3,492 research outputs found
Planar Matching in Streams Revisited
We present data stream algorithms for estimating the size or weight of the maximum matching in low arboricity graphs. A large body of work has focused on improving the constant approximation factor for general graphs when the data stream algorithm is permitted O(n polylog n) space where n is the number of nodes. This space is necessary if the algorithm must return the matching. Recently, Esfandiari et al. (SODA 2015) showed that it was possible to estimate the maximum cardinality of a matching in a planar graph up to a factor of 24+epsilon using O(epsilon^{-2} n^{2/3} polylog n) space. We first present an algorithm (with a simple analysis) that improves this to a factor 5+epsilon using the same space. We also improve upon the previous results for other graphs with bounded arboricity. We then present a factor 12.5 approximation for matching in planar graphs that can be implemented using O(log n) space in the adjacency list data stream model where the stream is a concatenation of the adjacency lists of the graph. The main idea behind our results is finding "local" fractional matchings, i.e., fractional matchings where the value of any edge e is solely determined by the edges sharing an endpoint with e. Our work also improves upon the results for the dynamic data stream model where the stream consists of a sequence of edges being inserted and deleted from the graph. We also extend our results to weighted graphs, improving over the bounds given by Bury and Schwiegelshohn (ESA 2015), via a reduction to the unweighted problem that increases the approximation by at most a factor of two
Almost-Smooth Histograms and Sliding-Window Graph Algorithms
We study algorithms for the sliding-window model, an important variant of the
data-stream model, in which the goal is to compute some function of a
fixed-length suffix of the stream. We extend the smooth-histogram framework of
Braverman and Ostrovsky (FOCS 2007) to almost-smooth functions, which includes
all subadditive functions. Specifically, we show that if a subadditive function
can be -approximated in the insertion-only streaming model, then
it can be -approximated also in the sliding-window model with
space complexity larger by factor , where is the
window size.
We demonstrate how our framework yields new approximation algorithms with
relatively little effort for a variety of problems that do not admit the
smooth-histogram technique. For example, in the frequency-vector model, a
symmetric norm is subadditive and thus we obtain a sliding-window
-approximation algorithm for it. Another example is for streaming
matrices, where we derive a new sliding-window
-approximation algorithm for Schatten -norm. We then
consider graph streams and show that many graph problems are subadditive,
including maximum submodular matching, minimum vertex-cover, and maximum
-cover, thereby deriving sliding-window -approximation algorithms for
them almost for free (using known insertion-only algorithms). Finally, we
design for every an artificial function, based on the
maximum-matching size, whose almost-smoothness parameter is exactly
A Simple, Space-Efficient, Streaming Algorithm for Matchings in Low Arboricity Graphs
We present a simple single-pass data stream algorithm using O((log n)/eps^2) space that returns an (alpha + 2)(1 + eps) approximation to the size of the maximum matching in a graph of arboricity alpha
Scalable and Energy-Efficient Millimeter Massive MIMO Architectures: Reflect-Array and Transmit-Array Antennas
Hybrid analog-digital architectures are considered as promising candidates
for implementing millimeter wave (mmWave) massive multiple-input
multiple-output (MIMO) systems since they enable a considerable reduction of
the required number of costly radio frequency (RF) chains by moving some of the
signal processing operations into the analog domain. However, the analog feed
network, comprising RF dividers, combiners, phase shifters, and line
connections, of hybrid MIMO architectures is not scalable due to its
prohibitively high power consumption for large numbers of transmit antennas.
Motivated by this limitation, in this paper, we study novel massive MIMO
architectures, namely reflect-array (RA) and transmit-array (TA) antennas. We
show that the precoders for RA and TA antennas have to meet different
constraints compared to those for conventional MIMO architectures. Taking these
constraints into account and exploiting the sparsity of mmWave channels, we
design an efficient precoder for RA and TA antennas based on the orthogonal
matching pursuit algorithm. Furthermore, in order to fairly compare the
performance of RA and TA antennas with conventional fully-digital and hybrid
MIMO architectures, we develop a unified power consumption model. Our
simulation results show that unlike conventional MIMO architectures, RA and TA
antennas are highly energy efficient and fully scalable in terms of the number
of transmit antennas.Comment: submitted to IEEE ICC 201
An Estimator for Matching Size in Low Arboricity Graphs with Two Applications
In this paper, we present a new simple degree-based estimator for the size of
maximum matching in bounded arboricity graphs. When the arboricity of the graph
is bounded by , the estimator gives a factor approximation
of the matching size. For planar graphs, we show the estimator does better and
returns a approximation of the matching size.
Using this estimator, we get new results for approximating the matching size
of planar graphs in the streaming and distributed models of computation. In
particular, in the vertex-arrival streams, we get a randomized
space algorithm for approximating the
matching size within factor in a planar graph on vertices.
Similarly, we get a simultaneous protocol in the vertex-partition model for
approximating the matching size within factor using
communication from each player.
In comparison with the previous estimators, the estimator in this paper does
not need to know the arboricity of the input graph and improves the
approximation factor for the case of planar graphs
Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling
A distance labeling scheme is an assignment of bit-labels to the vertices of
an undirected, unweighted graph such that the distance between any pair of
vertices can be decoded solely from their labels. An important class of
distance labeling schemes is that of hub labelings, where a node
stores its distance to the so-called hubs , chosen so that for
any there is belonging to some shortest
path. Notice that for most existing graph classes, the best distance labelling
constructions existing use at some point a hub labeling scheme at least as a
key building block. Our interest lies in hub labelings of sparse graphs, i.e.,
those with , for which we show a lowerbound of
for the average size of the hubsets.
Additionally, we show a hub-labeling construction for sparse graphs of average
size for some , where is the
so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced
matchings in dense graphs. This implies that further improving the lower bound
on hub labeling size to would require a
breakthrough in the study of lower bounds on , which have resisted
substantial improvement in the last 70 years. For general distance labeling of
sparse graphs, we show a lowerbound of , where is the communication complexity of the
Sum-Index problem over . Our results suggest that the best achievable
hub-label size and distance-label size in sparse graphs may be
for some
Streaming Kernelization
Kernelization is a formalization of preprocessing for combinatorially hard
problems. We modify the standard definition for kernelization, which allows any
polynomial-time algorithm for the preprocessing, by requiring instead that the
preprocessing runs in a streaming setting and uses
bits of memory on instances . We obtain
several results in this new setting, depending on the number of passes over the
input that such a streaming kernelization is allowed to make. Edge Dominating
Set turns out as an interesting example because it has no single-pass
kernelization but two passes over the input suffice to match the bounds of the
best standard kernelization
- …