\(
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% an undirected version of \rightarrow:
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\)
[This is a supplementary material to Chapter 2, Section 2.1]
Tropical convolution: a yet another application of Schnorr's bound
The (arithmetic) $n$-degree convolution is the set $P=\{P_0,P_1,\ldots,P_{2n-2}\}$ of the following degree-$2$ polynomials
of $2n$ variables $x_0,x_1,\ldots,x_{n-1}$ and $y_0,y_1,\ldots,y_{n-1}$:
\[
P_k(x,y)=\sum_{i+j=k} x_iy_j\,.
\]
Altogether, polynomials $P_0,P_1,\ldots,P_{2n-2}$ have $n^2$ monomials $x_iy_j$.
For example, if $n=3$, then we have $6$ variables $x_0,x_1,x_2,y_0,y_1,y_2$ and the $3$-degree convolution is $P=\{P_0,P_1,P_2,P_3,P_4\}$, where
\[
P_0=x_0y_0\ \ \ P_1=x_0y_1+x_1y_0\ \ \ P_2=x_0y_2+x_1y_1+x_2y_0
\ \ \ P_3=x_1y_2+x_2y_1\ \ \ P_4=x_2y_2\,.
\]
The tropical $(\min,+)$ $n$-degree convolution is the set
$T=\{T_0,T_1,\ldots,T_{2n-2}\}$ of $(\min,+)$ version of the arithmetic
$(+,\times)$ $n$-degree convolution $P=\{P_0,P_1,\ldots,P_{2n-2}\}$, where
\[
T_k(x,y)= \min\{x_i+y_j\colon i+j=k\}\,.
\]
In the case of maximization, we take $\max$ instead of $\min$.
All polynomials $P_0,P_1,\ldots,P_{2n-2}$ can be simultaneously computed by a monotone arithmetic $(+,\times)$ circuit (with $2n-1$ output gates) using only $n^2-2n+1$ addition gates: we have $n^2$ terms $x_iy_j$, each of which belongs to only one of the $2n-1$ polynomials $P_0,P_1,\ldots,P_{2n-2}$; since the sum of any $m$ terms can be computed using only $m-1$ addition gates, $n^2-(2n-1)=n^2-2n+1$ addition gates are enough.
So, the tropical versions $T_0,T_1,\ldots,T_{2n-2}$ can be also simultaneously computed by a tropical circuit using only $n^2-2n+1$
$\min$ or $\max$ gates. On the other hand, Schnorr's lower bound
(Theorem 2.1(1)) implies that
so many $\min$ or $\max$ gates are also necessary.
Theorem:
The minimal number of $\min$ or $\max$ gates in a
tropical $(\min,+)$ or $(\max,+)$ circuit computing the tropical $n$-degree convolution is $n^2-2n+1$.
Proof:
We prove the theorem for $(\min,+)$ circuits; the proof for $(\max,+)$ circuits is the same with $\min$ replaced by $\max$.
Consider the
sum-polynomial
\[
SP(x,y)=\sum_{k=0}^{2n-2} z_{k}\cdot P_k(x,y)
= \sum_{k=0}^{2n-2} \sum_{i=0}^k x_iy_{k-i}z_k
\]
of the (arithmetic) $n$-degree convolution, where $z_0,\ldots,z_{2n-2}$ are new $2n-1$ variables.
Let $A$ be the set of all $|A|=n^2$ exponent vectors of the sum-polynomial $SP$.
Every
monomial $ x_iy_{k-i}z_k$ of $SP$ is uniquely determined by
any two its variables. So, the set $A$ is cover-free, and
Schnorr's theorem (Theorem 2.1(1)) implies that every Minkowski $(\cup,+)$ circuit
producing the set $A$ must have at least $|A|-1=n^2-1$ union $(\cup)$ gates.
Since the set $A$ is homogeneous (of degree $3$),
Lemma 1.34(2) yields the same lower bound $n^2-1$ on
the number of $\min$ gates in every $(\min,+)$ circuit computing the tropical $(\min,+)$ version
\[
ST(x,y,z)= \min\Big\{x_i+y_{k-i}+z_k\colon k=0,1,\ldots, 2n-2 \mbox{ and } i=0,1,\ldots,k\Big\}
\]
of the sum-polynomial $SP$.
Now, given a $(\min,+)$ circuit for the $(\min,+)$ version $T=\{T_0,T_1,\ldots,T_{2n-2}\}$ of the $n$-degree convolution, we can solve $ST$ by adding
$2n-2$ additional $\min$ gates. Thus, the number of $\min$ gates in every $(\min,+)$ circuit computing $T$ is at least $(n^2-1)-(2n-2)=n^2-2n+1$.
∎
Footnotes:
(1)
Jump back ☝
(2)
Jump back ☝
References:
- Schnorr, C.P.: A lower bound on the number of additions in monotone computations. Theor.
Comput. Sci. 2(3), 305–315 (1976) Local copy
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