CONSTRUCTION OF LINEARLY TRANSFORMED PLANAR BDD BY WALSH
COEFFICIENTS
\author{
\begin{tabular}[t]{c@{\extracolsep{1em}}c@{\extracolsep{1em}}c}
Mark G. Karpovsky
& Radomir S. Stankovi\' c
\thanks{This
work was supported by the Academy of Finland, Finnish Center of Excellence Programme,
Grant No.
44876, and the EXSITE Project No. 51520.}
& Jaakko T. Astola \\
Dept. of Electrical and & Dept. of Computer Science & Tampere Int. Center \\
Computer Engineering & Faculty of Electronics & for Signal Processing \\
8 Saint Marry's Street & Beogradska 14 & Tampere University of \\
Boston University & 18 000 Ni\v s & Technology \\
Boston, Ma 02215 & Serbia & Tampere \\
USA & & Finland
\end{tabular}}
ABSTRACT
In VLSI design, crossings of interconnections occupy space and
cause delay. In particular, it is desirable to have planar
networks in FPGA synthesis and sub-micron LSIs, since delay in the
interconnections and crossings are comparable to the delay for
logic circuits. Decision diagrams (DDs) provide a simple
technology mapping, and planar DDs result in planar networks. In
this paper, we present a deterministic method to construct planar
Linearly Transformed Binary Decision Diagrams (BDDs) by Walsh
transform spectral coefficients.
1. INTRODUCTION
Binary Decision Diagrams (BDDs) are a standard part of many CAD
systems in logic design, signal processing, and other areas where
efficient, in terms of space and time, manipulation with switching
functions is required. Complexity of BDD representation of a
given function $f$ is usually estimated by the number of
non-terminal nodes in the BDD for $f$, denoted as the size of
BDD($f$). The size of BDDs is very sensitive to the order of
variables in the functions represented, ranging from the
polynomial to the exponential complexity for the same function for
different order of variables. Therefore, majority of approaches to
the reduction of size of BDDs is related to development of
efficient algorithms for reordering of variables, see for example,
\cite{[FujitaKukimotoBrayton]}, \cite{[Rudell]}. Linearly
transformed BDDs are defined by allowing linear combinations of
variables \cite{[GuntherDrechslerexactmethod]}. In this way, the
number of possible transformation over variables is increased from
$n!$ to $\prod_{i=0}^{n-1}(2^{n}-2^{i})$ for reordering and linear
transformation of variables, respectively.
There are many algorithms proposed to determine order and linear
transformation of variables, which exploit some good heuristics
and when integrated in widely used sifting algorithms for BDD
minimization \cite{[Rudell]} appear quite efficient see a
discussion in \cite{[MeinelSomenziTheobald2000]}. However, a
disadvantage of heuristic algorithms is that they cannot guarantee
the quality of the results produced. A deterministic algorithm for
construction of linearly transformed DDs exploiting properties of
switching functions and features of related operators for
switching functions in the original and spectral transform domain
is proposed in \cite{[KarpovskyStankovicAstola]}. An algorithm for
exact minimization of of BDDs by linear transformation of
variables has been proposed in
\cite{[GuntherDrechslerexactmethod]}. The algorithm takes
advantages from features of sifting procedures and properties of
linear transformation over the Galois field $GF(2)$. However, for
the huge search space, this method is limited to functions of a
small number of variables, 6 and in some examples 7 variables.
This number of variables is consider sufficient for applications
as FPGA synthesis and similar applications.
1.1 Planar decision diagrams
Networks without crossings are useful in synthesis with Field
Programmable Gate Arrays (FPGAs)
\cite{[MurgaiBraytonSangiovanni-Vincentelli]}, since crossings
produce considerable delay and in FPGAs delay of interconnections
is among the most important problems. Further, planar networks are
desirable in sub-micron LSIs, since delay in the interconnections
and crossings are comparable to the delay for logic circuits.
Decision diagrams provide a simple mapping to technology, since a
network is easily derived from a DD by replacing each node with
the logic elements realizing the decomposition rules used to
assign a given function $f$ to the corresponding DD for $f$.
Planar DDs \cite{[ButlerNowlinSasao]},
\cite{[SasaoButlerplanarbdd]} result in planar networks.
Definition 1
A Binary Decision Diagram (BDD) is planar if
there is no crossing of edges connecting non-terminal nodes, under
assumption that edges labelled by $\overline{x}_{i}$ and $x_{i}$
emerge to the left and right of a node, respectively, constant
node 0 is to the left of the constant node 1, and all edges are
directed down throughout their length, which precludes arcs that
extend around the root node or constant nodes.
Notice that a planar decision diagram can be derived by the
reduction of a decision tree if sharing of isomorphic subtrees is
restricted to subtrees rooted at neighboring nodes at a level in
the DT. The restrictions for order of labels and values of
constant nodes can be removed to enlarge the class of functions
with planar diagrams. However, in what follows, these restrictions
will be applied to the decision diagrams considered.
In this paper, we consider construction of linearly transformed
planar Binary Decision Diagrams (LT-BDDs) through Walsh transform
coefficients. We assume that switching functions are given in
$\{1,-1\}$ encoding to exploit properties of Walsh spectrum. When
a LT-BDD constructed, we replace values of constant nodes by logic
0 and 1.
2. BACKGROUND THEORY
Switching functions can be viewed as functions defined on finite
dyadic groups $C_{2}^{n}$, where $C_{2}=(\{0,1\}, \oplus)$, is the
basic cyclic group of order 2, with the group operation $\oplus$,
defined as the addition modulo 2, logic EXOR, and $n$ is the
number of variables.
Walsh functions are group characters of $C_{2}^{n}$, see for
example, \cite{[Karpovsky]}, and can be expressed as products of
the Rademacher functions $rad(i,x)$, $i=0,1, \ldots , n$, $w = 0,
1, \ldots, 2^{n}-1$, which can be related to switching variables
%\begin{eqnarray*}
$rad(i,w)= 1-2x_{i}(w)$,
%\end{eqnarray*}
where $x_{i}(w)$ is the switching function $f(x_{1}, \ldots ,
x_{n}) = x_{i}$. In this relation, it is assumed that logic values
0 and 1 for switching variables and functions are interpreted as
integers 0 and 1. Therefore, it may be said that Rademacher
functions are equal to switching variables in $(0,1)_{GF(2)}
\rightarrow (1,-1)_{Z}$ encoding, where $GF(2)$ is the Galois
field of order 2, and $Z$ is the set of integers.
It follows that Walsh functions can be expressed as
%\begin{eqnarray*}
$wal(r,w) = \prod_{i=1}^{n} (1-2x_{i})^{r_{i}}$,
%\end{eqnarray*}
where $r_{i}$ are coordinates in the binary representation of the
index $r = (r_{1}, \ldots , r_{n})$. Since the set of $2^{n}$
Walsh functions endorse the structure of a multiplicative group
$\Gamma$ with the group operations defined as multiplication in
$Z$, which is isomorphic to the finite dyadic group $C_{2}^{n}$, a
relationship between Walsh functions and linear switching
functions, defined as EXOR sum of switching variables, follows
directly \cite{[AgaianAstolaEgiazarian]}. We exploit this property
to determine linear combination of variables in LT-BDDs.
3. LT-BDD
Due to the relationship between Walsh functions and linear
switching functions, the following algorithm to construct LT-BDDs
can be deduced.
Algorithm (Construction of LT-BDD)
1. Given an $n$-variable function $f$ in $\{1,-1\}$ encoding.
Calculate the Walsh spectrum.
2. Find the Walsh coefficient $S_{f,max}(w)$ of the maximum absolute value.
Declare $w = w_{max}$ and write its binary representation, i.e.,
$w_{max} = (w_{1}, \ldots , w_{n})$.
3. Determine a linear function $\tau = \bigoplus_{r} x_{i}$,
where $r$ goes over the indices of $w_{i}=1$.
4. Determine co-factors for $f$ with respect to $\tau$, i.e.,
determine the subfunction of $n-1$ variables where $\tau = 0$ and
1, respectively.
5. Create a node whose outgoing edges point to the
co-factors $f_{0}$ and $f_{1}$ and label its edges by
$\overline{\tau}$ and $\tau$, respectively.
6. Repeat the Steps 1 to 5, for $i = 1$, to $n$.
Assignment of variables to the edges in LT-BDDs can be performed
by using the following algorithm.
Algorithm(Assignment of labels to the edges)
1. If at a node at the $i$-th level the decomposition with
respect to a linear combination $x_{k} \oplus x_{q} \oplus \cdots
\oplus x_{r}$, where $k < q < \ldots < r$, is performed, relate
the variable $x_{r}$ to the level $i$ and eliminate if from
further considerations.
2. Repeat the Step 1 to all the levels in the BDD starting
from the root node.
Example 1
For a four-variable function $f$, defined as ${\bf F} =
[1,1,-1,1,-1,1,1,1,-1,1,1,-1,1,-1,1,1]^{T}$, the Walsh spectrum is
${\bf S}_{f} = [6,-2,-2,-2,-2,-2,6,-2,2,-6,2,2,2,2,2,10]^{T}$, and
the coefficient with the maximum value is $S_{f_{1}}(15) = 10$,
and since $w_{max} = (1111)$, we determine a linear function
$\tau_{1} = x_{1} \oplus x_{2} \oplus x_{3} \oplus x_{4}$, where
each $x_{i}$ corresponds to the appearance of a coordinate with
the value 1 in the binary representation for $w_{max}$. The
cofactors of $f$ with respect to $\tau_{1}$ are ${\bf F}_{0} =
[1,1,1,1,1,1,1,1]^{T}$, and ${\bf F}_{1} =
[1,-1,-1,1,-1,-1,-1,1]^{T}$. We create a node where outgoing edges
point to subfunctions ${\bf F}_{0}$ and ${\bf F}_{1}$ and edges
are labelled by $\overline{x_{1} \oplus x_{2} \oplus x_{3} \oplus
x_{4}}$ and $x_{1} \oplus x_{2} \oplus x_{3} \oplus x_{4}$,
respectively. The variable $x_{4}$ is eliminated from further
considerations. Since ${\bf F}_{0}$ is a constant function, the
edge $\overline{x_{1} \oplus x_{2} \oplus x_{3} \oplus x_{4}}$
points to the constant node 1 directly. The Walsh spectrum for
${\bf F}_{1}$ is ${\bf S}_{f_{1}} = [-2,-2,-2,6,2,2,2,2]^{T}$. The
maximum Walsh coefficient for ${\bf F}_{1}$ is 6 for $w_{max} =
(011)$, we determine $\tau_{2} = x_{2} \oplus x_{3}$, and perform
decomposition of $f_{1}$ into subfunctions of two variables ${\bf
F}_{1,0} = [1,1,-1,1]^{T}$ and ${\bf F}_{1,1} =
[-1,-1,-1,-1]^{T}$.
Since $f_{1,1}$ is a constant function, we proceed with
decomposition of $f_{1,0}$. The cofactors depend on variables
$x_{1}$ and $x_{2}$, since $x_{3}$ was the last variable in
$\tau_{2}$. The Walsh spectrum of it is ${\bf S}_{f_{1,0}} =
[2,-2,2,2]^{T}$. Since all the coefficients have the equal
absolute value, we freely chose and determine $\tau_{3} = x_{1}$
and do not continue decomposition since we have subfunction of a
single variable. In this way, we derive the LT-BDD for $f$ as
shown in Fig.~\ref{fig-ltmtbdd}. This LT-BDD represents $f$ as $f
= (x_{1} \oplus x_{2} \oplus x_{3} \oplus x_{4})(\overline{(x_{2}
\oplus x_{3})}x_{1}\overline{x}_{2} \oplus (x_{2} \oplus x_{3}))$.
Since this LT-BDD has 4 nodes, the linearization by Walsh
coefficients provided for the reduction of 5 nodes compared to the
BDD for $f$ which has 9 nodes.
\begin{figure}
\begin{center}
\epsfysize=4.5cm \leavevmode \epsfbox{ltmtbddltar.eps}
%
\epsfile{file=structures.eps,height=10cm}
\caption{{\sf LT-BDD for $f$ in Example~\ref{ex-ltmtbdd}.}}
\label{fig-ltmtbdd}
\end{center}
\end{figure}
4. FEATURES OF LT-BDD
Table~\ref{tbl-benchmarks} compares the sizes of BDDs for the
initial (BDD) and optimal (BDDv) order of variables, planar BDDs
(BDDr), and planar LT-BDDs produced by decomposition with respect
to Walsh coefficients (LT-BDD). For BDDs and BDDs optimized by
reordering of variables, sizes are determined by allowing sharing
of all isomorphic subtrees, thus, these are not planar decision
diagrams. It should be noticed that planar decision diagrams
necessarily have larger sizes, since may contain some redundant
information at the price of avoiding crossing of edges, which may
be an important feature in circuit synthesis and some other
applications as explained in \cite{[SasaoButlerplanarbdd]}. With
this comparison we estimate impact of keeping decision diagrams
planar and efficiency of the linearization of BDDs by Walsh
coefficients under this restriction as in the definition of planar
DDs.
Each output of a multi-output benchmark function is considered as
a separate single output function $f-i$, where $f$ is the name of
the benchmark and $i$ is the number of the output. The experiments
are for small benchmarks, since we compared sizes of LT-BDDs with
the sizes of BDDs for the optimal order or variables, which are
determined by the brute force method examining all possible $n!$
combinations. However, there is no restrictions to the number of
variables in functions to which the method can be applied, due to
the existence of fast calculation methods for Walsh transform,
including methods over BDDs \cite{[ClarkeZhao]}, and various other
classes of decision diagrams, see a discussion in
\cite{[JankovicStankovicDrechsler]}. Notice that in applications
as FPGA synthesis, even a small gain in the number of reduced
nodes count may tremendously simplify the architecture
\cite{[MurgaiBraytonSangiovanni-Vincentelli]}.
It is worth mentioning that LT-BDD constructed by Walsh
coefficients have sizes comparable to the sizes of BDDs for the
optimal order of variables. This could be considered as an
important feature of LT-BDDs constructed by decomposition with
respect to Walsh coefficients.
For many functions planar LT-BDDs are smaller than planar BDDs, in
this example, for 18 functions. In some cases, the reduction is
considerable. Examples are clip-2, clip-3, clip-4, ex1010-2,
ex1010-8, rd84-2, sao2-1, sao2-2. There are exceptions where the
linearization by Walsh coefficients increases the size of BDDs
compared to the size of planar BDDs, since some regular structure
in the truth-vectors of these functions is destroyed by the
decomposition with respect to the Walsh coefficients. Under the
regular structure, we assume appearance of some constant or
identical subvectors with my repeat with some regularity. Examples
are 9sym, rd84-1, and rd84-4, however, for rd84-2 a considerable
reduction is achieved. In 12 cases, the size of planar LT-BDDs is
smaller than that of BDDs for the initial ordering of variables.
In 6 examples, size of planar LT-BDDs is smaller than size of BDDs
for optimal order of variables. Examples are clip-2, clip-4, and
sao2-2.
Table 1
\caption{\sf Sizes of BDDs, BDDs with optimal order of variables, planar BDD, and planar
LT-BDDs.}
\label{tbl-benchmarks}
\begin{center}
\begin{tabular}{lrrrrr} \hline
$f$ & $n$ & BDD & BDDv & BDDr & LT-BDD \\ \hline
5xp1-1 & 7 & 14 & 11 & 35 & 13 \\
5xp1-4 & 7 & 16 & 11 & 19 & 15 \\
9sym & 9 & 33 & 33 & 33 & 57 \\
apex4-10 & 9 & 95 & 91 & 193 & 148 \\
% apex4-14 & 9 & 94 & - & 213 & 59 \\
clip-1 & 9 & 37 & 34 & 67 & 35 \\
clip-2 & 9 & 58 & 42 & 156 & 45 \\
clip-3 & 9 & 73 & 32 & 196 & 51 \\
clip-4 & 9 & 76 & 36 & 169 & 27 \\
clip-5 & 9 & 36 & 36 & 36 & 22 \\
con1-1 & 7 & 12 & 11 & 11 & 16 \\
con1-2 & 7 & 8 & 7 & 8 & 8 \\
ex1010-2 & 10 & 155 & 148 & 390 & 156 \\
ex1010-8 & 10 & 154 & 147 & 377 & 209 \\
% misex1-3 & 8 & 13 & - & 13 & 14 \\
% msex1-6 & 8 & & 12 & 12 & 9 \\
rd73-3 & 7 & 16 & 16 & 16 & 16 \\
rd84-1 & 8 & 32 & 25 & 54 & 82 \\
rd84-2 & 8 & 25 & 22 & 169 & 15 \\
rd84-4 & 8 & 19 & 19 & 22 & 59 \\
sao2-1 & 10 & 46 & 32 & 75 & 32 \\
sao2-2 & 10 & 48 & 34 & 85 & 31 \\
squar5-1 & 5 & 5 & 5 & 6 & 5 \\
%squar5-2 & 5 & 6 & 6 & 4 \\
%squar5-3 & 5 & 8 & 7 & 8 \\
%squar5-4 & 5 & 11 & 9 & 11 \\
%squar5-5 & 5 & 11 & 9 & 12 \\
z5xp1-3 & 7 & 20 & 14 & 22 & 12 \\
z5xp1-6 & 7 & 15 & 9 & 21 & 9 \\ \hline
% z9sym & 9 & 33 & 33 & - & 44 \\ \hline
av. & & 45 & 36 & 96 & 46 \\ \hline
\end{tabular}
\end{center}
\end{table}
From Table~\ref{tbl-benchmarks}, it follows that on the average
LT-MTBDDs are for 48\% smaller than Planar BDD, and for 2\% and
26\% larger than BDDs for initial and optimal order of variables.
It should be noticed that compared to planar BDDs, when smaller,
planar LT-BDDs are considerably smaller, for 70\%, and when
larger, they are just for 20\% larger. This should be pointed out
as an useful feature of the method proposed.
6. CLOSING REMARKS
Experimental results show that planar LT-BDDs with linear
combination of variables determined from maximum value of Walsh
coefficients have interesting features, which can be considered as
a justification for application of this method.
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