383 research outputs found
Incremental Medians via Online Bidding
In the k-median problem we are given sets of facilities and customers, and
distances between them. For a given set F of facilities, the cost of serving a
customer u is the minimum distance between u and a facility in F. The goal is
to find a set F of k facilities that minimizes the sum, over all customers, of
their service costs.
Following Mettu and Plaxton, we study the incremental medians problem, where
k is not known in advance, and the algorithm produces a nested sequence of
facility sets where the kth set has size k. The algorithm is c-cost-competitive
if the cost of each set is at most c times the cost of the optimum set of size
k. We give improved incremental algorithms for the metric version: an
8-cost-competitive deterministic algorithm, a 2e ~ 5.44-cost-competitive
randomized algorithm, a (24+epsilon)-cost-competitive, poly-time deterministic
algorithm, and a (6e+epsilon ~ .31)-cost-competitive, poly-time randomized
algorithm.
The algorithm is s-size-competitive if the cost of the kth set is at most the
minimum cost of any set of size k, and has size at most s k. The optimal
size-competitive ratios for this problem are 4 (deterministic) and e
(randomized). We present the first poly-time O(log m)-size-approximation
algorithm for the offline problem and first poly-time O(log m)-size-competitive
algorithm for the incremental problem.
Our proofs reduce incremental medians to the following online bidding
problem: faced with an unknown threshold T, an algorithm submits "bids" until
it submits a bid that is at least the threshold. It pays the sum of all its
bids. We prove that folklore algorithms for online bidding are optimally
competitive.Comment: conference version appeared in LATIN 2006 as "Oblivious Medians via
Online Bidding
A really simple approximation of smallest grammar
In this paper we present a really simple linear-time algorithm constructing a
context-free grammar of size O(g log (N/g)) for the input string, where N is
the size of the input string and g the size of the optimal grammar generating
this string. The algorithm works for arbitrary size alphabets, but the running
time is linear assuming that the alphabet Sigma of the input string can be
identified with numbers from 1,ldots, N^c for some constant c. Algorithms with
such an approximation guarantee and running time are known, however all of them
were non-trivial and their analyses were involved. The here presented algorithm
computes the LZ77 factorisation and transforms it in phases to a grammar. In
each phase it maintains an LZ77-like factorisation of the word with at most l
factors as well as additional O(l) letters, where l was the size of the
original LZ77 factorisation. In one phase in a greedy way (by a left-to-right
sweep and a help of the factorisation) we choose a set of pairs of consecutive
letters to be replaced with new symbols, i.e. nonterminals of the constructed
grammar. We choose at least 2/3 of the letters in the word and there are O(l)
many different pairs among them. Hence there are O(log N) phases, each of them
introduces O(l) nonterminals to a grammar. A more precise analysis yields a
bound O(l log(N/l)). As l \leq g, this yields the desired bound O(g log(N/g)).Comment: Accepted for CPM 201
Solving k-center Clustering (with Outliers) in MapReduce and Streaming, almost as Accurately as Sequentially.
Center-based clustering is a fundamental primitive for data analysis and becomes very challenging for large datasets. In this paper, we focus on the popular k-center variant which, given a set S of points from some metric space and a parameter k0, the algorithms yield solutions whose approximation ratios are a mere additive term \u3f5 away from those achievable by the best known polynomial-time sequential algorithms, a result that substantially improves upon the state of the art. Our algorithms are rather simple and adapt to the intrinsic complexity of the dataset, captured by the doubling dimension D of the metric space. Specifically, our analysis shows that the algorithms become very space-efficient for the important case of small (constant) D. These theoretical results are complemented with a set of experiments on real-world and synthetic datasets of up to over a billion points, which show that our algorithms yield better quality solutions over the state of the art while featuring excellent scalability, and that they also lend themselves to sequential implementations much faster than existing ones
New Bell inequalities for the singlet state: Going beyond the Grothendieck bound
Contemporary versions of Bell's argument against local hidden variable (LHV)
theories are based on the Clauser Horne Shimony and Holt (CHSH) inequality, and
various attempts to generalize it. The amount of violation of these
inequalities cannot exceed the bound set by the Grothendieck constants.
However, if we go back to the original derivation by Bell, and use the perfect
anti-correlation embodied in the singlet spin state, we can go beyond these
bounds. In this paper we derive two-particle Bell inequalities for traceless
two-outcome observables, whose violation in the singlet spin state go beyond
the Grothendieck constants both for the two and three dimensional cases.
Moreover, creating a higher dimensional analog of perfect correlations, and
applying a recent result of Alon and his associates (Invent. Math. 163 499
(2006)) we prove that there are two-particle Bell inequalities for traceless
two-outcome observables whose violation increases to infinity as the dimension
and number of measurements grow. Technically these result are possible because
perfect correlations (or anti-correlations) allow us to transport the indices
of the inequality from the edges of a bipartite graph to those of the complete
graph. Finally, it is shown how to apply these results to mixed Werner states,
provided that the noise does not exceed 20%.Comment: 18 pages, two figures, some corrections and additional references,
published versio
One Table to Count Them All: Parallel Frequency Estimation on Single-Board Computers
Sketches are probabilistic data structures that can provide approximate
results within mathematically proven error bounds while using orders of
magnitude less memory than traditional approaches. They are tailored for
streaming data analysis on architectures even with limited memory such as
single-board computers that are widely exploited for IoT and edge computing.
Since these devices offer multiple cores, with efficient parallel sketching
schemes, they are able to manage high volumes of data streams. However, since
their caches are relatively small, a careful parallelization is required. In
this work, we focus on the frequency estimation problem and evaluate the
performance of a high-end server, a 4-core Raspberry Pi and an 8-core Odroid.
As a sketch, we employed the widely used Count-Min Sketch. To hash the stream
in parallel and in a cache-friendly way, we applied a novel tabulation approach
and rearranged the auxiliary tables into a single one. To parallelize the
process with performance, we modified the workflow and applied a form of
buffering between hash computations and sketch updates. Today, many
single-board computers have heterogeneous processors in which slow and fast
cores are equipped together. To utilize all these cores to their full
potential, we proposed a dynamic load-balancing mechanism which significantly
increased the performance of frequency estimation.Comment: 12 pages, 4 figures, 3 algorithms, 1 table, submitted to EuroPar'1
Mining Top-K Frequent Itemsets Through Progressive Sampling
We study the use of sampling for efficiently mining the top-K frequent
itemsets of cardinality at most w. To this purpose, we define an approximation
to the top-K frequent itemsets to be a family of itemsets which includes
(resp., excludes) all very frequent (resp., very infrequent) itemsets, together
with an estimate of these itemsets' frequencies with a bounded error. Our first
result is an upper bound on the sample size which guarantees that the top-K
frequent itemsets mined from a random sample of that size approximate the
actual top-K frequent itemsets, with probability larger than a specified value.
We show that the upper bound is asymptotically tight when w is constant. Our
main algorithmic contribution is a progressive sampling approach, combined with
suitable stopping conditions, which on appropriate inputs is able to extract
approximate top-K frequent itemsets from samples whose sizes are smaller than
the general upper bound. In order to test the stopping conditions, this
approach maintains the frequency of all itemsets encountered, which is
practical only for small w. However, we show how this problem can be mitigated
by using a variation of Bloom filters. A number of experiments conducted on
both synthetic and real bench- mark datasets show that using samples
substantially smaller than the original dataset (i.e., of size defined by the
upper bound or reached through the progressive sampling approach) enable to
approximate the actual top-K frequent itemsets with accuracy much higher than
what analytically proved.Comment: 16 pages, 2 figures, accepted for presentation at ECML PKDD 2010 and
publication in the ECML PKDD 2010 special issue of the Data Mining and
Knowledge Discovery journa
Maximum gradient embeddings and monotone clustering
Let (X,d_X) be an n-point metric space. We show that there exists a
distribution D over non-contractive embeddings into trees f:X-->T such that for
every x in X, the expectation with respect to D of the maximum over y in X of
the ratio d_T(f(x),f(y)) / d_X(x,y) is at most C (log n)^2, where C is a
universal constant. Conversely we show that the above quadratic dependence on
log n cannot be improved in general. Such embeddings, which we call maximum
gradient embeddings, yield a framework for the design of approximation
algorithms for a wide range of clustering problems with monotone costs,
including fault-tolerant versions of k-median and facility location.Comment: 25 pages, 2 figures. Final version, minor revision of the previous
one. To appear in "Combinatorica
Tree Compression with Top Trees Revisited
We revisit tree compression with top trees (Bille et al, ICALP'13) and
present several improvements to the compressor and its analysis. By
significantly reducing the amount of information stored and guiding the
compression step using a RePair-inspired heuristic, we obtain a fast compressor
achieving good compression ratios, addressing an open problem posed by Bille et
al. We show how, with relatively small overhead, the compressed file can be
converted into an in-memory representation that supports basic navigation
operations in worst-case logarithmic time without decompression. We also show a
much improved worst-case bound on the size of the output of top-tree
compression (answering an open question posed in a talk on this algorithm by
Weimann in 2012).Comment: SEA 201
New Approximability Results for the Robust k-Median Problem
We consider a robust variant of the classical -median problem, introduced
by Anthony et al. \cite{AnthonyGGN10}. In the \emph{Robust -Median problem},
we are given an -vertex metric space and client sets . The objective is to open a set of
facilities such that the worst case connection cost over all client sets is
minimized; in other words, minimize . Anthony
et al.\ showed an approximation algorithm for any metric and
APX-hardness even in the case of uniform metric. In this paper, we show that
their algorithm is nearly tight by providing
approximation hardness, unless . This hardness result holds even for uniform and line
metrics. To our knowledge, this is one of the rare cases in which a problem on
a line metric is hard to approximate to within logarithmic factor. We
complement the hardness result by an experimental evaluation of different
heuristics that shows that very simple heuristics achieve good approximations
for realistic classes of instances.Comment: 19 page
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