191 research outputs found
Finding the Median (Obliviously) with Bounded Space
We prove that any oblivious algorithm using space to find the median of a
list of integers from requires time . This bound also applies to the problem of determining whether the median
is odd or even. It is nearly optimal since Chan, following Munro and Raman, has
shown that there is a (randomized) selection algorithm using only
registers, each of which can store an input value or -bit counter,
that makes only passes over the input. The bound also implies
a size lower bound for read-once branching programs computing the low order bit
of the median and implies the analog of for length oblivious branching programs
The White-Box Adversarial Data Stream Model
We study streaming algorithms in the white-box adversarial model, where the
stream is chosen adaptively by an adversary who observes the entire internal
state of the algorithm at each time step. We show that nontrivial algorithms
are still possible. We first give a randomized algorithm for the -heavy
hitters problem that outperforms the optimal deterministic Misra-Gries
algorithm on long streams. If the white-box adversary is computationally
bounded, we use cryptographic techniques to reduce the memory of our
-heavy hitters algorithm even further and to design a number of additional
algorithms for graph, string, and linear algebra problems. The existence of
such algorithms is surprising, as the streaming algorithm does not even have a
secret key in this model, i.e., its state is entirely known to the adversary.
One algorithm we design is for estimating the number of distinct elements in a
stream with insertions and deletions achieving a multiplicative approximation
and sublinear space; such an algorithm is impossible for deterministic
algorithms.
We also give a general technique that translates any two-player deterministic
communication lower bound to a lower bound for {\it randomized} algorithms
robust to a white-box adversary. In particular, our results show that for all
, there exists a constant such that any -approximation
algorithm for moment estimation in insertion-only streams with a
white-box adversary requires space for a universe of size .
Similarly, there is a constant such that any -approximation algorithm
in an insertion-only stream for matrix rank requires space with a
white-box adversary. Our algorithmic results based on cryptography thus show a
separation between computationally bounded and unbounded adversaries.
(Abstract shortened to meet arXiv limits.)Comment: PODS 202
Source apportionment of carbonaceous chemical species to fossil fuel combustion, biomass burning and biogenic emissions by a coupled radiocarbon-levoglucosan marker method
An intensive aerosol measurement and sample collection campaign was conducted in central Budapest in a mild winter for 2 weeks. The online instruments included an FDMS-TEOM, RT-OC/EC analyser, DMPS, gas pollutant analysers and meteorological sensors. The aerosol samples were collected on quartz fibre filters by a low-volume sampler using the tandem filter method. Elemental carbon (EC), organic carbon (OC), levoglucosan, mannosan, galactosan, arabitol and mannitol were determined, and radiocarbon analysis was performed on the aerosol samples. Median atmospheric concentrations of EC, OC and PM2.5 mass were 0.97, 4.9 and 25 mu g m(-3), respectively. The EC and organic matter (1.6 x OC) accounted for 4.8 and 37 %, respectively, of the PM2.5 mass. Fossil fuel (FF) combustion represented 36% of the total carbon (TC = EC + OC) in the PM2.5 size fraction. Biomass burning (BB) was a major source (40 %) for the OC in the PM2.5 size fraction, and a substantial source (11 %) for the PM10 mass. We propose and apply here a novel, straightforward, coupled radiocarbon-levoglucosan marker method for source apportionment of the major carbonaceous chemical species. The contributions of EC and OC from FF combustion (ECFF and OCFF to the TC were 11.0 and 25 %, respectively, EC and OC from BB (ECBB and OCBB were responsible for 5.8 and 34 %, respectively, of the TC, while the OC from biogenic sources (OCBIO made up 24% of the TC. The overall relative uncertainty of the OCBIO and OCBB contributions was assessed to be up to 30 %, while the relative uncertainty for the other apportioned species is expected to be below 20 %. Evaluation of the apportioned atmospheric concentrations revealed some of their important properties and relationships among them. ECFF and OCFF were associated with different FF combustion sources. Most ECFF was emitted by vehicular road traffic, while the contribution of non-vehicular sources such as domestic and industrial heating or cooking using gas, oil or coal to OCFF was substantial. The mean contribution of BB to EC particles was smaller by a factor of approximately 2 than that of road traffic. The main formation processes of OCFF, OCBB and OCBIO from volatile organic compounds were jointly influenced by a common factor, which is most likely the atmospheric photochemistry, while primary organic emissions can also be important. Technological improvements and control measures for various BB appliances, together with efficient education and training of their users, in particular on the admissible fuel types, offer an important potential for improving the air quality in Budapest, and likely in other cities as well
Picture-Hanging Puzzles
We show how to hang a picture by wrapping rope around n nails, making a
polynomial number of twists, such that the picture falls whenever any k out of
the n nails get removed, and the picture remains hanging when fewer than k
nails get removed. This construction makes for some fun mathematical magic
performances. More generally, we characterize the possible Boolean functions
characterizing when the picture falls in terms of which nails get removed as
all monotone Boolean functions. This construction requires an exponential
number of twists in the worst case, but exponential complexity is almost always
necessary for general functions.Comment: 18 pages, 8 figures, 11 puzzles. Journal version of FUN 2012 pape
The early evolution of the H-free process
The H-free process, for some fixed graph H, is the random graph process
defined by starting with an empty graph on n vertices and then adding edges one
at a time, chosen uniformly at random subject to the constraint that no H
subgraph is formed. Let G be the random maximal H-free graph obtained at the
end of the process. When H is strictly 2-balanced, we show that for some c>0,
with high probability as , the minimum degree in G is at least
. This gives new lower bounds for
the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite
graphs with . When H is a complete graph with we show that for some C>0, with high probability the independence number of
G is at most . This gives new lower bounds
for Ramsey numbers R(s,t) for fixed and t large. We also obtain new
bounds for the independence number of G for other graphs H, including the case
when H is a cycle. Our proofs use the differential equations method for random
graph processes to analyse the evolution of the process, and give further
information about the structure of the graphs obtained, including asymptotic
formulae for a broad class of subgraph extension variables.Comment: 36 page
Bounds for graph regularity and removal lemmas
We show, for any positive integer k, that there exists a graph in which any
equitable partition of its vertices into k parts has at least ck^2/\log^* k
pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute
constants. This bound is tight up to the constant c and addresses a question of
Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma.
In order to gain some control over irregular pairs, another regularity lemma,
known as the strong regularity lemma, was developed by Alon, Fischer,
Krivelevich, and Szegedy. For this lemma, we prove a lower bound of
wowzer-type, which is one level higher in the Ackermann hierarchy than the
tower function, on the number of parts in the strong regularity lemma,
essentially matching the upper bound. On the other hand, for the induced graph
removal lemma, the standard application of the strong regularity lemma, we find
a different proof which yields a tower-type bound.
We also discuss bounds on several related regularity lemmas, including the
weak regularity lemma of Frieze and Kannan and the recently established regular
approximation theorem. In particular, we show that a weak partition with
approximation parameter \epsilon may require as many as
2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and
solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page
Computational Indistinguishability between Quantum States and Its Cryptographic Application
We introduce a computational problem of distinguishing between two specific
quantum states as a new cryptographic problem to design a quantum cryptographic
scheme that is "secure" against any polynomial-time quantum adversary. Our
problem, QSCDff, is to distinguish between two types of random coset states
with a hidden permutation over the symmetric group of finite degree. This
naturally generalizes the commonly-used distinction problem between two
probability distributions in computational cryptography. As our major
contribution, we show that QSCDff has three properties of cryptographic
interest: (i) QSCDff has a trapdoor; (ii) the average-case hardness of QSCDff
coincides with its worst-case hardness; and (iii) QSCDff is computationally at
least as hard as the graph automorphism problem in the worst case. These
cryptographic properties enable us to construct a quantum public-key
cryptosystem, which is likely to withstand any chosen plaintext attack of a
polynomial-time quantum adversary. We further discuss a generalization of
QSCDff, called QSCDcyc, and introduce a multi-bit encryption scheme that relies
on similar cryptographic properties of QSCDcyc.Comment: 24 pages, 2 figures. We improved presentation, and added more detail
proofs and follow-up of recent wor
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