What is the smallest prime?

What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers. To find the first prime, we must also know what the first positive integer is. Surprisingly, with the definitions used at various times throughout history, one was often not the first positive integer (some started with two, and a few with three). In this article, we survey the history of the primality of one, from the ancient Greeks to modern times. We will discuss some of the reasons definitions changed, and provide several examples. We will also discuss the last significant mathematicians to list the number one as prime.Comment: 11 pages, 5 figure

Parallel and Distributed Algorithms for the Housing Allocation Problem

We give parallel and distributed algorithms for the housing allocation problem. In this problem, there is a set of agents and a set of houses. Each agent has a strict preference list for a subset of houses. We need to find a matching such that some criterion is optimized. One such criterion is Pareto Optimality. A matching is Pareto optimal if no coalition of agents can be strictly better off by exchanging houses among themselves. We also study the housing market problem, a variant of the housing allocation problem, where each agent initially owns a house. In addition to Pareto optimality, we are also interested in finding the core of a housing market. A matching is in the core if there is no coalition of agents that can be better off by breaking away from other agents and switching houses only among themselves. In the first part of this work, we show that computing a Pareto optimal matching of a house allocation is in {\bf CC} and computing the core of a housing market is {\bf CC}-hard. Given a matching, we also show that verifying whether it is in the core can be done in {\bf NC}. We then give an algorithm to show that computing a maximum Pareto optimal matching for the housing allocation problem is in {\bf RNC}^2 and quasi-{\bf NC}^2. In the second part of this work, we present a distributed version of the top trading cycle algorithm for finding the core of a housing market. To that end, we first present two algorithms for finding all the disjoint cycles in a functional graph: a Las Vegas algorithm which terminates in $O(\log l)$ rounds with high probability, where $l$ is the length of the longest cycle, and a deterministic algorithm which terminates in $O(\log^* n \log l)$ rounds, where $n$ is the number of nodes in the graph. Both algorithms work in the synchronous distributed model and use messages of size $O(\log n)$

Toughness and hamiltonicity in kk-trees

We consider toughness conditions that guarantee the existence of a hamiltonian cycle in $k$-trees, a subclass of the class of chordal graphs. By a result of Chen et al.\ 18-tough chordal graphs are hamiltonian, and by a result of Bauer et al.\ there exist nontraceable chordal graphs with toughness arbitrarily close to $\frac{7}{4}$. It is believed that the best possible value of the toughness guaranteeing hamiltonicity of chordal graphs is less than 18, but the proof of Chen et al.\ indicates that proving a better result could be very complicated. We show that every 1-tough 2-tree on at least three vertices is hamiltonian, a best possible result since 1-toughness is a necessary condition for hamiltonicity. We generalize the result to $k$-trees for $k\ge 2$: Let $G$ be a $k$-tree. If $G$ has toughness at least $\frac{k+1}{3},$ then $G$ is hamiltonian. Moreover, we present infinite classes of nonhamiltonian 1-tough $k$-trees for each $k\ge 3

Linearizable Replicated State Machines With Lattice Agreement

This paper studies the lattice agreement problem in asynchronous systems and explores its application to building a linearizable replicated state machine (RSM). First, we propose an algorithm to solve the lattice agreement problem in O(log f) asynchronous rounds, where f is the number of crash failures that the system can tolerate. This is an exponential improvement over the previous best upper bound of O(f). Second, Faleiro et al have shown in [Faleiro et al. PODC, 2012] that combination of conflict-free data types and lattice agreement protocols can be applied to implement a linearizable RSM. They give a Paxos style lattice agreement protocol, which can be adapted to implement a linearizable RSM and guarantee that a command by a client can be learned in at most O(n) message delays, where n is the number of proposers. Later, Xiong et al in [Xiong et al. DISC, 2018] gave a lattice agreement protocol which improves the O(n) message delay guarantee to O(f). However, neither of the protocols is practical for building a linearizable RSM. Thus, in the second part of the paper, we first give an improved protocol based on the one proposed by Xiong et al. Then, we implement a simple linearizable RSM using our improved protocol and compare our implementation with an open source Java implementation of Paxos. Results show that better performance can be obtained by using lattice agreement based protocols to implement a linearizable RSM compared to traditional consensus based protocols

CSWA: Aggregation-Free Spatial-Temporal Community Sensing

In this paper, we present a novel community sensing paradigm -- {C}ommunity {S}ensing {W}ithout {A}ggregation}. CSWA is designed to obtain the environment information (e.g., air pollution or temperature) in each subarea of the target area, without aggregating sensor and location data collected by community members. CSWA operates on top of a secured peer-to-peer network over the community members and proposes a novel \emph{Decentralized Spatial-Temporal Compressive Sensing} framework based on \emph{Parallelized Stochastic Gradient Descent}. Through learning the \emph{low-rank structure} via distributed optimization, CSWA approximates the value of the sensor data in each subarea (both covered and uncovered) for each sensing cycle using the sensor data locally stored in each member's mobile device. Simulation experiments based on real-world datasets demonstrate that CSWA exhibits low approximation error (i.e., less than $0.2 ^\circ$C in city-wide temperature sensing task and $10$ units of PM2.5 index in urban air pollution sensing) and performs comparably to (sometimes better than) state-of-the-art algorithms based on the data aggregation and centralized computation.Comment: This paper has been accepted by AAAI 2018. First two authors are equally contribute

Sub-Nanosecond Time of Flight on Commercial Wi-Fi Cards

Time-of-flight, i.e., the time incurred by a signal to travel from transmitter to receiver, is perhaps the most intuitive way to measure distances using wireless signals. It is used in major positioning systems such as GPS, RADAR, and SONAR. However, attempts at using time-of-flight for indoor localization have failed to deliver acceptable accuracy due to fundamental limitations in measuring time on Wi-Fi and other RF consumer technologies. While the research community has developed alternatives for RF-based indoor localization that do not require time-of-flight, those approaches have their own limitations that hamper their use in practice. In particular, many existing approaches need receivers with large antenna arrays while commercial Wi-Fi nodes have two or three antennas. Other systems require fingerprinting the environment to create signal maps. More fundamentally, none of these methods support indoor positioning between a pair of Wi-Fi devices without~third~party~support. In this paper, we present a set of algorithms that measure the time-of-flight to sub-nanosecond accuracy on commercial Wi-Fi cards. We implement these algorithms and demonstrate a system that achieves accurate device-to-device localization, i.e. enables a pair of Wi-Fi devices to locate each other without any support from the infrastructure, not even the location of the access points.Comment: 14 page

Investigation of the Viscoelastic Effect on Optical- Fiber Sensing and Its Solution for 3D-Printed Sensor Packages

Viscoelasticity is an effect seen in a wide range of materials and it affects the reliability of static measurements made using Fiber Bragg Grating-based sensors, because either the target structure, the adhesive used, or the fiber itself could be viscoelastic. The effect of viscoelasticity on FBG-based sensing has been comprehensively researched through theoretical analysis and simulation using a finite-element approach and a further data processing method to reconstruct the graphical data has been developed. An integrated sensor package comprising of an FBG-based sensor in a polymer host and manufactured by using three-dimensional printing was investigated and examined through tensile testing to validate the approach. The application of the 3D-printed FBG-based sensor package, coupled to the data process method has been explored to monitor the height of a railway pantograph, a critical measurement requirement to monitor elongation, employing a method that can be used in the presence of electromagnetic interference. The results show that the effect of viscoelasticity can be effectively eliminated, and the graphical system response allows results that are sufficiently precise for field use to be generated