8,340 research outputs found

### Exploring the environment, discovering learning resources and creating low cost training and development

School of Managemen

### Hyperelliptic jacobians with real multiplication

Let $K$ be a field of characteristic $p \neq 2$, and let $f(x)$ be a sextic
polynomial irreducible over $K$ with no repeated roots, whose Galois group is
isomorphic to \A_5. If the jacobian $J(C)$ of the hyperelliptic curve
$C:y^2=f(x)$ admits real multiplication over the ground field from an order of
a real quadratic field $D$, then either its endomorphism algebra is isomorphic
to $D$, or $p > 0$ and $J(C)$ is a supersingular abelian variety. The
supersingular outcome cannot occur when $p$ splits in $D$.Comment: Corrected typos; clarified proofs; added more examples in positive
characteristi

### Artificial in its own right

Artificial Cells, , Artificial Ecologies, Artificial Intelligence, Bio-Inspired Hardware Systems, Computational Autopoiesis, Computational Biology, Computational Embryology, Computational Evolution, Morphogenesis, Cyborgization, Digital Evolution, Evolvable Hardware, Cyborgs, Mathematical Biology, Nanotechnology, Posthuman, Transhuman

### Optimal Euclidean spanners: really short, thin and lanky

In a seminal STOC'95 paper, titled "Euclidean spanners: short, thin and
lanky", Arya et al. devised a construction of Euclidean (1+\eps)-spanners
that achieves constant degree, diameter $O(\log n)$, and weight $O(\log^2 n)
\cdot \omega(MST)$, and has running time $O(n \cdot \log n)$. This construction
applies to $n$-point constant-dimensional Euclidean spaces. Moreover, Arya et
al. conjectured that the weight bound can be improved by a logarithmic factor,
without increasing the degree and the diameter of the spanner, and within the
same running time.
This conjecture of Arya et al. became a central open problem in the area of
Euclidean spanners.
In this paper we resolve the long-standing conjecture of Arya et al. in the
affirmative. Specifically, we present a construction of spanners with the same
stretch, degree, diameter, and running time, as in Arya et al.'s result, but
with optimal weight $O(\log n) \cdot \omega(MST)$.
Moreover, our result is more general in three ways. First, we demonstrate
that the conjecture holds true not only in constant-dimensional Euclidean
spaces, but also in doubling metrics. Second, we provide a general tradeoff
between the three involved parameters, which is tight in the entire range.
Third, we devise a transformation that decreases the lightness of spanners in
general metrics, while keeping all their other parameters in check. Our main
result is obtained as a corollary of this transformation.Comment: A technical report of this paper was available online from April 4,
201

### On Efficient Distributed Construction of Near Optimal Routing Schemes

Given a distributed network represented by a weighted undirected graph
$G=(V,E)$ on $n$ vertices, and a parameter $k$, we devise a distributed
algorithm that computes a routing scheme in $(n^{1/2+1/k}+D)\cdot n^{o(1)}$
rounds, where $D$ is the hop-diameter of the network. The running time matches
the lower bound of $\tilde{\Omega}(n^{1/2}+D)$ rounds (which holds for any
scheme with polynomial stretch), up to lower order terms. The routing tables
are of size $\tilde{O}(n^{1/k})$, the labels are of size $O(k\log^2n)$, and
every packet is routed on a path suffering stretch at most $4k-5+o(1)$. Our
construction nearly matches the state-of-the-art for routing schemes built in a
centralized sequential manner. The previous best algorithms for building
routing tables in a distributed small messages model were by \cite[STOC
2013]{LP13} and \cite[PODC 2015]{LP15}. The former has similar properties but
suffers from substantially larger routing tables of size $O(n^{1/2+1/k})$,
while the latter has sub-optimal running time of
$\tilde{O}(\min\{(nD)^{1/2}\cdot n^{1/k},n^{2/3+2/(3k)}+D\})$

### Distributed Deterministic Edge Coloring using Bounded Neighborhood Independence

We study the {edge-coloring} problem in the message-passing model of
distributed computing. This is one of the most fundamental and well-studied
problems in this area. Currently, the best-known deterministic algorithms for
(2Delta -1)-edge-coloring requires O(Delta) + log-star n time \cite{PR01},
where Delta is the maximum degree of the input graph. Also, recent results of
\cite{BE10} for vertex-coloring imply that one can get an
O(Delta)-edge-coloring in O(Delta^{epsilon} \cdot \log n) time, and an
O(Delta^{1 + epsilon})-edge-coloring in O(log Delta log n) time, for an
arbitrarily small constant epsilon > 0.
In this paper we devise a drastically faster deterministic edge-coloring
algorithm. Specifically, our algorithm computes an O(Delta)-edge-coloring in
O(Delta^{epsilon}) + log-star n time, and an O(Delta^{1 +
epsilon})-edge-coloring in O(log Delta) + log-star n time. This result improves
the previous state-of-the-art {exponentially} in a wide range of Delta,
specifically, for 2^{Omega(\log-star n)} \leq Delta \leq polylog(n). In
addition, for small values of Delta our deterministic algorithm outperforms all
the existing {randomized} algorithms for this problem.
On our way to these results we study the {vertex-coloring} problem on the
family of graphs with bounded {neighborhood independence}. This is a large
family, which strictly includes line graphs of r-hypergraphs for any r = O(1),
and graphs of bounded growth. We devise a very fast deterministic algorithm for
vertex-coloring graphs with bounded neighborhood independence. This algorithm
directly gives rise to our edge-coloring algorithms, which apply to {general}
graphs.
Our main technical contribution is a subroutine that computes an
O(Delta/p)-defective p-vertex coloring of graphs with bounded neighborhood
independence in O(p^2) + \log-star n time, for a parameter p, 1 \leq p \leq
Delta

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