Almost Optimal Explicit Selectors

Let MA(n,d) denote the maximum number of group tests for a group testing algorithm A to identify d defectives from a set of n items when d is known, and let MA(n∣d) denote the number when d is unknown. Define M(n,d) = minA MA(n,d). An algorithm A is called a competitive algorithm if there exist constants c and a such that for all n > d ≥ 0, MA(n∣d) ≤ cM(n,d) + a. In this paper, we present some competitive group testing algorithms.

In the k-junta testing problem, a tester has to efficiently decide whether a given function f: {0, 1}n → {0, 1} is a k-junta (i.e., depends on at most k of its input bits) or is ∊-far from any k-junta. Our main result is a quantum algorithm for this problem with query complexity and time complexity . This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atıcı and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of Ω(k1/3) queries for junta-testing (for constant ∊).

The goal of group testing is to efficiently identify a few specific items, called positives, in a large population of items via tests. A test is an action on a subset of items that returns positive if the subset contains at least one positive and negative otherwise. In non-adaptive group testing, all tests are independent, can be performed in parallel, and represented as a measurement matrix. In this work, we consider non-adaptive group testing with consecutive positives in which the items are linearly ordered and the positives are consecutive in that order. We present two algorithms for efficiently identifying consecutive positives. In particular, without storing measurement matrices, we can identify up to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d$</tex> consecutive positives with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2 \log_{2}\frac{\mathrm{n}}{d}+2d (4\log_{2}\frac{n}{d}+2d,\ resp.)$</tex> tests in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log_{2}^{2}\frac{n}{d}+d)\ (O(\log_{2}\frac{n}{d}+d)$</tex> , resp.) time. These results significantly improve the state-of-the-art scheme in which it takes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$5 \log_{2}\frac{n}{d} +2d+21$</tex> tests to identify the positives in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\frac{n}{d}\log_{2}\frac{n}{d}+d^{2})$</tex> time with the measurement matrices associated with the scheme stored somewhere.

A non-adaptive group testing (NGT) algorithm is useful in many areas. One of the examples is the problem of DNA library screening. While a mathematical model of NGT algorithm is a d-disjunct matrix. In this paper, we construct a d-disjunct matrix in a simple way. By discussing the Hamming distance of B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> (delta <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">**</sup> (n, d, k)), where delta**(n, d, k) is the matrix by augmenting the matrix delta**(n, d, k) with delta <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sup> (n, alpha, k), 1 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">les</sub> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">alpha</sub> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">les</sub> m + 1 and alpha isin Z, we imply that our construction gives one error-correcting and two error-detecting NGT algorithm.

A cognitive radio (CR) network refers to a secondary network operating in a frequency band originally licensed/allocated to a primary network consisting of one or multiple primary users (PUs). A fundamental challenge for realizing such a system is to ensure the quality of service (QoS) of the PUs as well as to maximize the throughput or ensure the QoS, such as signal-to-interference-plus-noise ratios (SINRs), of the secondary users (SUs). In this paper, we study single-input multiple output multiple access channels (SIMO-MAC) for the CR network. Subject to interference constraints for the PUs as well as peak power constraints for the SUs, two optimization problems involving a joint beamforming and power allocation for the CR network are considered: the sum-rate maximization problem and the SINR balancing problem. For the sum-rate maximization problem, zero-forcing based decision feedback equalizers are used to decouple the SIMO-MAC, and a capped multi-level (CML) water-filling algorithm is proposed to maximize the achievable sum-rate of the SUs for the single PU case. When multiple PUs exist, a recursive decoupled power allocation algorithm is proposed to derive the optimal power allocation solution. For the SINR balancing problem, it is shown that, using linear minimum mean-square-error receivers, each of the interference constraints and peak power constraints can be completely decoupled, and thus the multi-constraint optimization problem can be solved through multiple single-constraint sub-problems. Theoretical analysis for the proposed algorithms is presented, together with numerical simulations which compare the performances of different power allocation schemes.

In this paper we consider a time slotted cognitive radio (CR) network with N wireless channels and M secondary users (SUs). We focus on a random access policy where each SU stochastically decides whether to access idle channels or not based on the given access probability (AP) that is adapted to the channel state information (CSI), if possible. The AP plays an important role in the random access policy because it can control the number of SUs who can access idle channels in a simple manner and hence alleviate packet collisions among SUs. We assume that each SU can access at most L idle channels simultaneously at a time slot whenever possible.

A selection problem is among the basic communication primitives in networks. In this problem at most k participating stations have to broadcast successfully their messages. This problem is especially important in packet radio networks, where simultaneous transmissions of many neighbors result in interference among delivered messages. This work focuses on a single-hop radio networks with n-stations, also called a multiple access channel, and considers both static and dynamic versions of the selection problem. We construct a family of efficient oblivious deterministic protocols based on selectors, one of them with selection time cO(klog(n/k)), and the second explicit construction with selection time cO(k polylog n). The first construction matches the lower bound Ω(klog(n/k)) on deterministic oblivious selection, while the second one is the first known explicit construction better than Θ(k2). In the dynamic case we introduce the model of dynamic requests, called k-streams, which generalizes the static model and the dynamic requests with at most k participants. We prove that each oblivious deterministic protocol has latency Ω(k2/log k), and on the other hand we prove the existence of the oblivious deterministic protocol with latency cO(k2log n). In view of the existence of the randomized oblivious protocol with expected latency cO(klog(n/k)), this shows that randomization is substantially better than determinism for dynamic setting. Selection problem can be applied to implement other communication primitives --- we demonstrate it in the example of broadcast problem in multi-hop ad-hoc radio networks. In particular, we design an adaptive deterministic protocol broadcasting in time cO(nlog D) in every D-hop radio network of n-stations.

We propose two selection protols that run on multiple access channels in log-logarithmic expected time, and establish a complementary lower bound showing that the first protocols falls within an additive constant of optimality and that the second differs from optimality by less than any multiplicative factor infinitesimally greater than 1 as the size of the problem approaches infinity. It is difficult to second-guess the fast-changing electronics industry, but our mathematical analysis could be relevant outside the traditional interests of communications protocols to semaphore-like problems.

Group testing is an efficient algorithmic approach to the infection identification problem, based on mixing the test samples and testing the mixed samples instead of individually testing each sample. In this paper, we consider the dynamic infection spread model that is based on the discrete SIR model, which assumes the disease to be spread over time via infected and non-isolated individuals. In our system, the main objective is not to minimize the number of required tests to identify every infection, but instead, to utilize the available, given testing capacity T at each time instance to efficiently control the infection spread. We introduce and study a novel performance metric, which we coin as ϵ-disease control time. This metric can be used to measure how fast a given algorithm can control the spread of a disease. We characterize the performance of the dynamic individual testing algorithm and introduce a novel dynamic SAFFRON-based group testing algorithm. We present theoretical results and implement the proposed algorithms to compare their performances.

Consider a finite population of N items, where item i has a probability pi to be defective. The goal is to identify all items by means of group testing. This is the generalized group testing problem (hereafter GGTP). In the case of p1 = … = pN = p, Yao and Hwang (1990) proved that the pairwise testing algorithm is the optimal nested algorithm, with respect to the expected number of tests, for all N if and only if (R‐range hereafter) (an optimal at the boundary values). In this note, we present a result that helps to define the generalized pairwise testing algorithm (hereafter GPTA) for the GGTP. We present two conjectures: (a) when all pi,i = 1,…,N belong to the R‐range, GPTA is the optimal procedure among nested procedures applied to pi of nondecreasing order; and (b) if all pi,i = 1,…,N belong to the R‐range, GPTA is the optimal nested procedure, that is, minimizes the expected total number of tests with respect to all possible testing orders in the class of nested procedures. Although these conjectures are logically reasonable, we were only able to empirically verify the first one up to a particular level of N. We also provide a short survey of GGTP.

In this paper, we propose an efficient multi-stage algorithm for non-adaptive Group Testing (GT) with general correlated prior statistics. The proposed solution can be applied to any correlated statistical prior represented in trellis, e.g., finite state machines and Markov processes. We introduce a variation of List Viterbi Algorithm (LVA) to enable accurate recovery using much fewer tests than objectives, which efficiently gains from the correlated prior statistics structure. Our numerical results demonstrate that the proposed Multi-Stage GT (MSGT) algorithm can obtain the optimal Maximum A Posteriori (MAP) performance with feasible complexity in practical regimes, such as with COVID-19 and sparse signal recovery applications, and reduce in the scenarios tested the number of pooled tests by at least 25% compared to existing classical low complexity GT algorithms. Moreover, we analytically characterize the complexity of the proposed MSGT algorithm that guarantees its efficiency.

With the rapid development of mobile Internet and the Internet of Things, the demand for high-speed highquality data applications, such as wireless video streaming, mobile interactive gaming, social networking, and machine-to-machine communications, has been growing dramatically. To meet this rapidly growing demand, there is an urgent need for revolutionary approaches involving new wireless network architectures as well as advanced networking technologies. Heterogeneous networks (HetNets) have attracted significant interest from both academia and industry as a means to potentially improve spatial reuse and coverage, thus allowing cellular systems to achieve higher data rates, while retaining the seamless connectivity and mobility of cellular networks. However, considering the severe inter-tier interference and limited cooperative gains, new techniques for improving both the spectral efficiency and energy efficiency through suppressing inter-tier interference and enhancing cooperative processing capabilities are needed. On the other hand, cloud computing technologies have emerged as a promising solution for providing high energy efficiency together with flexibility through software defined virtualization of communication networks and protocols. Consequently, heterogeneous cloud radio access networks (H-CRANs) that combine HetNets with cloud computing have been recognized as a potential cost-effective solution for alleviating inter-tier interference and improving cooperative processing gains in HetNets. © 2015 IEEE.

In Shannon’s classical model of transmitting a message over a noisy channel we have the following situation:There are two persons called sender and receiver. Sender and receiver can communicate via a channel. In the simplest case the sender just puts some input letters into the channel and the receiver gets some output letters. Usually the channel is noisy, i.e. the channel output is a random variable whose distribution is governed by the input letters. This model can be extended in several ways: Channels with passive feedback for example give the output letters back to the sender. Multiuser channels like multiple access channels or broadcast channels (which will not be considered in this paper) have several senders or receivers which want to communicate simultaneously. Common to all these models of transmission is the task that sender and receiver have to perform: Both have a common message setM and the sender is given a messagei∈M. He has to encode the message (i.e. transform it into a sequence of input letters for the channel) in such a way, that the receiver can decode the sequence of output letters so that he can decide with a small probability of error what the message i was. The procedures for encoding and decoding are called a code for the channel and the number of times the channel is used to transmit one message is called the blocklength of the code.KeywordsConditional EntropyStochastic MatrixFeedback ChannelMultiple Access ChannelInput AlphabetThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Cooperative content caching and power allocation strategy for fog radio access networks

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2013 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Near Optimal Leader Election in Multi-Hop Radio NetworksMohsen Ghaffari and Bernhard HaeuplerMohsen GhaffariMITMITSearch for more papers by this author and Bernhard HaeuplerMITMITSearch for more papers by this authorpp.748 - 766Chapter DOI:https://doi.org/10.1137/1.9781611973105.54PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We design leader election protocols for multi-hop radio networks that elect a leader in almost the same time TBC that it takes for broadcasting one message (one ID). For the setting without collision detection our algorithm runs whp. in rounds on any n-node network with diameter D. Since is a lower bound, our upper bound is optimal up to a factor of at most log log n and the extra log n factor on the additive term. Our algorithm is furthermore the first O(n) time algorithm for this setting. Our algorithm improves over a 23 year old simulation approach of Bar-Yehuda, Goldreich and Itai with a O(TBC log n) running time: In 1987 they designed a fast broadcast protocol and subsequently in 1989 they showed how it can be used to simulate one round of a single-hop network that has collision detection in TBC time. The prime application of this simulation was to simulate Willards single-hop leader election protocol, which elects a leader in O(log n) rounds whp. and O(log log n) rounds in expectation. While it was subsequently shown that Willards bounds are tight, it was unclear whether the simulation approach is optimal. Our results break this barrier and essentially remove the logarithmic slowdown over the broadcast time TBC. This is achieved by going away from the simulation approach. We also give an leader election algorithm for the setting with collision detection (even with single-bit messages). This is optimal up to log log n factors and improves over a deterministic algorithm that requires Θ(n) rounds independently of D. Our almost optimal leader election protocols are especially important because countless communication protocols in radio networks use leader election as a crucial first step to solve various, seemingly unrelated, communication primitives such as gathering, multiple unicasts or multiple broadcasts. Even though leader election seems easier than these tasks, its best-known O(TBC log n) running time had become a bottleneck, preventing optimal algorithms. Breaking the simulation barrier for leader election in this paper has subsequently led to the development of near optimal protocols for these communication primitives. Previous chapter Next chapter RelatedDetails Published:2013ISBN:978-1-61197-251-1eISBN:978-1-61197-310-5 https://doi.org/10.1137/1.9781611973105Book Series Name:ProceedingsBook Code:PR143Book Pages:xix + 1915