609 research outputs found
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
Towards Tight Bounds for the Streaming Set Cover Problem
We consider the classic Set Cover problem in the data stream model. For
elements and sets () we give a -pass algorithm with a
strongly sub-linear space and logarithmic
approximation factor. This yields a significant improvement over the earlier
algorithm of Demaine et al. [DIMV14] that uses exponentially larger number of
passes. We complement this result by showing that the tradeoff between the
number of passes and space exhibited by our algorithm is tight, at least when
the approximation factor is equal to . Specifically, we show that any
algorithm that computes set cover exactly using passes
must use space in the regime of .
Furthermore, we consider the problem in the geometric setting where the
elements are points in and sets are either discs, axis-parallel
rectangles, or fat triangles in the plane, and show that our algorithm (with a
slight modification) uses the optimal space to find a
logarithmic approximation in passes.
Finally, we show that any randomized one-pass algorithm that distinguishes
between covers of size 2 and 3 must use a linear (i.e., ) amount of
space. This is the first result showing that a randomized, approximate
algorithm cannot achieve a space bound that is sublinear in the input size.
This indicates that using multiple passes might be necessary in order to
achieve sub-linear space bounds for this problem while guaranteeing small
approximation factors.Comment: A preliminary version of this paper is to appear in PODS 201
Communication Complexity of Permutation-Invariant Functions
Motivated by the quest for a broader understanding of communication
complexity of simple functions, we introduce the class of
"permutation-invariant" functions. A partial function is permutation-invariant if for every bijection
and every , it is the case that . Most of the commonly studied functions
in communication complexity are permutation-invariant. For such functions, we
present a simple complexity measure (computable in time polynomial in given
an implicit description of ) that describes their communication complexity
up to polynomial factors and up to an additive error that is logarithmic in the
input size. This gives a coarse taxonomy of the communication complexity of
simple functions. Our work highlights the role of the well-known lower bounds
of functions such as 'Set-Disjointness' and 'Indexing', while complementing
them with the relatively lesser-known upper bounds for 'Gap-Inner-Product'
(from the sketching literature) and 'Sparse-Gap-Inner-Product' (from the recent
work of Canonne et al. [ITCS 2015]). We also present consequences to the study
of communication complexity with imperfectly shared randomness where we show
that for total permutation-invariant functions, imperfectly shared randomness
results in only a polynomial blow-up in communication complexity after an
additive overhead
Distributed Minimum Cut Approximation
We study the problem of computing approximate minimum edge cuts by
distributed algorithms. We use a standard synchronous message passing model
where in each round, bits can be transmitted over each edge (a.k.a.
the CONGEST model). We present a distributed algorithm that, for any weighted
graph and any , with high probability finds a cut of size
at most in
rounds, where is the size of the minimum cut. This algorithm is based
on a simple approach for analyzing random edge sampling, which we call the
random layering technique. In addition, we also present another distributed
algorithm, which is based on a centralized algorithm due to Matula [SODA '93],
that with high probability computes a cut of size at most
in rounds for any .
The time complexities of both of these algorithms almost match the
lower bound of Das Sarma et al. [STOC '11], thus
leading to an answer to an open question raised by Elkin [SIGACT-News '04] and
Das Sarma et al. [STOC '11].
Furthermore, we also strengthen the lower bound of Das Sarma et al. by
extending it to unweighted graphs. We show that the same lower bound also holds
for unweighted multigraphs (or equivalently for weighted graphs in which
bits can be transmitted in each round over an edge of weight ),
even if the diameter is . For unweighted simple graphs, we show
that even for networks of diameter , finding an -approximate minimum cut
in networks of edge connectivity or computing an
-approximation of the edge connectivity requires rounds
A Lower Bound for Sampling Disjoint Sets
Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}
Cell-probe Lower Bounds for Dynamic Problems via a New Communication Model
In this paper, we develop a new communication model to prove a data structure
lower bound for the dynamic interval union problem. The problem is to maintain
a multiset of intervals over with integer coordinates,
supporting the following operations:
- insert(a, b): add an interval to , provided that
and are integers in ;
- delete(a, b): delete a (previously inserted) interval from
;
- query(): return the total length of the union of all intervals in
.
It is related to the two-dimensional case of Klee's measure problem. We prove
that there is a distribution over sequences of operations with
insertions and deletions, and queries, for which any data
structure with any constant error probability requires time
in expectation. Interestingly, we use the sparse set disjointness protocol of
H\aa{}stad and Wigderson [ToC'07] to speed up a reduction from a new kind of
nondeterministic communication games, for which we prove lower bounds.
For applications, we prove lower bounds for several dynamic graph problems by
reducing them from dynamic interval union
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
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