3,512 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
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
The Distributed Complexity of Locally Checkable Labeling Problems Beyond Paths and Trees
We consider locally checkable labeling LCL problems in the LOCAL model of
distributed computing. Since 2016, there has been a substantial body of work
examining the possible complexities of LCL problems. For example, it has been
established that there are no LCL problems exhibiting deterministic
complexities falling between and . This line of
inquiry has yielded a wealth of algorithmic techniques and insights that are
useful for algorithm designers.
While the complexity landscape of LCL problems on general graphs, trees, and
paths is now well understood, graph classes beyond these three cases remain
largely unexplored. Indeed, recent research trends have shifted towards a
fine-grained study of special instances within the domains of paths and trees.
In this paper, we generalize the line of research on characterizing the
complexity landscape of LCL problems to a much broader range of graph classes.
We propose a conjecture that characterizes the complexity landscape of LCL
problems for an arbitrary class of graphs that is closed under minors, and we
prove a part of the conjecture.
Some highlights of our findings are as follows.
1. We establish a simple characterization of the minor-closed graph classes
sharing the same deterministic complexity landscape as paths, where ,
, and are the only possible complexity classes.
2. It is natural to conjecture that any minor-closed graph class shares the
same complexity landscape as trees if and only if the graph class has bounded
treewidth and unbounded pathwidth. We prove the "only if" part of the
conjecture.
3. In addition to the well-known complexity landscapes for paths, trees, and
general graphs, there are infinitely many different complexity landscapes among
minor-closed graph classes
Ortho-Radial Drawing in Near-Linear Time
An orthogonal drawing is an embedding of a plane graph into a grid. In a seminal work of Tamassia (SIAM Journal on Computing 1987), a simple combinatorial characterization of angle assignments that can be realized as bend-free orthogonal drawings was established, thereby allowing an orthogonal drawing to be described combinatorially by listing the angles of all corners. The characterization reduces the need to consider certain geometric aspects, such as edge lengths and vertex coordinates, and simplifies the task of graph drawing algorithm design.
Barth, Niedermann, Rutter, and Wolf (SoCG 2017) established an analogous combinatorial characterization for ortho-radial drawings, which are a generalization of orthogonal drawings to cylindrical grids. The proof of the characterization is existential and does not result in an efficient algorithm. Niedermann, Rutter, and Wolf (SoCG 2019) later addressed this issue by developing quadratic-time algorithms for both testing the realizability of a given angle assignment as an ortho-radial drawing without bends and constructing such a drawing.
In this paper, we improve the time complexity of these tasks to near-linear time. We establish a new characterization for ortho-radial drawings based on the concept of a good sequence. Using the new characterization, we design a simple greedy algorithm for constructing ortho-radial drawings
Efficient Distributed Decomposition and Routing Algorithms in Minor-Free Networks and Their Applications
In the LOCAL model, low-diameter decomposition is a useful tool in designing
algorithms, as it allows us to shift from the general graph setting to the
low-diameter graph setting, where brute-force information gathering can be done
efficiently. Recently, Chang and Su [PODC 2022] showed that any
high-conductance network excluding a fixed minor contains a high-degree vertex,
so the entire graph topology can be gathered to one vertex efficiently in the
CONGEST model using expander routing. Therefore, in networks excluding a fixed
minor, many problems that can be solved efficiently in LOCAL via low-diameter
decomposition can also be solved efficiently in CONGEST via expander
decomposition.
In this work, we show improved decomposition and routing algorithms for
networks excluding a fixed minor in the CONGEST model. Our algorithms cost
rounds deterministically. For bounded-degree
graphs, our algorithms finish in
rounds.
Our algorithms have a wide range of applications, including the following
results in CONGEST.
1. A -approximate maximum independent set in a network
excluding a fixed minor can be computed deterministically in
rounds, nearly matching the
lower bound of Lenzen and Wattenhofer [DISC
2008].
2. Property testing of any additive minor-closed property can be done
deterministically in rounds if is a constant or
rounds if the maximum degree
is a constant, nearly matching the lower
bound of Levi, Medina, and Ron [PODC 2018].Comment: To appear in PODC 202
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