33,429 research outputs found
Conditional Lower Bounds for Dynamic Geometric Measure Problems
We give new polynomial lower bounds for a number of dynamic measure problems
in computational geometry. These lower bounds hold in the Word-RAM model,
conditioned on the hardness of either 3SUM, APSP, or the Online Matrix-Vector
Multiplication problem [Henzinger et al., STOC 2015]. In particular we get
lower bounds in the incremental and fully-dynamic settings for counting maximal
or extremal points in R^3, different variants of Klee's Measure Problem,
problems related to finding the largest empty disk in a set of points, and
querying the size of the i'th convex layer in a planar set of points. We also
answer a question of Chan et al. [SODA 2022] by giving a conditional lower
bound for dynamic approximate square set cover. While many conditional lower
bounds for dynamic data structures have been proven since the seminal work of
Patrascu [STOC 2010], few of them relate to computational geometry problems.
This is the first paper focusing on this topic. Most problems we consider can
be solved in O(n log n) time in the static case and their dynamic versions have
only been approached from the perspective of improving known upper bounds. One
exception to this is Klee's measure problem in R^2, for which Chan [CGTA 2010]
gave an unconditional lower bound on the worst-case update
time. By a similar approach, we show that such a lower bound also holds for an
important special case of Klee's measure problem in R^3 known as the
Hypervolume Indicator problem, even for amortized runtime in the incremental
setting.Comment: Improved presentation, improved the reduction for the Hypervolume
Indicator problem and added a reduction for dynamic approximate square set
cove
Depth-4 Lower Bounds, Determinantal Complexity : A Unified Approach
Tavenas has recently proved that any n^{O(1)}-variate and degree n polynomial
in VP can be computed by a depth-4 circuit of size 2^{O(\sqrt{n}\log n)}. So to
prove VP not equal to VNP, it is sufficient to show that an explicit polynomial
in VNP of degree n requires 2^{\omega(\sqrt{n}\log n)} size depth-4 circuits.
Soon after Tavenas's result, for two different explicit polynomials, depth-4
circuit size lower bounds of 2^{\Omega(\sqrt{n}\log n)} have been proved Kayal
et al. and Fournier et al. In particular, using combinatorial design Kayal et
al.\ construct an explicit polynomial in VNP that requires depth-4 circuits of
size 2^{\Omega(\sqrt{n}\log n)} and Fournier et al.\ show that iterated matrix
multiplication polynomial (which is in VP) also requires 2^{\Omega(\sqrt{n}\log
n)} size depth-4 circuits.
In this paper, we identify a simple combinatorial property such that any
polynomial f that satisfies the property would achieve similar circuit size
lower bound for depth-4 circuits. In particular, it does not matter whether f
is in VP or in VNP. As a result, we get a very simple unified lower bound
analysis for the above mentioned polynomials.
Another goal of this paper is to compare between our current knowledge of
depth-4 circuit size lower bounds and determinantal complexity lower bounds. We
prove the that the determinantal complexity of iterated matrix multiplication
polynomial is \Omega(dn) where d is the number of matrices and n is the
dimension of the matrices. So for d=n, we get that the iterated matrix
multiplication polynomial achieves the current best known lower bounds in both
fronts: depth-4 circuit size and determinantal complexity. To the best of our
knowledge, a \Theta(n) bound for the determinantal complexity for the iterated
matrix multiplication polynomial was known only for constant d>1 by Jansen.Comment: Extension of the previous uploa
On the rank of matrix multiplication
For every positive integer we obtain the lower bound
for the
rank of the matrix multiplication. This bound improves the previous
one due to Landsberg.
Furthermore our bound improves the classic bound , due to
Bl\"aser, for every . Finally, for , with a sligtly different
strategy we menage to obtain the lower bound which improves
Bl\"aser's bound for any .Comment: 10 pages. New version, title and main result changed. Linear Algebra
and its Applications 201
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
Faster all-pairs shortest paths via circuit complexity
We present a new randomized method for computing the min-plus product
(a.k.a., tropical product) of two matrices, yielding a faster
algorithm for solving the all-pairs shortest path problem (APSP) in dense
-node directed graphs with arbitrary edge weights. On the real RAM, where
additions and comparisons of reals are unit cost (but all other operations have
typical logarithmic cost), the algorithm runs in time
and is correct with high probability.
On the word RAM, the algorithm runs in time for edge weights in . Prior algorithms used either time for
various , or time for various
and .
The new algorithm applies a tool from circuit complexity, namely the
Razborov-Smolensky polynomials for approximately representing
circuits, to efficiently reduce a matrix product over the algebra to
a relatively small number of rectangular matrix products over ,
each of which are computable using a particularly efficient method due to
Coppersmith. We also give a deterministic version of the algorithm running in
time for some , which utilizes the
Yao-Beigel-Tarui translation of circuits into "nice" depth-two
circuits.Comment: 24 pages. Updated version now has slightly faster running time. To
appear in ACM Symposium on Theory of Computing (STOC), 201
Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication
Understanding the query complexity for testing linear-invariant properties
has been a central open problem in the study of algebraic property testing.
Triangle-freeness in Boolean functions is a simple property whose testing
complexity is unknown. Three Boolean functions , and are said to be triangle free if there is no such that . This property
is known to be strongly testable (Green 2005), but the number of queries needed
is upper-bounded only by a tower of twos whose height is polynomial in 1 /
\epsislon, where \epsislon is the distance between the tested function
triple and triangle-freeness, i.e., the minimum fraction of function values
that need to be modified to make the triple triangle free. A lower bound of for any one-sided tester was given by Bhattacharyya and
Xie (2010). In this work we improve this bound to .
Interestingly, we prove this by way of a combinatorial construction called
\emph{uniquely solvable puzzles} that was at the heart of Coppersmith and
Winograd's renowned matrix multiplication algorithm
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