21,488 research outputs found
Energy Complexity of Distance Computation in Multi-hop Networks
Energy efficiency is a critical issue for wireless devices operated under
stringent power constraint (e.g., battery). Following prior works, we measure
the energy cost of a device by its transceiver usage, and define the energy
complexity of an algorithm as the maximum number of time slots a device
transmits or listens, over all devices. In a recent paper of Chang et al. (PODC
2018), it was shown that broadcasting in a multi-hop network of unknown
topology can be done in energy. In this paper, we continue
this line of research, and investigate the energy complexity of other
fundamental graph problems in multi-hop networks. Our results are summarized as
follows.
1. To avoid spending energy, the broadcasting protocols of Chang
et al. (PODC 2018) do not send the message along a BFS tree, and it is open
whether BFS could be computed in energy, for sufficiently large . In
this paper we devise an algorithm that attains energy
cost.
2. We show that the framework of the round lower bound proof
for computing diameter in CONGEST of Abboud et al. (DISC 2017) can be adapted
to give an energy lower bound in the wireless network model
(with no message size constraint), and this lower bound applies to -arboricity graphs. From the upper bound side, we show that the energy
complexity of can be attained for bounded-genus graphs
(which includes planar graphs).
3. Our upper bounds for computing diameter can be extended to other graph
problems. We show that exact global minimum cut or approximate -- minimum
cut can be computed in energy for bounded-genus graphs
Toward a unified interpretation of quark and lepton mixing from flavor and CP symmetries
We discussed the scenario that a discrete flavor group combined with CP
symmetry is broken to in both neutrino and charged lepton
sectors. All lepton mixing angles and CP violation phases are predicted to
depend on two free parameters and varying in the
range of . As an example, we comprehensively study the lepton mixing
patterns which can be derived from the flavor group and CP
symmetry. Three kinds of phenomenologically viable lepton mixing matrices are
obtained up to row and column permutations. We further extend this approach to
the quark sector. The precisely measured quark mixing angles and CP invariant
can be accommodated for certain values of the free parameters and
. A simultaneous description of quark and lepton flavor mixing
structures can be achieved from a common flavor group and CP,
and accordingly the smallest value of the group index is .Comment: 40 pages, 8 figure
A Time Hierarchy Theorem for the LOCAL Model
The celebrated Time Hierarchy Theorem for Turing machines states, informally,
that more problems can be solved given more time. The extent to which a time
hierarchy-type theorem holds in the distributed LOCAL model has been open for
many years. It is consistent with previous results that all natural problems in
the LOCAL model can be classified according to a small constant number of
complexities, such as , etc.
In this paper we establish the first time hierarchy theorem for the LOCAL
model and prove that several gaps exist in the LOCAL time hierarchy.
1. We define an infinite set of simple coloring problems called Hierarchical
-Coloring}. A correctly colored graph can be confirmed by simply
checking the neighborhood of each vertex, so this problem fits into the class
of locally checkable labeling (LCL) problems. However, the complexity of the
-level Hierarchical -Coloring problem is ,
for . The upper and lower bounds hold for both general graphs
and trees, and for both randomized and deterministic algorithms.
2. Consider any LCL problem on bounded degree trees. We prove an
automatic-speedup theorem that states that any randomized -time
algorithm solving the LCL can be transformed into a deterministic -time algorithm. Together with a previous result, this establishes that on
trees, there are no natural deterministic complexities in the ranges
--- or ---.
3. We expose a gap in the randomized time hierarchy on general graphs. Any
randomized algorithm that solves an LCL problem in sublogarithmic time can be
sped up to run in time, which is the complexity of the distributed
Lovasz local lemma problem, currently known to be and
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