57 research outputs found
Building Efficient and Compact Data Structures for Simplicial Complexes
The Simplex Tree (ST) is a recently introduced data structure that can
represent abstract simplicial complexes of any dimension and allows efficient
implementation of a large range of basic operations on simplicial complexes. In
this paper, we show how to optimally compress the Simplex Tree while retaining
its functionalities. In addition, we propose two new data structures called the
Maximal Simplex Tree (MxST) and the Simplex Array List (SAL). We analyze the
compressed Simplex Tree, the Maximal Simplex Tree, and the Simplex Array List
under various settings.Comment: An extended abstract appeared in the proceedings of SoCG 201
Superpolynomial lower bounds for general homogeneous depth 4 arithmetic circuits
In this paper, we prove superpolynomial lower bounds for the class of
homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP
of degree in variables such that any homogeneous depth 4 arithmetic
circuit computing it must have size .
Our results extend the works of Nisan-Wigderson [NW95] (which showed
superpolynomial lower bounds for homogeneous depth 3 circuits),
Gupta-Kamath-Kayal-Saptharishi and Kayal-Saha-Saptharishi [GKKS13, KSS13]
(which showed superpolynomial lower bounds for homogeneous depth 4 circuits
with bounded bottom fan-in), Kumar-Saraf [KS13a] (which showed superpolynomial
lower bounds for homogeneous depth 4 circuits with bounded top fan-in) and
Raz-Yehudayoff and Fournier-Limaye-Malod-Srinivasan [RY08, FLMS13] (which
showed superpolynomial lower bounds for multilinear depth 4 circuits). Several
of these results in fact showed exponential lower bounds.
The main ingredient in our proof is a new complexity measure of {\it bounded
support} shifted partial derivatives. This measure allows us to prove
exponential lower bounds for homogeneous depth 4 circuits where all the
monomials computed at the bottom layer have {\it bounded support} (but possibly
unbounded degree/fan-in), strengthening the results of Gupta et al and Kayal et
al [GKKS13, KSS13]. This new lower bound combined with a careful "random
restriction" procedure (that transforms general depth 4 homogeneous circuits to
depth 4 circuits with bounded support) gives us our final result
Log-concavity and lower bounds for arithmetic circuits
One question that we investigate in this paper is, how can we build
log-concave polynomials using sparse polynomials as building blocks? More
precisely, let be a
polynomial satisfying the log-concavity condition a\_i^2 \textgreater{} \tau
a\_{i-1}a\_{i+1} for every where \tau
\textgreater{} 0. Whenever can be written under the form where the polynomials have at most
monomials, it is clear that . Assuming that the
have only non-negative coefficients, we improve this degree bound to if \tau \textgreater{} 1,
and to if .
This investigation has a complexity-theoretic motivation: we show that a
suitable strengthening of the above results would imply a separation of the
algebraic complexity classes VP and VNP. As they currently stand, these results
are strong enough to provide a new example of a family of polynomials in VNP
which cannot be computed by monotone arithmetic circuits of polynomial size
On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields
Recently, Gupta et.al. [GKKS2013] proved that over Q any -variate
and -degree polynomial in VP can also be computed by a depth three
circuit of size . Over fixed-size
finite fields, Grigoriev and Karpinski proved that any
circuit that computes (or ) must be of size
[GK1998]. In this paper, we prove that over fixed-size finite fields, any
circuit for computing the iterated matrix multiplication
polynomial of generic matrices of size , must be of size
. The importance of this result is that over fixed-size
fields there is no depth reduction technique that can be used to compute all
the -variate and -degree polynomials in VP by depth 3 circuits of
size . The result [GK1998] can only rule out such a possibility
for depth 3 circuits of size .
We also give an example of an explicit polynomial () in
VNP (not known to be in VP), for which any circuit computing
it (over fixed-size fields) must be of size . The
polynomial we consider is constructed from the combinatorial design. An
interesting feature of this result is that we get the first examples of two
polynomials (one in VP and one in VNP) such that they have provably stronger
circuit size lower bounds than Permanent in a reasonably strong model of
computation.
Next, we prove that any depth 4
circuit computing
(over any field) must be of size . To the best of our knowledge, the polynomial is the
first example of an explicit polynomial in VNP such that it requires
size depth four circuits, but no known matching
upper bound
On Σ A Σ A Σ Circuits: The Role of Middle Σ Fan-in, Homogeneity and Bottom Degree.
We study polynomials computed by depth five Σ ∧ Σ ∧ Σ arithmetic circuits where ‘Σ’ and ‘∧’ represent gates that compute sum and power of their inputs respectively. Such circuits compute polynomials of the form Pt i=1 Q αi i , where Qi = Pri j=1 ` dij ij where `ij are linear forms and ri, αi, t > 0. These circuits are a natural generalization of the well known class of Σ ∧ Σ circuits and received significant attention recently. We prove an exponential lower bound for the monomial x1 · · · xn against depth five Σ ∧ Σ [≤n] ∧ [≥21] Σ and Σ ∧ Σ [≤2 √n/1000] ∧ [≥ √n] Σ arithmetic circuits where the bottom Σ gate is homogeneous. Our results show that the fan-in of the middle Σ gates, the degree of the bottom powering gates and the homogeneity at the bottom Σ gates play a crucial role in the computational power of Σ ∧ Σ ∧ Σ circuits
Les concepts d'état limite et d'état critique et leurs applications pratiques à l'étude des argiles
Les notions d'état limite et d'état critique constituent un outil très puissant pour la compréhension de la plupart des problèmes pratiques de géotechnique des argiles molles.On montre d'abord comment l'état limite d'une argile naturelle représente en fait l'ensemble de ses propriétés de résistance, de déformabilité et de fluage.On analyse ensuite les implications pratiques de ces caractéristiques de comportement quant à l'interprétation des essais de laboratoire et des essais en place courants, le calcul des ouvrages de soutènement et surtout l'étude de la stabilité et des tassements des remblais sur sols compressibles
Building Efficient and Compact Data Structures for Simplicial Complexes
International audienceThe Simplex Tree (ST) is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows efficient implementation of a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the Simplex Tree while retaining its functionalities. In addition, we propose two new data structures called Maximal Simplex Tree (MxST) and Simplex Array List (SAL). We analyze the compressed Simplex Tree, the Maximal Simplex Tree, and the Simplex Array List under various settings
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