32 research outputs found
Recognizing well-parenthesized expressions in the streaming model
Motivated by a concrete problem and with the goal of understanding the sense
in which the complexity of streaming algorithms is related to the complexity of
formal languages, we investigate the problem Dyck(s) of checking matching
parentheses, with different types of parenthesis.
We present a one-pass randomized streaming algorithm for Dyck(2) with space
\Order(\sqrt{n}\log n), time per letter \polylog (n), and one-sided error.
We prove that this one-pass algorithm is optimal, up to a \polylog n factor,
even when two-sided error is allowed. For the lower bound, we prove a direct
sum result on hard instances by following the "information cost" approach, but
with a few twists. Indeed, we play a subtle game between public and private
coins. This mixture between public and private coins results from a balancing
act between the direct sum result and a combinatorial lower bound for the base
case.
Surprisingly, the space requirement shrinks drastically if we have access to
the input stream in reverse. We present a two-pass randomized streaming
algorithm for Dyck(2) with space \Order((\log n)^2), time \polylog (n) and
one-sided error, where the second pass is in the reverse direction. Both
algorithms can be extended to Dyck(s) since this problem is reducible to
Dyck(2) for a suitable notion of reduction in the streaming model.Comment: 20 pages, 5 figure
Quantum Chebyshev's Inequality and Applications
In this paper we provide new quantum algorithms with polynomial speed-up for
a range of problems for which no such results were known, or we improve
previous algorithms. First, we consider the approximation of the frequency
moments of order in the multi-pass streaming model with
updates (turnstile model). We design a -pass quantum streaming algorithm
with memory satisfying a tradeoff of ,
whereas the best classical algorithm requires . Then,
we study the problem of estimating the number of edges and the number
of triangles given query access to an -vertex graph. We describe optimal
quantum algorithms that perform and
queries respectively. This is
a quadratic speed-up compared to the classical complexity of these problems.
For this purpose we develop a new quantum paradigm that we call Quantum
Chebyshev's inequality. Namely we demonstrate that, in a certain model of
quantum sampling, one can approximate with relative error the mean of any
random variable with a number of quantum samples that is linear in the ratio of
the square root of the variance to the mean. Classically the dependency is
quadratic. Our algorithm subsumes a previous result of Montanaro [Mon15]. This
new paradigm is based on a refinement of the Amplitude Estimation algorithm of
Brassard et al. [BHMT02] and of previous quantum algorithms for the mean
estimation problem. We show that this speed-up is optimal, and we identify
another common model of quantum sampling where it cannot be obtained. For our
applications, we also adapt the variable-time amplitude amplification technique
of Ambainis [Amb10] into a variable-time amplitude estimation algorithm.Comment: 27 pages; v3: better presentation, lower bound in Theorem 4.3 is ne
Information Cost Tradeoffs for Augmented Index and Streaming Language Recognition
This paper makes three main contributions to the theory of communication
complexity and stream computation. First, we present new bounds on the
information complexity of AUGMENTED-INDEX. In contrast to analogous results for
INDEX by Jain, Radhakrishnan and Sen [J. ACM, 2009], we have to overcome the
significant technical challenge that protocols for AUGMENTED-INDEX may violate
the "rectangle property" due to the inherent input sharing. Second, we use
these bounds to resolve an open problem of Magniez, Mathieu and Nayak [STOC,
2010] that asked about the multi-pass complexity of recognizing Dyck languages.
This results in a natural separation between the standard multi-pass model and
the multi-pass model that permits reverse passes. Third, we present the first
passive memory checkers that verify the interaction transcripts of priority
queues, stacks, and double-ended queues. We obtain tight upper and lower bounds
for these problems, thereby addressing an important sub-class of the memory
checking framework of Blum et al. [Algorithmica, 1994]
Incidence Geometries and the Pass Complexity of Semi-Streaming Set Cover
Set cover, over a universe of size , may be modelled as a data-streaming
problem, where the sets that comprise the instance are to be read one by
one. A semi-streaming algorithm is allowed only space to process this stream. For each , we give a very
simple deterministic algorithm that makes passes over the input stream and
returns an appropriately certified -approximation to the
optimum set cover. More importantly, we proceed to show that this approximation
factor is essentially tight, by showing that a factor better than
is unachievable for a -pass semi-streaming
algorithm, even allowing randomisation. In particular, this implies that
achieving a -approximation requires
passes, which is tight up to the factor. These results extend to a
relaxation of the set cover problem where we are allowed to leave an
fraction of the universe uncovered: the tight bounds on the best
approximation factor achievable in passes turn out to be
. Our lower bounds are based
on a construction of a family of high-rank incidence geometries, which may be
thought of as vast generalisations of affine planes. This construction, based
on algebraic techniques, appears flexible enough to find other applications and
is therefore interesting in its own right.Comment: 20 page
On the communication complexity of sparse set disjointness and exists-equal problems
In this paper we study the two player randomized communication complexity of
the sparse set disjointness and the exists-equal problems and give matching
lower and upper bounds (up to constant factors) for any number of rounds for
both of these problems. In the sparse set disjointness problem, each player
receives a k-subset of [m] and the goal is to determine whether the sets
intersect. For this problem, we give a protocol that communicates a total of
O(k\log^{(r)}k) bits over r rounds and errs with very small probability. Here
we can take r=\log^{*}k to obtain a O(k) total communication \log^{*}k-round
protocol with exponentially small error probability, improving on the O(k)-bits
O(\log k)-round constant error probability protocol of Hastad and Wigderson
from 1997.
In the exist-equal problem, the players receive vectors x,y\in [t]^n and the
goal is to determine whether there exists a coordinate i such that x_i=y_i.
Namely, the exists-equal problem is the OR of n equality problems. Observe that
exists-equal is an instance of sparse set disjointness with k=n, hence the
protocol above applies here as well, giving an O(n\log^{(r)}n) upper bound. Our
main technical contribution in this paper is a matching lower bound: we show
that when t=\Omega(n), any r-round randomized protocol for the exists-equal
problem with error probability at most 1/3 should have a message of size
\Omega(n\log^{(r)}n). Our lower bound holds even for super-constant r <=
\log^*n, showing that any O(n) bits exists-equal protocol should have \log^*n -
O(1) rounds