307 research outputs found
Faster Isomorphism for -Groups of Class 2 and Exponent
The group isomorphism problem determines whether two groups, given by their
Cayley tables, are isomorphic. For groups with order , an algorithm with
running time, attributed to Tarjan, was proposed in the
1970s [Mil78]. Despite the extensive study over the past decades, the current
best group isomorphism algorithm has an running time
[Ros13].
The isomorphism testing for -groups of (nilpotent) class 2 and exponent
has been identified as a major barrier to obtaining an time
algorithm for the group isomorphism problem. Although the -groups of class 2
and exponent have much simpler algebraic structures than general groups,
the best-known isomorphism testing algorithm for this group class also has an
running time.
In this paper, we present an isomorphism testing algorithm for -groups of
class 2 and exponent with running time for any
prime . Our result is based on a novel reduction to the skew-symmetric
matrix tuple isometry problem [IQ19]. To obtain the reduction, we develop
several tools for matrix space analysis, including a matrix space
individualization-refinement method and a characterization of the low rank
matrix spaces.Comment: Accepted to STOC 202
Learning mixtures of structured distributions over discrete domains
Let be a class of probability distributions over the discrete
domain We show that if satisfies a rather
general condition -- essentially, that each distribution in can
be well-approximated by a variable-width histogram with few bins -- then there
is a highly efficient (both in terms of running time and sample complexity)
algorithm that can learn any mixture of unknown distributions from
We analyze several natural types of distributions over , including
log-concave, monotone hazard rate and unimodal distributions, and show that
they have the required structural property of being well-approximated by a
histogram with few bins. Applying our general algorithm, we obtain
near-optimally efficient algorithms for all these mixture learning problems.Comment: preliminary full version of soda'13 pape
A composition theorem for parity kill number
In this work, we study the parity complexity measures
and .
is the \emph{parity kill number} of , the
fewest number of parities on the input variables one has to fix in order to
"kill" , i.e. to make it constant. is the depth
of the shortest \emph{parity decision tree} which computes . These
complexity measures have in recent years become increasingly important in the
fields of communication complexity \cite{ZS09, MO09, ZS10, TWXZ13} and
pseudorandomness \cite{BK12, Sha11, CT13}.
Our main result is a composition theorem for .
The -th power of , denoted , is the function which results
from composing with itself times. We prove that if is not a parity
function, then In other words, the parity kill number of
is essentially supermultiplicative in the \emph{normal} kill number of
(also known as the minimum certificate complexity).
As an application of our composition theorem, we show lower bounds on the
parity complexity measures of and . Here is the sort function due to Ambainis \cite{Amb06},
and is Kushilevitz's hemi-icosahedron function \cite{NW95}. In
doing so, we disprove a conjecture of Montanaro and Osborne \cite{MO09} which
had applications to communication complexity and computational learning theory.
In addition, we give new lower bounds for conjectures of \cite{MO09,ZS10} and
\cite{TWXZ13}
Fully Dynamic Min-Cut of Superconstant Size in Subpolynomial Time
We present a deterministic fully dynamic algorithm with subpolynomial
worst-case time per graph update such that after processing each update of the
graph, the algorithm outputs a minimum cut of the graph if the graph has a cut
of size at most for some . Previously, the best update
time was for any and [Thorup,
Combinatorica'07].Comment: SODA 202
Recommended from our members
On the isomorphism testing of graphs
Graph Isomorphism is one of the very few classical problems in NP of unsettled complexity status. The families of highly regular structures, for example Steiner 2-designs, strongly regular graphs and primitive coherent configurations, have been perceived as difficult cases for graph isomorphism. These highly regular structures arise naturally as obstacles for both the classical group theory and combinatorial approaches for the graph isomorphism problem.
In this thesis we investigate the isomorphism problem of highly regular structures. We present new results to understand the combinatorial structure of highly regular structures, and propose some new algorithms to compute the canonical forms (and thus isomorphism testing) of highly regular structures based on the structural theorems.
We also give an algorithm solving the isomorphism problem of two unknown graphs in the property testing setting. Our new algorithm has sample complexity matching the information theoretical lower bound up to some multiplicative subpolynomial factor
Dynamic Kernel Sparsifiers
A geometric graph associated with a set of points and a fixed kernel function
is a
complete graph on such that the weight of edge is
. We present a fully-dynamic data structure that
maintains a spectral sparsifier of a geometric graph under updates that change
the locations of points in one at a time. The update time of our data
structure is with high probability, and the initialization time is
. Under certain assumption, we can provide a fully dynamic spectral
sparsifier with the robostness to adaptive adversary.
We further show that, for the Laplacian matrices of these geometric graphs,
it is possible to maintain random sketches for the results of matrix vector
multiplication and inverse-matrix vector multiplication in time,
under updates that change the locations of points in or change the query
vector by a sparse difference
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