329 research outputs found
Oblivious channels
Let C = {x_1,...,x_N} \subset {0,1}^n be an [n,N] binary error correcting
code (not necessarily linear). Let e \in {0,1}^n be an error vector. A codeword
x in C is said to be "disturbed" by the error e if the closest codeword to x +
e is no longer x. Let A_e be the subset of codewords in C that are disturbed by
e. In this work we study the size of A_e in random codes C (i.e. codes in which
each codeword x_i is chosen uniformly and independently at random from
{0,1}^n). Using recent results of Vu [Random Structures and Algorithms 20(3)]
on the concentration of non-Lipschitz functions, we show that |A_e| is strongly
concentrated for a wide range of values of N and ||e||.
We apply this result in the study of communication channels we refer to as
"oblivious". Roughly speaking, a channel W(y|x) is said to be oblivious if the
error distribution imposed by the channel is independent of the transmitted
codeword x. For example, the well studied Binary Symmetric Channel is an
oblivious channel.
In this work, we define oblivious and partially oblivious channels and
present lower bounds on their capacity. The oblivious channels we define have
connections to Arbitrarily Varying Channels with state constraints.Comment: Submitted to the IEEE International Symposium on Information Theory
(ISIT) 200
Distributed Broadcasting and Mapping Protocols in Directed Anonymous Networks
We initiate the study of distributed protocols over directed anonymous networks that are not necessarily strongly connected. In such networks, nodes are aware only of their incoming and outgoing edges, have no unique identity, and have no knowledge of the network topology or even bounds on its parameters, like the number of nodes or the network diameter. Anonymous networks are of interest in various settings such as wireless ad-hoc networks and peer to peer networks. Our goal is to create distributed protocols that reduce the uncertainty by distributing the knowledge of the network topology to all the nodes.
We consider two basic protocols: broadcasting and unique label assignment. These two protocols enable a complete mapping of the network and can serve as key building blocks in more advanced protocols. We develop distributed asynchronous protocols as well as derive lower bounds on their communication complexity, total bandwidth complexity, and node label complexity. The resulting lower bounds are sometimes surprisingly high, exhibiting the complexity of topology extraction in directed anonymous networks
Graphs with tiny vector chromatic numbers and huge chromatic numbers
Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246-265] introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly Δ^(1 - 2/k) colors. Here Δ is the maximum degree in the graph and is assumed to be of the order of n^5 for some 0 < δ < 1. Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets. We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than n/Δ^(1- 2/k) (and hence cannot be colored with significantly fewer than Δ^(1-2/k) colors). For k = O(log n/log log n) we show vector k-colorable graphs that do not have independent sets of size (log n)^c, for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylogn.
As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653-750] for this problem
The Capacity of Online (Causal) -ary Error-Erasure Channels
In the -ary online (or "causal") channel coding model, a sender wishes to
communicate a message to a receiver by transmitting a codeword symbol by symbol via a channel
limited to at most errors and/or erasures. The channel is
"online" in the sense that at the th step of communication the channel
decides whether to corrupt the th symbol or not based on its view so far,
i.e., its decision depends only on the transmitted symbols .
This is in contrast to the classical adversarial channel in which the
corruption is chosen by a channel that has a full knowledge on the sent
codeword .
In this work we study the capacity of -ary online channels for a combined
corruption model, in which the channel may impose at most {\em errors} and
at most {\em erasures} on the transmitted codeword. The online
channel (in both the error and erasure case) has seen a number of recent
studies which present both upper and lower bounds on its capacity. In this
work, we give a full characterization of the capacity as a function of ,
and .Comment: This is a new version of the binary case, which can be found at
arXiv:1412.637
Optimal Unviersal Schedules for Discrete Broadcast
In this paper we study the scenario in which a server sends dynamic data over a single broadcast channel to
a number of passive clients. We consider the data to consist of discrete packets, where each update is sent in a
separate packet. On demand, each client listens to the channel in order to obtain the most recent data packet. Such
scenarios arise in many practical applications such as the distribution of weather and traffic updates to wireless
mobile devices and broadcasting stock price information over the Internet.
To satisfy a request, a client must listen to at least one packet from beginning to end. We thus consider the design
of a broadcast schedule which minimizes the time that passes between a clients request and the time that it hears a
new data packet, i.e., the waiting time of the client. Previous studies have addressed this objective, assuming that
client requests are distributed uniformly over time. However, in the general setting, the clients behavior is difficult
to predict and might not be known to the server. In this work we consider the design of universal schedules that
guarantee a short waiting time for any possible client behavior. We define the model of dynamic broadcasting in
the universal setting, and prove various results regarding the waiting time achievable in this framework
Optimal Schedules for Asynchronous Transmission of Discrete Packets
In this paper we study the distribution of dynamic data over a broadcast channel to a large number of
passive clients. Clients obtain the information by accessing the channel and listening for the next available
packet. This scenario, referred to as packet-based or discrete broadcast, has many practical applications such
as the distribution of weather and traffic updates to wireless mobile devices, reconfiguration and reprogramming
of wireless sensors and downloading dynamic task information in battlefield networks.
The optimal broadcast protocols require a high degree of synchronization between the server and the
wireless clients. However, in typical wireless settings such degree of synchronization is difficult to achieve
due to the inaccuracy of internal clocks. Moreover, in some settings, such as military applications, synchronized
transmission is not desirable due to jamming. The lack of synchronization leads to large delays
and excessive power consumption. Accordingly, in this work we focus on the design of optimal broadcast
schedules that are robust to clock inaccuracy. We present universal schedules for delivery of up-to-date
information with minimum waiting time in asynchronous settings
Optimal Universal Schedules for Discrete Broadcast
We study broadcast systems that distribute a series of data updates to a large number of passive clients. The updates are sent over a broadcast channel in the form of discrete packets. We assume that clients periodically access the channel to obtain the most recent update. Such scenarios arise in many practical applications, such as distribution of traffic information and market updates to mobile wireless devices
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