10.4230/LIPICS.FSTTCS.2010.296
Jansen, Maurice
Maurice
Jansen
Qiao, Youming
Youming
Qiao
Sarma M.N., Jayalal
Jayalal
Sarma M.N.
Deterministic Black-Box Identity Testing $pi$-Ordered Algebraic Branching Programs
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
2010
ordered algebraic branching program
polynomial identity testing
Lodaya, Kamal
Kamal
Lodaya
Mahajan, Meena
Meena
Mahajan
2010
2010-12-14
2010-12-14
2010-12-14
en
urn:nbn:de:0030-drops-28728
10.4230/LIPIcs.FSTTCS.2010
978-3-939897-23-1
1868-8969
10.4230/LIPIcs.FSTTCS.2010
LIPIcs, Volume 8, FSTTCS 2010
IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)
2013
8
25
296
307
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Lodaya, Kamal
Kamal
Lodaya
Mahajan, Meena
Meena
Mahajan
1868-8969
Leibniz International Proceedings in Informatics (LIPIcs)
2010
8
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
12 pages
542461 bytes
application/pdf
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license
info:eu-repo/semantics/openAccess
In this paper we study algebraic branching programs (ABPs) with restrictions on the order and the number of reads of variables in the program. An ABP is given by a layered directed acyclic graph with source $s$ and sink $t$, whose edges are labeled by variables taken from the set $\{x_1, x_2, \ldots, x_n\}$ or field constants. It computes the sum of weights of all paths from $s$ to $t$, where the weight of a path is defined as the product of edge-labels on the path.
Given a permutation $\pi$ of the $n$ variables, for a $\pi$-ordered ABP ($\pi$-OABP), for any directed path $p$ from $s$ to $t$, a variable can appear at most once on $p$, and the order in which variables appear on $p$ must respect $\pi$. One can think of OABPs as being the arithmetic analogue of ordered binary decision diagrams (OBDDs). We say an ABP $A$ is of read $r$, if any variable appears at most $r$ times in $A$.
Our main result pertains to the polynomial identity testing problem, i.e. the problem of deciding whether a given $n$-variate polynomial is identical to the zero polynomial or not. We prove that over any field $\F$, and in the black-box model, i.e. given only query access to the polynomial, read $r$ $\pi$-OABP computable polynomials can be tested in $\DTIME[2^{O(r\log r \cdot \log^2 n \log\log n)}]$. In case $\F$ is a finite field, the above time bound holds provided the identity testing algorithm is allowed to make queries to extension fields of $\F$. To establish this result, we combine some basic tools from algebraic geometry with ideas from derandomization in the Boolean domain.
Our next set of results investigates the computational limitations of OABPs. It is shown that any OABP computing the determinant or permanent requires size $\Omega(2^n/n)$ and read $\Omega(2^n/n^2)$. We give a multilinear polynomial $p$ in $2n+1$ variables over some specifically selected field $\mathbb{G}$, such that any OABP computing $p$ must read some variable at least $2^n$ times. We prove a strict separation for the computational power of read $(r-1)$ and read $r$ OABPs. Namely, we show that the elementary symmetric polynomial of degree $r$ in $n$ variables can be computed by a size $O(rn)$ read $r$ OABP, but not by a read $(r-1)$ OABP, for any $0 < 2r-1 \leq n$. Finally, we give an example of a polynomial $p$ and two variables orders $\pi \neq \pi'$, such that $p$ can be computed by a read-once $\pi$-OABP, but where any $\pi'$-OABP computing $p$ must read some variable at least $2^n$ times.
LIPIcs, Vol. 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010), pages 296-307