35,123 research outputs found
Optimal paths on the road network as directed polymers
We analyze the statistics of the shortest and fastest paths on the road
network between randomly sampled end points. To a good approximation, these
optimal paths are found to be directed in that their lengths (at large scales)
are linearly proportional to the absolute distance between them. This motivates
comparisons to universal features of directed polymers in random media. There
are similarities in scalings of fluctuations in length/time and transverse
wanderings, but also important distinctions in the scaling exponents, likely
due to long-range correlations in geographic and man-made features. At short
scales the optimal paths are not directed due to circuitous excursions governed
by a fat-tailed (power-law) probability distribution.Comment: 5 pages, 7 figure
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201
Relations between automata and the simple k-path problem
Let be a directed graph on vertices. Given an integer , the
SIMPLE -PATH problem asks whether there exists a simple -path in . In
case is weighted, the MIN-WT SIMPLE -PATH problem asks for a simple
-path in of minimal weight. The fastest currently known deterministic
algorithm for MIN-WT SIMPLE -PATH by Fomin, Lokshtanov and Saurabh runs in
time for graphs with integer weights in
the range . This is also the best currently known deterministic
algorithm for SIMPLE k-PATH- where the running time is the same without the
factor. We define to be the set of words of
length whose symbols are all distinct. We show that an explicit
construction of a non-deterministic automaton (NFA) of size for implies an algorithm of running time for MIN-WT SIMPLE -PATH when the weights are
non-negative or the constructed NFA is acyclic as a directed graph. We show
that the algorithm of Kneis et al. and its derandomization by Chen et al. for
SIMPLE -PATH can be used to construct an acylic NFA for of size
.
We show, on the other hand, that any NFA for must be size at least
. We thus propose closing this gap and determining the smallest NFA for
as an interesting open problem that might lead to faster algorithms
for MIN-WT SIMPLE -PATH.
We use a relation between SIMPLE -PATH and non-deterministic xor automata
(NXA) to give another direction for a deterministic algorithm with running time
for SIMPLE -PATH
Towards a complexity theory for the congested clique
The congested clique model of distributed computing has been receiving
attention as a model for densely connected distributed systems. While there has
been significant progress on the side of upper bounds, we have very little in
terms of lower bounds for the congested clique; indeed, it is now know that
proving explicit congested clique lower bounds is as difficult as proving
circuit lower bounds.
In this work, we use various more traditional complexity-theoretic tools to
build a clearer picture of the complexity landscape of the congested clique:
-- Nondeterminism and beyond: We introduce the nondeterministic congested
clique model (analogous to NP) and show that there is a natural canonical
problem family that captures all problems solvable in constant time with
nondeterministic algorithms. We further generalise these notions by introducing
the constant-round decision hierarchy (analogous to the polynomial hierarchy).
-- Non-constructive lower bounds: We lift the prior non-uniform counting
arguments to a general technique for proving non-constructive uniform lower
bounds for the congested clique. In particular, we prove a time hierarchy
theorem for the congested clique, showing that there are decision problems of
essentially all complexities, both in the deterministic and nondeterministic
settings.
-- Fine-grained complexity: We map out relationships between various natural
problems in the congested clique model, arguing that a reduction-based
complexity theory currently gives us a fairly good picture of the complexity
landscape of the congested clique
One More Weight is Enough: Toward the Optimal Traffic Engineering with OSPF
Traffic Engineering (TE) leverages information of network traffic to generate
a routing scheme optimizing the traffic distribution so as to advance network
performance. However, optimize the link weights for OSPF to the offered traffic
is an known NP-hard problem. In this paper, motivated by the fairness concept
of congestion control, we firstly propose a new generic objective function,
where various interests of providers can be extracted with different parameter
settings. And then, we model the optimal TE as the utility maximization of
multi-commodity flows with the generic objective function and theoretically
show that any given set of optimal routes corresponding to a particular
objective function can be converted to shortest paths with respect to a set of
positive link weights. This can be directly configured on OSPF-based protocols.
On these bases, we employ the Network Entropy Maximization(NEM) framework and
develop a new OSPF-based routing protocol, SPEF, to realize a flexible way to
split traffic over shortest paths in a distributed fashion. Actually, comparing
to OSPF, SPEF only needs one more weight for each link and provably achieves
optimal TE. Numerical experiments have been done to compare SPEF with the
current version of OSPF, showing the effectiveness of SPEF in terms of link
utilization and network load distribution
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