1,845 research outputs found
Dependent Random Graphs And Multi-Party Pointer Jumping
We initiate a study of a relaxed version of the standard Erdos-Renyi random graph model, where each edge may depend on a few other edges. We call such graphs dependent random graphs . Our main result in this direction is a thorough understanding of the clique number of dependent random graphs. We also obtain bounds for the chromatic number. Surprisingly, many of the standard properties of random graphs also hold in this relaxed setting. We show that with high probability, a dependent random graph will contain a clique of size ((1-o(1))log(n))/log(1/p), and the chromatic number will be at most (nlog(1/(1-p)))/log(n). We expect these results to be of independent interest. As an application and second main result, we give a new communication protocol for the k-player Multi-Party Pointer Jumping problem (MPJk) in the number-on-the-forehead (NOF) model. Multi-Party Pointer Jumping is one of the canonical NOF communication problems, yet even for three players, its communication complexity is not well understood. Our protocol for MPJ3 costs O((n * log(log(n)))/log(n)) communication, improving on a bound from [BrodyChakrabarti08]. We extend our protocol to the non-Boolean pointer jumping problem, achieving an upper bound which is o(n) for any k \u3e= 4 players. This is the first o(n) protocol and improves on a bound of Damm, Jukna, and Sgall, which has stood for almost twenty years
Sublinear Communication Protocols for Multi-Party Pointer Jumping and a Related Lower Bound
We study the one-way number-on-the-forehead (NOF) communication complexity of
the -layer pointer jumping problem with vertices per layer. This classic
problem, which has connections to many aspects of complexity theory, has seen a
recent burst of research activity, seemingly preparing the ground for an
lower bound, for constant . Our first result is a surprising
sublinear -- i.e., -- upper bound for the problem that holds for , dashing hopes for such a lower bound. A closer look at the protocol
achieving the upper bound shows that all but one of the players involved are
collapsing, i.e., their messages depend only on the composition of the layers
ahead of them. We consider protocols for the pointer jumping problem where all
players are collapsing. Our second result shows that a strong
lower bound does hold in this case. Our third result is another upper bound
showing that nontrivial protocols for (a non-Boolean version of) pointer
jumping are possible even when all players are collapsing. Our lower bound
result uses a novel proof technique, different from those of earlier lower
bounds that had an information-theoretic flavor. We hope this is useful in
further study of the problem
On total communication complexity of collapsing protocols for pointer jumping problem
This paper focuses on bounding the total communication complexity of
collapsing protocols for multiparty pointer jumping problem (). Brody
and Chakrabati in \cite{bc08} proved that in such setting one of the players
must communicate at least bits. Liang in \cite{liang} has
shown protocol matching this lower bound on maximum complexity. His protocol,
however, was behaving worse than the trivial one in terms of total complexity
(number of bits sent by all players). He conjectured that achieving total
complexity better then the trivial one is impossible. In this paper we prove
this conjecture. Namely, we show that for a collapsing protocol for ,
the total communication complexity is at least which closes the gap
between lower and upper bound for total complexity of in collapsing
setting
Dependent Random Graphs and Multiparty Pointer Jumping
We initiate a study of a relaxed version of the standard Erdos-Renyi random
graph model, where each edge may depend on a few other edges. We call such
graphs "dependent random graphs". Our main result in this direction is a
thorough understanding of the clique number of dependent random graphs. We also
obtain bounds for the chromatic number. Surprisingly, many of the standard
properties of random graphs also hold in this relaxed setting. We show that
with high probability, a dependent random graph will contain a clique of size
, and the chromatic number will be at most
.
As an application and second main result, we give a new communication
protocol for the k-player Multiparty Pointer Jumping (MPJ_k) problem in the
number-on-the-forehead (NOF) model. Multiparty Pointer Jumping is one of the
canonical NOF communication problems, yet even for three players, its
communication complexity is not well understood. Our protocol for MPJ_3 costs
communication, improving on a bound of Brody
and Chakrabarti [BC08]. We extend our protocol to the non-Boolean pointer
jumping problem , achieving an upper bound which is o(n) for
any players. This is the first o(n) bound for and
improves on a bound of Damm, Jukna, and Sgall [DJS98] which has stood for
almost twenty years.Comment: 18 page
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
Some Communication Complexity Results and their Applications
Communication Complexity represents one of the premier techniques for proving lower bounds in theoretical computer science. Lower bounds on communication problems can be leveraged to prove lower bounds in several different areas. In this work, we study three different communication complexity problems. The lower bounds for these problems have applications in circuit complexity, wireless sensor networks, and streaming algorithms. First, we study the multiparty pointer jumping problem. We present the first nontrivial upper bound for this problem. We also provide a suite of strong lower bounds under several restricted classes of protocols. Next, we initiate the study of several non-monotone functions in the distributed functional monitoring setting and provide several lower bounds. In particular, we give a generic adversarial technique and show that when deletions are allowed, no nontrivial protocol is possible. Finally, we study the Gap-Hamming-Distance problem and give tight lower bounds for protocols that use a constant number of messages. As a result, we take a well-known lower bound for one-pass streaming algorithms for a host of problems and extend it so it applies to streaming algorithms that use a constant number of passes
Robust lower bounds for communication and stream computation
We study the communication complexity of evaluating functions when the input data is randomly allocated (according to some known distribution) amongst two or more players, possibly with information overlap. This naturally extends previously studied variable partition models such as the best-case and worst-case partition models. We aim to understand whether the hardness of a communication problem holds for almost every allocation of the input, as opposed to holding for perhaps just a few atypical partitions.
A key application is to the heavily studied data stream model. There is a strong connection between our communication lower bounds and lower bounds in the data stream model that are “robust” to the ordering of the data. That is, we prove lower bounds for when the order of the items in the stream is chosen not adversarially but rather uniformly (or near-uniformly) from the set of all permutations. This random-order data stream model has attracted recent interest, since lower bounds here give stronger evidence for the inherent hardness of streaming problems.
Our results include the first random-partition communication lower bounds for problems including multi-party set disjointness and gap-Hamming-distance. Both are tight. We also extend and improve previous results for a form of pointer jumping that is relevant to the problem of selection (in particular, median finding). Collectively, these results yield lower bounds for a variety of problems in the random-order data stream model, including estimating the number of distinct elements, approximating frequency moments, and quantile estimation.
A short version of this article is available in the Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC'08), ACM, pp. 641-650. Compared to the conference presentation, this version considerably expands the detail of the discussion and in the proofs, and substantially changes some of the proof techniques
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