spreading
method, “placement migration based on diffusion technique models the density map as an electric field
process”. The placement migration is the whereby every region of the density map has some
movement of cells in an existing placement to attraction or repulsion to every cell in the design. In
address a variety of post layout design issues, contrast to this global technique, the process of
which performs the smooth spreading and diffusion is local, only requiring immediate bin
preserves the Integrity of the original placement. neighbors. Thus, it is actually a simpler technique.
This approach can address the problem of post One can directly apply the diffusion velocity field as
placement optimization for objectives such as the spreading force, which satisfies all the four
timing, routing congestion, signal Integrity and requirements for the spreading force [6]. Besides, it is
heat distribution. This method is useful as generic hard to apply the force-directed approach to
spreading technique to be used in conjunction placement migration, which does not start from
with analytic or force directed placement scratch but from an existing placement.
methods. To perform this, we use the diffusion
algorithm to address the problem of placement Among all the placement migration
legalization. Our experimental results show applications, the most straightforward one is
significant improvements in wavelength and legalization. Therefore we will use legalization to
timing. describe the detail of diffusion method.
Keywords: placement migration, routing
congestion, smooth spreading, signal integrity, II. PROBLEM FORMULATION
placement legalization.. Placement Migration for Legalization
Suppose we divide the chip area into N
I. INTRODUCTION equal sized bins. If the chip has a width of W and
During placement and physical synthesis of height of H, the density dj,k of each bin (j,k) can be
VLSI circuits, one is often faced with tasks such as defined as:
cell spreading, legalization of overlapping cells, and dj,k (1)
manipulating the placement to address other physical
objectives like power and routing congestion. These where is the overlapping area of cell i and bin (j,
tasks share a common theme of starting with an
k). For simplicity, we assume the fixed macros either
initial placement that is “good” and perturbing it so
totally occupy a bin or not, therefore the density for a
that it is improved in some way while still preserving
bin on a fixed macro is always 1.
the essential nature (cell ordering, wirelength, etc.) of
The problem of placement migration for
the original placement. We call these sets of tasks
legalization can be described as: Given an existing
“placement migration”.
placement (xi, yi) for each cell i, how to gradually
In this paper, we propose a new technique
for placement migration based on the physical move cells to produce a new placement
process of diffusion. Diffusion is a well-understood such that the maximum density dj,k is less than or
process that moves a physical elements (such as air equal to dmax.
molecules) from a state with non-zero potential This process is similar to the diffusion
energy to a state of equilibrium. The process can be process, which moves material from high
modeled by taking several small finite time steps and concentration area to less concentrated area.
moving each element the distance it would be Naturally, we can formulate the placement migration
expected to move in that time step. Our approach to process under the diffusion law, which is given in
placement migration does just that, it moves each cell next section.
a small amount in a given time step according Diffusion Process
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2. R.PRABHAKAR, Dr K E Sreenivasa Murthy, Dr K Soundara Rajan / International Journal of
Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.1181-1184
The dopant diffusion process on chip
substrate is a well known diffusion process.
Intuitively, materials from highly concentrated areas
would flow into less concentrated areas. Diffusion is
driven by the concentration gradient, which is the
slope and steepness of the concentration difference at
a given point. And the increase in concentration in a
cross section of unit area with time is simply the
difference of the material flow into the cross section
and the material flow out of it. The final equilibrium
of diffusion is an equal concentration distribution.
Mathematically, we can describe the
relationship of material concentration with time and
space using following equation.
Figure 1: Velocity Interpolation inside Bin.
(2) Meanwhile a cell just across the bin
where d is the material concentration, D is the boundary will get a totally different velocity. So
diffusivity which determines the speed of diffusion before we assign the velocity of a bin to a velocity of
(For the rest of the work, we set D to 1 for the a cell, we use interpolation. As shown in Fig. 1, the
simplicity of presentation). It states that the speed of bin velocity will be marked at the lower left corner of
density change is linear to its second order gradient each bin. The velocity for a point inside of a bin is
over space. interpolated by the velocities at the four corners of
In the context of placement, material this bin. Given a cell at (x, y) which is inside of bin (j,
concentration can be de- fined as the placement k), where j<x<j+1, k<y<k+1, we can compute vxx,y
density dx,y (t). and vyx,y using following interpolation:
We can define a velocity field dx,y = (vxx,y,
vyx,y) of diffusion at time t, which can be computed vxx,y = vxj,k + 𝛼(vxj+1,k -vxj,k) + β(vxj,k+1 -
vxj,k)+𝛼β(vxj,k+vxj+1,k+1 -vxj+1,k -vxj,k+1)
vyx,y = vyj,k + 𝛼(vyj+1,k -vyj,k) + β(vyj,k+1 -
as: vyj,k)+𝛼β(vyj,k+vyj+1,k+1 -vyj+1,k -vyj,k+1) (5)
where 𝛼 = x-j and β= y-k.
(3)
For the example shown in Fig 1, We calculate the
Therefore, starting from a initial location velocity at (x = 1.5, y = 1.4) with 𝛼 = 0.5, β = 0.4,
(x(0), y(0)), the cell location (x(t), y(t)) at time t can
be calculated by integrating the velocity field thus: vx1.5,1.4 = vx1,1+0.5(vx2,1-vx1,1)+0.4(vx1,2-
vx1,1)+0.2(vx1,1+vx2,2-vx2,1-vx1,2)=0.3
vy1.5,1.4 = vy1,1+0.5(vy2,1-vy1,1)+0.4(vy1,2-
(4) vy1,1)+0.2(vy1,1+vy2,2-vy2,1-vy1,2)=0.13
With (2), (4) and (3), we can incrementally change a IV. Diffusion Based Legalization Algorithm
placement based on the continuous density The input of the diffusion-based legalization
distribution. algorithm is locations (xi, yi) of each cell i, maximum
bin density dmax, bin number N and diffusion time T.
III. Velocity Interpolation It first computes the initial bin density using the
One problem with the proposed approach is given placement, then manipulates the density map to
that every cell within a bin has the same velocity and avoid over spreading. Starting from time 0, it
will thus get the same displacement. recursively compute bin density, bin velocity and cell
locations for each time step n. It stops after T
iterations or when the maximum bin density is less
than dmax. The complete diffusion algorithm is given
in Algorithm 1.
After diffusion, the placement should have a
max density of dmax and is roughly legal. We need to
run a final legalization step to put cells onto circuit
rows without overlap. Any legalizer can be used at
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3. R.PRABHAKAR, Dr K E Sreenivasa Murthy, Dr K Soundara Rajan / International Journal of
Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.1181-1184
this step. It will only take the legalizer a little effort Initial Density Generation
to remove those overlaps. Here we use the IBM Since the diffusion process will generate an
CPlace internal legalizer. equal density placement when it reaches the
equilibrium, we can expect the final density after
Algorithm1 Diffusion-based Legalization Algorithm
Inputs: cell locations (xi, yi), max density dmax, bin diffusion is average density j,k of the initial
number N, diffusion time T densities, which means not only the cells in bins
1: map cells onto bins and compute dj,k for each bin above 1 will expand, those in bins above average will
(j,k) expand as well. Suppose we want to achieve the
2: compute j, k using (11), the average bin density is maximum density dmax for the equilibrium, the total
now dmax area Ao that need to spread out of bins over dmax are:
3: dj, k (0) ← j, k Ao = ∑max (dj,k - dmax,0) (5)
The total slack As that can be used to hold Ao is: As =
4: n ← 0 ∑max (dmax -dj,k, 0) (6)
5: repeat
6: compute vxj,k(n), vyj,k(n) for each bin (j,k) using (6) If we can change dj,k for those bins under dmax to
7: compute xi(n), yi(n) for each cell i using (7) and make As = Ao, then at the equilibrium only the
velocity interpolation (8) overlaps Ao will move to As, and the densities of all
8: compute dj,k(n + 1) for each bin (j,k) using (6) the bins will be under dmax. One way to adjust dj,k is
9: n ← n + 1
j,k =
10: until n = T OR max (dj, k (n)) ≤ dmax+
(7)
We can validate that the new As = ∑max (dmax -dj,k, 0)
= Ao.
V. EXPERIMENTAL RESULTS
In this section, we report the experimental
results of diffusion based legalizer (DIFF). We first
evaluate its merit by comparing it with other
legalizers, i.e. a greedy legalizer (GREED) which
uses slide-and-spiral techniques to place cells onto
their nearest legal locations, and a network flow
legalizer (FLOW) which uses min-cost flow
algorithm to direct cell movements. Then we
characterize its performance based on different
parameter settings.
Comparison with Other Legalizers
FLOW includes two steps: first cells are
roughly spread out by the min-cost flow algorithm,
then, in a second step they are moved to their final
positions such that all overlaps are removed. GREED
sorts all the cells and place them sequentially. It first
Figure 2: Diffusion-based Legalization Example.
tries to place a cell at the original location. If that
location is occupied, it performs a spiral search
Fig 2 shows an example of diffusion-based
starting from the original location. During a spiral
legalization in a small region surrounded by fixed
search, it could slide other placed cells a little bit in
blocks. The left picture shows the initial illegal
order to fit in. All three legalizers are implemented in
placement. The right picture is the placement out of
C and run on a IBM P690 server. The timing results
legalization. Cells are colored to represent their
are reported by IBM Einstimer.
relative order. We can see after diffusion, the relative
orders are not changed.
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4. R.PRABHAKAR, Dr K E Sreenivasa Murthy, Dr K Soundara Rajan / International Journal of
Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.1181-1184
Table 1. TWL Comparison of Three Legalizers (m) overlapping area to under occupied area, congestion
test Base GREE FLO DIF mitigation which move cells from congested area to
cases D W F %impro non-congested area, etc. make the diffusion method
v very attractive. The experiment result on legalization
ckt1 11.4 13.23 13.40 12.4 44 problem has demonstrated very significant
8 6 improvements on timing and wire length over
ckt2 15.0 17.03 17.33 19 conventional methods.
6 16.6
5 VII. REFERENCES
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