Figure I. Architecture of a dynamic comparator. The transistor
switches M5, M6, and M7 are driven by a clock signal eLK.
symmetry axis
Figure 2. A symmetry island consists of one symmetry pair (bb bl)
and three self-symmetry modules bos, b2s and b3s. The module b2s with
a clock pin is placed inside the island.
(a)
b'
,
b'
2
(b)
Figure 3. (a) A placement in 1D vertical symmetry. (b) The feasible
conditions for a symmetry pair served as a boundary module.
compact placement is obtained. Their algorithm first divides
the whole placement region into an nxn mesh and calculates
the vertical and horizontal wire capacities for each room
according to the minimum wire width. Then, the congestion of
each room is estimated and every room is expanded based on
the congestion map. The area and HPWL of this expanded
placement will be used in the computation of the cost function
for the current candidate solution, which is more accurate than
the original one.
Although the adjustment algorithm by Xiao et al. [13] can
keep enough spaces between devices for routing nets, they
ignore that the routing spaces of a symmetry pair should be
identical such that each symmetry pair in a symmetry group
can keep symmetric with respect to the symmetry axis after
adjustment. To deal with this problem, we suggest employing
ASF-B*-trees [5] to represent symmetry groups. Since only
the representatives on a half of the placement plane need to be
considered, the maximum routing space of each symmetry pair
is selected for the representative. According to the definition
of ASF-B*-tree representation, the routing spaces in the other
side of the placement plane duplicate those of the
representatives while packing. Therefore, the routing spaces
for a symmetry pair will be the same and each symmetry pair
in a symmetry group will keep symmetric with respect to the
symmetry axis after the placement regions are expanded.
IV. COMMON-CENTROID CAPACITOR PLACEMENT
In this section, a common-centroid placement algorithm
considering mismatches for capacitor arrays is discussed since