On the Hardness of Set Disjointness and Set Intersection with Bounded Universe

In the SetDisjointness problem, a collection of m sets S_1,S_2,...,S_m from some universe U is preprocessed in order to answer queries on the emptiness of the intersection of some two query sets from the collection. In the SetIntersection variant, all the elements in the intersection of the query sets are required to be reported. These are two fundamental problems that were considered in several papers from both the upper bound and lower bound perspective. Several conditional lower bounds for these problems were proven for the tradeoff between preprocessing and query time or the tradeoff between space and query time. Moreover, there are several unconditional hardness results for these problems in some specific computational models. The fundamental nature of the SetDisjointness and SetIntersection problems makes them useful for proving the conditional hardness of other problems from various areas. However, the universe of the elements in the sets may be very large, which may cause the reduction to some other problems to be inefficient and therefore it is not useful for proving their conditional hardness. In this paper, we prove the conditional hardness of SetDisjointness and SetIntersection with bounded universe. This conditional hardness is shown for both the interplay between preprocessing and query time and the interplay between space and query time. Moreover, we present several applications of these new conditional lower bounds. These applications demonstrates the strength of our new conditional lower bounds as they exploit the limited universe size. We believe that this new framework of conditional lower bounds with bounded universe can be useful for further significant applications

Orthogonal Vectors Indexing

In the recent years, intensive research work has been dedicated to prove conditional lower bounds in order to reveal the inner structure of the class P. These conditional lower bounds are based on many popular conjectures on well-studied problems. One of the most heavily used conjectures is the celebrated Strong Exponential Time Hypothesis (SETH). It turns out that conditional hardness proved based on SETH goes, in many cases, through an intermediate problem - the Orthogonal Vectors (OV) problem. Almost all research work regarding conditional lower bound was concentrated on time complexity. Very little attention was directed toward space complexity. In a recent work, Goldstein et al.[WADS \u2717] set the stage for proving conditional lower bounds regarding space and its interplay with time. In this spirit, it is tempting to investigate the space complexity of a data structure variant of OV which is called OV indexing. In this problem n boolean vectors of size clogn are given for preprocessing. As a query, a vector v is given and we are required to verify if there is an input vector that is orthogonal to it or not. This OV indexing problem is interesting in its own, but it also likely to have strong implications on problems known to be conditionally hard, in terms of time complexity, based on OV. Having this in mind, we study OV indexing in this paper from many aspects. We give some space-efficient algorithms for the problem, show a tradeoff between space and query time, describe how to solve its reporting variant, shed light on an interesting connection between this problem and the well-studied SetDisjointness problem and demonstrate how it can be solved more efficiently on random input

Improved Space-Time Tradeoffs for kSUM

In the kSUM problem we are given an array of numbers a_1,a_2,...,a_n and we are required to determine if there are k different elements in this array such that their sum is 0. This problem is a parameterized version of the well-studied SUBSET-SUM problem, and a special case is the 3SUM problem that is extensively used for proving conditional hardness. Several works investigated the interplay between time and space in the context of SUBSET-SUM. Recently, improved time-space tradeoffs were proven for kSUM using both randomized and deterministic algorithms. In this paper we obtain an improvement over the best known results for the time-space tradeoff for kSUM. A major ingredient in achieving these results is a general self-reduction from kSUM to mSUM where m1. (iv) An algorithm for 6SUM running in O(n^4) time using just O(n^{2/3}) space. (v) A solution to 3SUM on random input using O(n^2) time and O(n^{1/3}) space, under the assumption of a random read-only access to random bits

How Hard is it to Find (Honest) Witnesses?

In recent years much effort has been put into developing polynomial-time conditional lower bounds for algorithms and data structures in both static and dynamic settings. Along these lines we introduce a framework for proving conditional lower bounds based on the well-known 3SUM conjecture. Our framework creates a compact representation of an instance of the 3SUM problem using hashing and domain specific encoding. This compact representation admits false solutions to the original 3SUM problem instance which we reveal and eliminate until we find a true solution. In other words, from all witnesses (candidate solutions) we figure out if an honest one (a true solution) exists. This enumeration of witnesses is used to prove conditional lower bounds on reporting problems that generate all witnesses. In turn, these reporting problems are then reduced to various decision problems using special search data structures which are able to enumerate the witnesses while only using solutions to decision variants. Hence, 3SUM-hardness of the decision problems is deduced. We utilize this framework to show conditional lower bounds for several variants of convolutions, matrix multiplication and string problems. Our framework uses a strong connection between all of these problems and the ability to find witnesses. Specifically, we prove conditional lower bounds for computing partial outputs of convolutions and matrix multiplication for sparse inputs. These problems are inspired by the open question raised by Muthukrishnan 20 years ago. The lower bounds we show rule out the possibility (unless the 3SUM conjecture is false) that almost linear time solutions to sparse input-output convolutions or matrix multiplications exist. This is in contrast to standard convolutions and matrix multiplications that have, or assumed to have, almost linear solutions. Moreover, we improve upon the conditional lower bounds of Amir et al. for histogram indexing, a problem that has been of much interest recently. The conditional lower bounds we show apply for both reporting and decision variants. For the well-studied decision variant, we show a full tradeoff between preprocessing and query time for every alphabet size > 2. At an extreme, this implies that no solution to this problem exists with subquadratic preprocessing time and ~O(1) query time for every alphabet size > 2, unless the 3SUM conjecture is false. This is in contrast to a recent result by Chan and Lewenstein for a binary alphabet. While these specific applications are used to demonstrate the techniques of our framework, we believe that this novel framework is useful for many other problems as well

Semiparametric inference of effective reproduction number dynamics from wastewater pathogen surveillance data

Concentrations of pathogen genomes measured in wastewater have recently become available as a new data source to use when modeling the spread of infectious diseases. One promising use for this data source is inference of the effective reproduction number, the average number of individuals a newly infected person will infect. We propose a model where new infections arrive according to a time-varying immigration rate which can be interpreted as a compound parameter equal to the product of the proportion of susceptibles in the population and the transmission rate. This model allows us to estimate the effective reproduction number from concentrations of pathogen genomes while avoiding difficult to verify assumptions about the dynamics of the susceptible population. As a byproduct of our primary goal, we also produce a new model for estimating the effective reproduction number from case data using the same framework. We test this modeling framework in an agent-based simulation study with a realistic data generating mechanism which accounts for the time-varying dynamics of pathogen shedding. Finally, we apply our new model to estimating the effective reproduction number of SARS-CoV-2 in Los Angeles, California, using pathogen RNA concentrations collected from a large wastewater treatment facility.Comment: 23 pages, 6 figures in main te

Fruit load governs transpiration of olive trees

We tested the hypothesis that whole-tree water consumption of olives (Olea europaea L.) is fruit load-dependent and investigated the driving physiological mechanisms. Fruit load was manipulated in mature olives grown in weighing-drainage lysimeters. Fruit was thinned or entirely removed from trees at three separate stages of growth: early, mid and late in the season. Tree-scale transpiration, calculated from lysimeter water balance, was found to be a function of fruit load, canopy size and weather conditions. Fruit removal caused an immediate decline in water consumption, measured as whole-plant transpiration normalized to tree size, which persisted until the end of the season. The later the execution of fruit removal, the greater was the response. The amount of water transpired by a fruit-loaded tree was found to be roughly 30% greater than that of an equivalent low- or nonyielding tree. The tree-scale response to fruit was reflected in stem water potential but was not mirrored in leaf-scale physiological measurements of stomatal conductance or photosynthesis. Trees with low or no fruit load had higher vegetative growth rates. However, no significant difference was observed in the overall aboveground dry biomass among groups, when fruit was included. This case, where carbon sources and sinks were both not limiting, suggests that the role of fruit on water consumption involves signaling and alterations in hydraulic properties of vascular tissues and tree organs.</p

Semi-parametric modeling of SARS-CoV-2 transmission in Orange County, California using tests, cases, deaths, and seroprevalence data

Mechanistic modeling of SARS-CoV-2 transmission dynamics and frequently estimating model parameters using streaming surveillance data are important components of the pandemic response toolbox. However, transmission model parameter estimation can be imprecise, and sometimes even impossible, because surveillance data are noisy and not informative about all aspects of the mechanistic model. To partially overcome this obstacle, we propose a Bayesian modeling framework that integrates multiple surveillance data streams. Our model uses both SARS-CoV-2 diagnostics test and mortality time series to estimate our model parameters, while also explicitly integrating seroprevalence data from cross-sectional studies. Importantly, our data generating model for incidence data takes into account changes in the total number of tests performed. We model transmission rate, infection-to-fatality ratio, and a parameter controlling a functional relationship between the true case incidence and the fraction of positive tests as time-varying quantities and estimate changes of these parameters nonparameterically. We apply our Bayesian data integration method to COVID-19 surveillance data collected in Orange County, California between March, 2020 and March, 2021 and find that 33-62% of the Orange County residents experienced SARS-CoV-2 infection by the end of February, 2021. Despite this high number of infections, our results show that the abrupt end of the winter surge in January, 2021, was due to both behavioral changes and a high level of accumulated natural immunity.Comment: 37 pages, 16 pages of main text, including 5 figures, 1 tabl

Results from the centers for disease control and prevention's predict the 2013-2014 Influenza Season Challenge

Background: Early insights into the timing of the start, peak, and intensity of the influenza season could be useful in planning influenza prevention and control activities. To encourage development and innovation in influenza forecasting, the Centers for Disease Control and Prevention (CDC) organized a challenge to predict the 2013-14 Unites States influenza season. Methods: Challenge contestants were asked to forecast the start, peak, and intensity of the 2013-2014 influenza season at the national level and at any or all Health and Human Services (HHS) region level(s). The challenge ran from December 1, 2013-March 27, 2014; contestants were required to submit 9 biweekly forecasts at the national level to be eligible. The selection of the winner was based on expert evaluation of the methodology used to make the prediction and the accuracy of the prediction as judged against the U.S. Outpatient Influenza-like Illness Surveillance Network (ILINet). Results: Nine teams submitted 13 forecasts for all required milestones. The first forecast was due on December 2, 2013; 3/13 forecasts received correctly predicted the start of the influenza season within one week, 1/13 predicted the peak within 1 week, 3/13 predicted the peak ILINet percentage within 1 %, and 4/13 predicted the season duration within 1 week. For the prediction due on December 19, 2013, the number of forecasts that correctly forecasted the peak week increased to 2/13, the peak percentage to 6/13, and the duration of the season to 6/13. As the season progressed, the forecasts became more stable and were closer to the season milestones. Conclusion: Forecasting has become technically feasible, but further efforts are needed to improve forecast accuracy so that policy makers can reliably use these predictions. CDC and challenge contestants plan to build upon the methods developed during this contest to improve the accuracy of influenza forecasts. © 2016 The Author(s)

Intracellular machinery for the transport of AMPA receptors

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73173/1/sj.bjp.0707525.pd