50 research outputs found
Certification of Compact Low-Stretch Routing Schemes
On the one hand, the correctness of routing protocols in networks is an issue of utmost importance for guaranteeing the delivery of messages from any source to any target. On the other hand, a large collection of routing schemes have been proposed during the last two decades, with the objective of transmitting messages along short routes, while keeping the routing tables small. Regrettably, all these schemes share the property that an adversary may modify the content of the routing tables with the objective of, e.g., blocking the delivery of messages between some pairs of nodes, without being detected by any node.
In this paper, we present a simple certification mechanism which enables the nodes to locally detect any alteration of their routing tables. In particular, we show how to locally verify the stretch 3 routing scheme by Thorup and Zwick [SPAA 2001] by adding certificates of ~O(sqrt(n)) bits at each node in n-node networks, that is, by keeping the memory size of the same order of magnitude as the original routing tables. We also propose a new name-independent routing scheme using routing tables of size ~O(sqrt(n)) bits. This new routing scheme can be locally verified using certificates on ~O(sqrt(n)) bits. Its stretch is 3 if using handshaking, and 5 otherwise
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
Improved Distributed Fractional Coloring Algorithms
We prove new bounds on the distributed fractional coloring problem in the
LOCAL model. Fractional -colorings can be understood as multicolorings as
follows. For some natural numbers and such that , each node
is assigned a set of at least colors from such that
adjacent nodes are assigned disjoint sets of colors. The minimum for which
a fractional -coloring of a graph exists is called the fractional
chromatic number of .
Recently, [Bousquet, Esperet, and Pirot; SIROCCO '21] showed that for any
constant , a fractional -coloring can be
computed in rounds. We show that
such a coloring can be computed in only rounds, without any
dependency on .
We further show that in rounds, it is
possible to compute a fractional -coloring, even if the
fractional chromatic number is not known. That is, this problem can
be approximated arbitrarily well by an efficient algorithm in the LOCAL model.
For the standard coloring problem, it is only known that an -approximation can be computed in polylogarithmic time in
the LOCAL model. We also show that our distributed fractional coloring
approximation algorithm is best possible. We show that in trees, which have
fractional chromatic number , computing a fractional -coloring
requires at least rounds.
We finally study fractional colorings of regular grids. In [Bousquet,
Esperet, and Pirot; SIROCCO '21], it is shown that in regular grids of bounded
dimension, a fractional -coloring can be computed in time
. We show that such a coloring can even be computed in
rounds in the LOCAL model
Distributed Lower Bounds for Ruling Sets
Given a graph , an -ruling set is a subset such that the distance between any two vertices in is at least
, and the distance between any vertex in and the closest vertex in
is at most . We present lower bounds for distributedly computing
ruling sets.
More precisely, for the problem of computing a -ruling set in the
LOCAL model, we show the following, where denotes the number of vertices,
the maximum degree, and is some universal constant independent of
and .
Any deterministic algorithm requires
rounds, for all . By optimizing , this implies a
deterministic lower bound of for all .
Any randomized algorithm requires rounds, for all . By optimizing
, this implies a randomized lower bound of
for all
.
For , this improves on the previously best lower bound of
rounds that follows from the 30-year-old bounds of Linial
[FOCS'87] and Naor [J.Disc.Math.'91]. For , i.e., for the problem of
computing a maximal independent set, our results improve on the previously best
lower bound of on trees, as our bounds already hold on
trees
A Big Data Analyzer for Large Trace Logs
Current generation of Internet-based services are typically hosted on large
data centers that take the form of warehouse-size structures housing tens of
thousands of servers. Continued availability of a modern data center is the
result of a complex orchestration among many internal and external actors
including computing hardware, multiple layers of intricate software, networking
and storage devices, electrical power and cooling plants. During the course of
their operation, many of these components produce large amounts of data in the
form of event and error logs that are essential not only for identifying and
resolving problems but also for improving data center efficiency and
management. Most of these activities would benefit significantly from data
analytics techniques to exploit hidden statistical patterns and correlations
that may be present in the data. The sheer volume of data to be analyzed makes
uncovering these correlations and patterns a challenging task. This paper
presents BiDAl, a prototype Java tool for log-data analysis that incorporates
several Big Data technologies in order to simplify the task of extracting
information from data traces produced by large clusters and server farms. BiDAl
provides the user with several analysis languages (SQL, R and Hadoop MapReduce)
and storage backends (HDFS and SQLite) that can be freely mixed and matched so
that a custom tool for a specific task can be easily constructed. BiDAl has a
modular architecture so that it can be extended with other backends and
analysis languages in the future. In this paper we present the design of BiDAl
and describe our experience using it to analyze publicly-available traces from
Google data clusters, with the goal of building a realistic model of a complex
data center.Comment: 26 pages, 10 figure