61 research outputs found
Every property is testable on a natural class of scale-free multigraphs
In this paper, we introduce a natural class of multigraphs called
hierarchical-scale-free (HSF) multigraphs, and consider constant-time
testability on the class. We show that a very wide subclass, specifically, that
in which the power-law exponent is greater than two, of HSF is hyperfinite.
Based on this result, an algorithm for a deterministic partitioning oracle can
be constructed. We conclude by showing that every property is constant-time
testable on the above subclass of HSF. This algorithm utilizes findings by
Newman and Sohler of STOC'11. However, their algorithm is based on the
bounded-degree model, while it is known that actual scale-free networks usually
include hubs, which have a very large degree. HSF is based on scale-free
properties and includes such hubs. This is the first universal result of
constant-time testability on the general graph model, and it has the potential
to be applicable on a very wide range of scale-free networks.Comment: 13 pages, one figure. Difference from ver. 1: Definitions of HSF and
SF become more general. Typos were fixe
How to solve the cake-cutting problem in sublinear time
In this paper, we show algorithms for solving the cake-cutting problem in
sublinear-time. More specifically, we preassign (simple) fair portions to o(n)
players in o(n)-time, and minimize the damage to the rest of the players. All
currently known algorithms require Omega(n)-time, even when assigning a portion
to just one player, and it is nontrivial to revise these algorithms to run in
-time since many of the remaining players, who have not been asked any
queries, may not be satisfied with the remaining cake. To challenge this
problem, we begin by providing a framework for solving the cake-cutting problem
in sublinear-time. Generally speaking, solving a problem in sublinear-time
requires the use of approximations. However, in our framework, we introduce the
concept of "eps n-victims," which means that eps n players (victims) may not
get fair portions, where 0< eps =< 1 is an arbitrary constant. In our
framework, an algorithm consists of the following two parts: In the first
(Preassigning) part, it distributes fair portions to r < n players in
o(n)-time. In the second (Completion) part, it distributes fair portions to the
remaining n-r players except for the eps n victims in poly}(n)-time. There are
two variations on the r players in the first part. Specifically, whether they
can or cannot be designated. We will then present algorithms in this framework.
In particular, an O(r/eps)-time algorithm for r =< eps n/127 undesignated
players with eps n-victims, and an O~(r^2/eps)-time algorithm for r =< eps
e^{{sqrt{ln{n}}}/{7}} designated players and eps =< 1/e with eps n-victims are
presented.Comment: 15 pages, no figur
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 Detection of Cycles
Distributed property testing in networks has been introduced by Brakerski and
Patt-Shamir (2011), with the objective of detecting the presence of large dense
sub-networks in a distributed manner. Recently, Censor-Hillel et al. (2016)
have shown how to detect 3-cycles in a constant number of rounds by a
distributed algorithm. In a follow up work, Fraigniaud et al. (2016) have shown
how to detect 4-cycles in a constant number of rounds as well. However, the
techniques in these latter works were shown not to generalize to larger cycles
with . In this paper, we completely settle the problem of cycle
detection, by establishing the following result. For every , there
exists a distributed property testing algorithm for -freeness, performing
in a constant number of rounds. All these results hold in the classical CONGEST
model for distributed network computing. Our algorithm is 1-sided error. Its
round-complexity is where is the property
testing parameter measuring the gap between legal and illegal instances
Quick Detection of High-degree Entities in Large Directed Networks
In this paper, we address the problem of quick detection of high-degree
entities in large online social networks. Practical importance of this problem
is attested by a large number of companies that continuously collect and update
statistics about popular entities, usually using the degree of an entity as an
approximation of its popularity. We suggest a simple, efficient, and easy to
implement two-stage randomized algorithm that provides highly accurate
solutions for this problem. For instance, our algorithm needs only one thousand
API requests in order to find the top-100 most followed users in Twitter, a
network with approximately a billion of registered users, with more than 90%
precision. Our algorithm significantly outperforms existing methods and serves
many different purposes, such as finding the most popular users or the most
popular interest groups in social networks. An important contribution of this
work is the analysis of the proposed algorithm using Extreme Value Theory -- a
branch of probability that studies extreme events and properties of largest
order statistics in random samples. Using this theory, we derive an accurate
prediction for the algorithm's performance and show that the number of API
requests for finding the top-k most popular entities is sublinear in the number
of entities. Moreover, we formally show that the high variability among the
entities, expressed through heavy-tailed distributions, is the reason for the
algorithm's efficiency. We quantify this phenomenon in a rigorous mathematical
way
Distributed Testing of Excluded Subgraphs
We study property testing in the context of distributed computing, under the
classical CONGEST model. It is known that testing whether a graph is
triangle-free can be done in a constant number of rounds, where the constant
depends on how far the input graph is from being triangle-free. We show that,
for every connected 4-node graph H, testing whether a graph is H-free can be
done in a constant number of rounds too. The constant also depends on how far
the input graph is from being H-free, and the dependence is identical to the
one in the case of testing triangles. Hence, in particular, testing whether a
graph is K_4-free, and testing whether a graph is C_4-free can be done in a
constant number of rounds (where K_k denotes the k-node clique, and C_k denotes
the k-node cycle). On the other hand, we show that testing K_k-freeness and
C_k-freeness for k>4 appear to be much harder. Specifically, we investigate two
natural types of generic algorithms for testing H-freeness, called DFS tester
and BFS tester. The latter captures the previously known algorithm to test the
presence of triangles, while the former captures our generic algorithm to test
the presence of a 4-node graph pattern H. We prove that both DFS and BFS
testers fail to test K_k-freeness and C_k-freeness in a constant number of
rounds for k>4
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