98 research outputs found
Stochastic majorisation: exploding some myths
The analysis of many randomised algorithms involves random variables that are not independent, and hence many of the standard tools from classical probability theory that would be useful in the analysis, such as the Chernoff--Hoeffding bounds are rendered inapplicable. However, in many instances, the random variables involved are, nevertheless {\em negatively related\/} in the intuitive sense that when one of the variables is ``large'', another is likely to be ``small''. (this notion is made precise and analysed in [1].) In such situations, one is tempted to conjecture that these variables are in some sense {\em stochastically dominated\/} by a set of {\em independent\/} random variables with the same marginals. Thereby, one hopes to salvage tools such as the Chernoff--Hoeffding bound also for analysis involving the dependent set of variables. The analysis in [6, 7, 8] seems to strongly hint in this direction. In this note, we explode myths of this kind, and argue that stochastic majorisation in conjunction with an independent set of variables is actually much less useful a notion than it might have appeared
Some correlation inequalities for probabilistic analysis of algorithms
The analysis of many randomized algorithms, for example in dynamic load balancing, probabilistic divide-and-conquer paradigm and distributed edge-coloring, requires ascertaining the precise nature of the correlation between the random variables arising in the following prototypical ``balls-and-bins'' experiment. Suppose a certain number of balls are thrown uniformly and independently at random into bins. Let be the random variable denoting the number of balls in the th bin, . These variables are clearly not independent and are intuitively negatively related. We make this mathematically precise by proving the following type of correlation inequalities: \begin{itemize} \item For index sets such that or , and any non--negative integers , \prob[\sum_{i \in I} X_i \geq t_I \mid \sum_{j \in J} X_j \geq t_J] \-5mm] \[\leq \prob[\sum_{i \in I} X_i \geq t_I] . \item For any disjoint index sets , any and any non--negative integers and , \prob[\bigwedge_{i \in I}X_i \geq t_i \mid \bigwedge_{j \in J} X_j \geq t_j]\-5mm]\[ \leq \prob[\bigwedge_{i \in I'}X_i \geq t_i \mid \bigwedge_{j \in J'} X_j \geq t_j] . \end{itemize} Although these inequalities are intuitively appealing, establishing them is non--trivial; in particular, direct counting arguments become intractable very fast. We prove the inequalities of the first type by an application of the celebrated FKG Correlation Inequality. The proof for the second uses only elementary methods and hinges on some {\em monotonicity} properties. More importantly, we then introduce a general methodology that may be applicable whenever the random variables involved are negatively related. Precisely, we invoke a general notion of {\em negative assocation\/} of random variables and show that: \begin{itemize} \item The variables are negatively associated. This yields most of the previous results in a uniform way. \item For a set of negatively associated variables, one can apply the Chernoff-Hoeffding bounds to the sum of these variables. This provides a tool that facilitates analysis of many randomized algorithms, for example, the ones mentioned above
Searching, sorting and randomised algorithms for central elements and ideal counting in posets
By the Central Element Theorem of Linial and Saks, it follows that for the problem of (generalised) searching in posets, the information--theoretic lower bound of comparisons (where is the number of order--ideals in the poset) is tight asymptotically. We observe that this implies that the problem of (generalised) sorting in posets has complexity (where is the number of elements in the poset). We present schemes for (efficiently) transforming a randomised generation procedure for central elements (which often exists for some classes of posets) into randomised procedures for approximately counting ideals in the poset and for testing if an arbitrary element is central
A lower bound for area-universal graphs
We establish a lower bound on the efficiency of area--universal circuits. The area of every graph that can host any graph of area (at most) with dilation , and congestion satisfies the tradeoff In particular, if then
Pure Exploration in Bandits with Linear Constraints
We address the problem of identifying the optimal policy with a fixed
confidence level in a multi-armed bandit setup, when \emph{the arms are subject
to linear constraints}. Unlike the standard best-arm identification problem
which is well studied, the optimal policy in this case may not be deterministic
and could mix between several arms. This changes the geometry of the problem
which we characterize via an information-theoretic lower bound. We introduce
two asymptotically optimal algorithms for this setting, one based on the
Track-and-Stop method and the other based on a game-theoretic approach. Both
these algorithms try to track an optimal allocation based on the lower bound
and computed by a weighted projection onto the boundary of a normal cone.
Finally, we provide empirical results that validate our bounds and visualize
how constraints change the hardness of the problem
Bidirectional PageRank Estimation: From Average-Case to Worst-Case
We present a new algorithm for estimating the Personalized PageRank (PPR)
between a source and target node on undirected graphs, with sublinear
running-time guarantees over the worst-case choice of source and target nodes.
Our work builds on a recent line of work on bidirectional estimators for PPR,
which obtained sublinear running-time guarantees but in an average-case sense,
for a uniformly random choice of target node. Crucially, we show how the
reversibility of random walks on undirected networks can be exploited to
convert average-case to worst-case guarantees. While past bidirectional methods
combine forward random walks with reverse local pushes, our algorithm combines
forward local pushes with reverse random walks. We also discuss how to modify
our methods to estimate random-walk probabilities for any length distribution,
thereby obtaining fast algorithms for estimating general graph diffusions,
including the heat kernel, on undirected networks.Comment: Workshop on Algorithms and Models for the Web-Graph (WAW) 201
Distributed Symmetry Breaking in Hypergraphs
Fundamental local symmetry breaking problems such as Maximal Independent Set
(MIS) and coloring have been recognized as important by the community, and
studied extensively in (standard) graphs. In particular, fast (i.e.,
logarithmic run time) randomized algorithms are well-established for MIS and
-coloring in both the LOCAL and CONGEST distributed computing
models. On the other hand, comparatively much less is known on the complexity
of distributed symmetry breaking in {\em hypergraphs}. In particular, a key
question is whether a fast (randomized) algorithm for MIS exists for
hypergraphs.
In this paper, we study the distributed complexity of symmetry breaking in
hypergraphs by presenting distributed randomized algorithms for a variety of
fundamental problems under a natural distributed computing model for
hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can
be solved in rounds ( is the number of nodes of the
hypergraph) in the LOCAL model. We then present a key result of this paper ---
an -round hypergraph MIS algorithm in
the CONGEST model where is the maximum node degree of the hypergraph
and is any arbitrarily small constant.
To demonstrate the usefulness of hypergraph MIS, we present applications of
our hypergraph algorithm to solving problems in (standard) graphs. In
particular, the hypergraph MIS yields fast distributed algorithms for the {\em
balanced minimal dominating set} problem (left open in Harris et al. [ICALP
2013]) and the {\em minimal connected dominating set problem}. We also present
distributed algorithms for coloring, maximal matching, and maximal clique in
hypergraphs.Comment: Changes from the previous version: More references adde
Secrecy Results for Compound Wiretap Channels
We derive a lower bound on the secrecy capacity of the compound wiretap
channel with channel state information at the transmitter which matches the
general upper bound on the secrecy capacity of general compound wiretap
channels given by Liang et al. and thus establishing a full coding theorem in
this case. We achieve this with a stronger secrecy criterion and the maximum
error probability criterion, and with a decoder that is robust against the
effect of randomisation in the encoding. This relieves us from the need of
decoding the randomisation parameter which is in general not possible within
this model. Moreover we prove a lower bound on the secrecy capacity of the
compound wiretap channel without channel state information and derive a
multi-letter expression for the capacity in this communication scenario.Comment: 25 pages, 1 figure. Accepted for publication in the journal "Problems
of Information Transmission". Some of the results were presented at the ITW
2011 Paraty [arXiv:1103.0135] and published in the conference paper available
at the IEEE Xplor
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