Approximate Counting Research Articles

Approximate Nearest Neighbor for Curves: Simple, Efficient, and Deterministic

In the \((1+{\varepsilon },r)\)-approximate near-neighbor problem for curves (ANNC) under some similarity measure \(\delta \), the goal is to construct a data structure for a given set \(\mathcal {C}\) of curves that supports approximate near-neighbor queries: Given a query curve Q, if there exists a curve \(C\in \mathcal {C}\) such that \(\delta (Q,C)\le r\), then return a curve \(C'\in \mathcal {C}\) with \(\delta (Q,C')\le (1+{\varepsilon })r\). There exists an efficient reduction from the \((1+{\varepsilon })\)-approximate nearest-neighbor problem to ANNC, where in the former problem the answer to a query is a curve \(C\in \mathcal {C}\) with \(\delta (Q,C)\le (1+{\varepsilon })\cdot \delta (Q,C^*)\), where \(C^*\) is the curve of \(\mathcal {C}\) most similar to Q. Given a set \(\mathcal {C}\) of n curves, each consisting of m points in d dimensions, we construct a data structure for ANNC that uses \(n\cdot O(\frac{1}{{\varepsilon }})^{md}\) storage space and has O(md) query time (for a query curve of length m), where the similarity measure between two curves is their discrete Fréchet or dynamic time warping distance. Our method is simple to implement, deterministic, and results in an exponential improvement in both query time and storage space compared to all previous bounds. Further, we also consider the asymmetric version of ANNC, where the length of the query curves is \(k \ll m\), and obtain essentially the same storage and query bounds as above, except that m is replaced by k. Finally, we apply our method to a version of approximate range counting for curves and achieve similar bounds.

Counting the Answers to a Query

Counting the answers to a query is a fundamental problem in databases, with several applications in the evaluation, optimization, and visualization of queries. Unfortunately, counting query answers is a #P-hard problem in most cases, so it is unlikely to be solvable in polynomial time. Recently, new results on approximate counting have been developed, specifically by showing that some problems in automata theory admit fully polynomial-time randomized approximation schemes. These results have several implications for the problem of counting the answers to a query; in particular, for graph and conjunctive queries. In this work, we present the main ideas of these approximation results, by using labeled DAGs instead of automata to simplify the presentation. In addition, we review how to apply these results to count query answers in different areas of databases.

Quantum approximate counting for Markov chains and collision counting

In this paper we show how to generalize the quantum approximate counting technique developed by Brassard, H{\o}yer and Tapp [ICALP 1998] to a more general setting: estimating the number of marked states of a Markov chain (a Markov chain can be seen as a random walk over a graph with weighted edges). This makes it possible to construct quantum approximate counting algorithms from quantum search algorithms based on the powerful ``quantum walk based search'' framework established by Magniez, Nayak, Roland and Santha [SIAM Journal on Computing 2011]. As an application, we apply this approach to the quantum element distinctness algorithm by Ambainis [SIAM Journal on Computing 2007]: for two injective functions over a set of $N$ elements, we obtain a quantum algorithm that estimates the number $m$ of collisions of the two functions within relative error $\epsilon$ by making $\tilde{O}\left(\frac{1}{\epsilon^{25/24}}\big(\frac{N}{\sqrt{m}}\big)^{2/3}\right)$ queries, which gives an improvement over the $\Theta\big(\frac{1}{\epsilon}\frac{N}{\sqrt{m}}\big)$-query classical algorithm based on random sampling when $m\ll N$.

Parallel Five-cycle Counting Algorithms

Counting the frequency of subgraphs in large networks is a classic research question that reveals the underlying substructures of these networks for important applications. However, subgraph counting is a challenging problem, even for subgraph sizes as small as five, due to the combinatorial explosion in the number of possible occurrences. This article focuses on the five-cycle, which is an important special case of five-vertex subgraph counting and one of the most difficult to count efficiently. We design two new parallel five-cycle counting algorithms and prove that they are work efficient and achieve polylogarithmic span. Both algorithms are based on computing low out-degree orientations, which enables the efficient computation of directed two-paths and three-paths, and the algorithms differ in the ways in which they use this orientation to eliminate double-counting. Additionally, we present new parallel algorithms for obtaining unbiased estimates of five-cycle counts using graph sparsification. We develop fast multicore implementations of the algorithms and propose a work scheduling optimization to improve their performance. Our experiments on a variety of real-world graphs using a 36-core machine with two-way hyper-threading show that our best exact parallel algorithm achieves 10–46× self-relative speedup, outperforms our serial benchmarks by 10–32×, and outperforms the previous state-of-the-art serial algorithm by up to 818×. Our best approximate algorithm, for a reasonable probability parameter, achieves up to 20× self-relative speedup and is able to approximate five-cycle counts 9–189× faster than our best exact algorithm, with between 0.52% and 11.77% error.

ACM: Accuracy-Aware Collaborative Monitoring for Software-Defined Network-Wide Measurement.

Software-defined measurement (SDM) is a simple and efficient way to deploy measurement tasks and collect measurement data. With SDM, it is convenient for operators to implement fine-grained network-wide measurements at the flow level, from which many important functions can benefit. The prior work provides mechanisms to distribute flows to monitors, such that each monitor can identify its non-overlapped subset of flows to measure, and a certain global performance criterion is optimized, such as load balance or flow coverage. Many applications of network management can benefit from a function that can find large flows efficiently, such as congestion control by dynamically scheduling large flows, caching of forwarding table entries, and network capacity planning. However, the current network-wide measurements neglect the diversity of different flows as they treat large flows and small flows equally. In this paper, we present a mechanism of accuracy-aware collaborative monitoring (ACM) to improve the measurement accuracies of large flows in network-wide measurements at the flow level. The structure of the sketch is an approximate counting algorithm, and a high-measurement accuracy can be achieved by merging the results from multiple monitors with sketches, which is termed as collaborative monitoring. The core idea of our method is to allocate more monitors to large flows and achieve the load balance to provide accuracy-aware monitoring. We modeled our problem as an integer–linear programming problem, which is NP-hard. Thus, we propose an approximation algorithm, named the improved longest processing time algorithm (iLPTA); we proved that its approximation ratio is . We propose a two-stage online distribution algorithm (TODA). Moreover, we proved that its approximation ratio is . The iLPTA is an offline approximation algorithm used to assign monitors for each flow, which prove the validity and feasibility of the core idea. The TODA is an online algorithm that attempts to achieve the load balance by selecting the monitor with the smallest load to a large flow. Our extensional experiment results verify the effectiveness of our proposed algorithms.

Data structures for categorical path counting queries

Consider an ordinal tree T on n nodes, each of which is assigned a category from an alphabet [σ]={1,2,…,σ}. We preprocess the tree T in order to support categorical path counting queries, which ask for the number of distinct categories occurring on the path in T between two query nodes x and y. For this problem, we propose a linear-space data structure with query time O(nlg⁡lg⁡σlg⁡w), where w=Ω(lg⁡n) is the word size in the word-RAM. As shown in our proof, from the assumption that matrix multiplication cannot be solved in time polynomially faster than cubic (with only combinatorial methods), our result is optimal, save for polylogarithmic speed-ups. For a trade-off parameter 1≤t≤n, we propose an O(n+n2t2)-word, O(tlg⁡lg⁡σlg⁡w) query time data structure. We also consider c-approximate categorical path counting queries, which must return an approximation to the number of distinct categories occurring on the query path, by counting each such category at least once and at most c times. We describe a linear-space data structure that supports 2-approximate categorical path counting queries in O(lg⁡n/lg⁡lg⁡n) time.Next, we generalize the categorical path counting queries to weighted trees. Here, a query specifies two nodes x,y and an orthogonal range Q. The answer to thus formed categorical path range counting query is the number of distinct categories occurring on the path from x to y, if only the nodes with weights falling inside Q are considered. We propose an O(nlg⁡lg⁡n+(n/t)4)-word data structure with O(tlg⁡lg⁡n) query time, or an O(n+(n/t)4)-word data structure with O(tlgϵ⁡n) query time. For an appropriate choice of the trade-off parameter t, this implies a linear-space data structure with O(n3/4lgϵ⁡n) query time. We then extend the approach to the trees weighted with vectors from [n]d, where d is a constant integer greater than or equal to 2. We present a data structure with O(nlgd−1+ϵ⁡n+(n/t)2d+2) words of space and O(tlgd−1⁡n(lg⁡lg⁡n)d−2) query time. For an O(n⋅polylogn)-space solution, one thus has O(n2d+12d+2⋅polylogn) query time.The inherent difficulty revealed by the lower bound we proved motivated us to consider data structures based on sketching. In unweighted trees, we propose a sketching data structure to solve the approximate categorical path counting problem which asks for a (1±ϵ)-approximation (i.e. within 1±ϵ of the true answer) of the number of distinct categories on the given path, with probability 1−δ, where 0<ϵ,δ<1 are constants. The data structure occupies O(n+ntlg⁡n) words of space, for the query time of O(tlg⁡n). For trees weighted with d-dimensional weight vectors (d≥1), we propose a data structure with O((n+ntlg⁡n)lgd⁡n) words of space and O(tlgd+1⁡n) query time.All these problems generalize the corresponding categorical range counting problems in Euclidean space Rd+1, for respective d, by replacing one of the dimensions with a tree topology.

Combining bounded solving and controllable randomization for approximate model counting

ABSTRACT Propositional model counting is the problem of computing the satisfying assignments count of a given CNF formula. Due to the weak solving ability of the existing exact model counters in large-scale problems, approximate model counting has been proposed as a practical alternative to exact model counting. Most of the best current approaches are based on XOR constraints for approximate model counting, also known as the hashing-based method. In this paper, a new approximate model counter using short XOR constraints is proposed, which integrates bounded solving and controllable randomisation. By constantly adding short XOR constants, the solution space has been reduced to a smaller one. However, a smaller scale of solution space may result in worse result accuracy. Therefore, by limiting the scale of solution space that applies XOR constraints, bounded solving effectively increases the accuracy of a model counter. Controllable randomisation makes use of backbone variables and the constraint level among variables, such that the short XOR constraints can reach the same reduction effect on solution space as the long ones. It improves the quality of XOR constraints as well as the SAT solving efficiency. Experimentally, we demonstrate that in almost every benchmark used in well-known ApproxMC2 and STAC_CNF, our approach outperforms the existing approximate model counters in both accuracy and efficiency.

Approximate counting of standard set-valued tableaux

We present a randomized algorithm for generating standard set-valued tableaux by extending the Green-Nijenhuis-Wilf hook walk algorithm. In the case of asymptotically rank two partitions, we use this algorithm to give a fully polynomial almost uniform sampler (FPAUS) for standard set-valued tableaux. This FPAUS is then used to construct a fully polynomial randomized approximation scheme (FPRAS) for counting the number of standard set-valued tableaux for such shapes. We also construct a FPAUS and FPRAS for standard set-valued tableaux when either the size of the partition or the difference between the maximum value and the size of the partition is fixed. Our methods build on the work of Jerrum-Valiant-Vazirani and provide a framework for constructing FPAUS's and FPRAS's for other counting problems in algebraic combinatorics.

Approximate counting and NP search problems

We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory [Formula: see text] of [E. Jeřábek, Approximate counting by hashing in bounded arithmetic, J. Symb. Log. 74(3) (2009) 829–860]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. We give a purely computational characterization of this class and show that, relative to an oracle, it does not contain the problem CPLS, a strengthening of PLS. As CPLS is provably total in the theory [Formula: see text], this shows that [Formula: see text] does not prove every [Formula: see text] sentence which is provable in bounded arithmetic. This answers the question posed in [S. Buss, L. A. Kołodziejczyk and N. Thapen, Fragments of approximate counting, J. Symb. Log. 79(2) (2014) 496–525] and represents some progress in the program of separating the levels of the bounded arithmetic hierarchy by low-complexity sentences. Our main technical tool is an extension of the “fixing lemma” from [P. Pudlák and N. Thapen, Random resolution refutations, Comput. Complexity, 28(2) (2019) 185–239], a form of switching lemma, which we use to show that a random partial oracle from a certain distribution will, with high probability, determine an entire computation of a [Formula: see text] oracle machine. The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic.

Spatiotemporal nonhomogeneous poisson model with a seasonal component applied to the analysis of extreme rainfall

This paper develops an extension of spatiotemporal models that handle count data using nonhomogeneous Poisson processes. In this new proposal, we incorporate a seasonal cycle component in the definition of the intensity function to control possible effects produced by the occurrence of the event of interest in regular periods. The seasonal cycle can cause problems in estimating the shape parameter of the Weibull and generalized Goel intensity functions. This shape parameter serves to confront the research hypothesis that seeks to identify a trend in the occurrence rate of an event of interest. In the case of the Weibull intensity function, a value significantly equal to one of the shape parameters indicates a constant rate of occurrence, less than one indicates a decreasing rate, and greater than one indicates an increasing rate. In the case of the Goel intensity function, parameter values less than or equal to one indicate a decreasing occurrence rate, and values greater than one indicate the presence of a change point. We also built a spatial model using the Musa-Okumoto intensity function as an alternative to approximate counting processes for which there is a decreasing trend in the occurrence rate of the event of interest. We estimated the parameters of the proposed method from a Bayesian perspective. Finally, we fitted the proposed model and compared it with other approximations to analyze the frequency of extreme rainfall in the northern region of the states of Maranhão and Piauí in northeastern Brazil over ten years. Among the main results, we found that (1) the proposed method has proven superior in terms of fit and prediction performance than the other models, and (2) unlike other approximations, the proposed model does not detect changes in the rate of extreme rainfall occurrences.

Zero-freeness and approximation of real Boolean Holant problems

Holant problems provide a novel framework to study the complexity of counting problems. It is a refinement to counting constraint satisfaction problems (#CSP) with a more explicit role for the constraint functions. Both graph homomorphisms and #CSP can be viewed as special cases of Holant problems. For approximation algorithms on Holant problems, the attention is focused on proving zero-freeness of the partition functions and establishing fully polynomial-time approximation schemes (FPTAS) using the Taylor expansion method. In this paper, we study the Holant problems defined by a real constraint function satisfying a generalized second-order recurrence. We present fully polynomial-time (deterministic or randomized) approximation schemes for the Holant problems except for a couple of cases. Our algorithms are established in two ways: 1) we construct holographic transformations from the Holant problems to the Ising model, and obtain the algorithms using the approaches with respect to the Ising model; 2) we prove the zero-freeness of the Holant problems, and present an algorithm based on the Taylor expansion method. In addition, for most of the other cases, there exist approximation-preserving reductions between the Holant problems and the problem of counting perfect matchings, which is a central open problem in approximate counting.

The complexity of approximating the complex-valued Potts model

We study the complexity of approximating the partition function ofthe q-state Potts model and the closely related Tutte polynomialfor complex values of the underlying parameters. Apart from theclassical connections with quantum computing and phase transitionsin statistical physics, recent work in approximate counting hasshown that the behaviour in the complex plane, and more preciselythe location of zeros, is strongly connected with the complexity ofthe approximation problem, even for positive real-valued parameters.Previous work in the complex plane by Goldberg and Guo focused onq = 2, which corresponds to the case of the Ising model; for q > 2,the behaviour in the complex plane is not as well understood andmost work applies only to the real-valued Tutte plane. Our mainresult is a complete classification of the complexity of theapproximation problems for all non-real values of the parameters, byestablishing #P-hardness results that apply even when restricted toplanar graphs. Our techniques apply to all q geq 2 and furthercomplement/refine previous results both for the Ising model and theTutte plane, answering in particular a question raised by Bordewich,Freedman, Lovász and Welsh in the context of quantumcomputations.

DCTNets: Deep crowd transfer networks for an approximate crowd counting

Due to the numerous real-world applications of the crowd counting job, it has become a popular research topic. Modern crowd counting systems have a sophisticated structure and employ a filter on a big image size, making them difficult to use. Because these technologies are computationally intensive and difficult to implement in small surveillance systems, they are not appropriate for use in small surveillance systems. They also function poorly in a variety of sizes and densities, as well. Transfer learning and deep convolutional neural network architecture are used to create a modest but efficient network, which we describe herein. We named the proposed crowd counting architecture deep crowd transfer network (DCTNets) since it incorporates both deep learning and transfer learning principles into a single system. Among DCTNets’ key components are a detection module that is based on mask R-CNNs and an estimate module that is based on deep convolutional neural networks. In the first step, we apply transfer learning to the Mask R-CNN model using the datasets ShanghaiTech, JHU-CROWD++, and UCF-QNRF. After that, we train and evaluate the complete architecture on these datasets using the transfer learning results. Input images are sent through a Mask R-CNN, which counts individuals and segments the counted region, then through an estimation network, which estimates the population size, and finally through a merge of the outputs from the two models. According to the findings of comparative tests, the proposed model outperforms existing state-of-the-art approaches on the ShanghaiTech, JHU-CROWD++, and UCF-QNRF datasets.

Log-concave polynomials, I: Entropy and a deterministic approximation algorithm for counting bases of matroids

We give a deterministic polynomial-time 2O(r)-approximation algorithm for the number of bases of a given matroid of rank r and the number of common bases of any two matroids of rank r. To the best of our knowledge, this is the first nontrivial deterministic approximation algorithm that works for arbitrary matroids. Based on a lower bound of Azar, Broder, and Frieze, this is almost the best possible result assuming oracle access to independent sets of the matroid. There are two main ingredients in our result. For the first, we build upon recent results of Adiprasito, Huh, Katz, and Wang on combinatorial Hodge theory to show that the basis generating polynomial of any matroid is a (completely) log-concave polynomial. Formally, we prove that the multivariate generating polynomial of the bases of any matroid is (and all of its directional derivatives along the positive orthant are) log-concave as functions over the positive orthant. For the second ingredient, we develop a general framework for approximate counting in discrete problems, based on convex optimization. The connection goes through subadditivity of the entropy. For matroids, we prove that an approximate superadditivity of the entropy holds by relying on the log-concavity of the basis generating polynomial.

#NFA Admits an FPRAS: Efficient Enumeration, Counting, and Uniform Generation for Logspace Classes

In this work, we study two simple yet general complexity classes, based on logspace Turing machines, that provide a unifying framework for efficient query evaluation in areas such as information extraction and graph databases, among others. We investigate the complexity of three fundamental algorithmic problems for these classes: enumeration, counting, and uniform generation of solutions, and show that they have several desirable properties in this respect. Both complexity classes are defined in terms of non-deterministic logspace transducers (NL-transducers). For the first class, we consider the case of unambiguous NL-transducers, and we prove constant delay enumeration and both counting and uniform generation of solutions in polynomial time. For the second class, we consider unrestricted NL-transducers, and we obtain polynomial delay enumeration, approximate counting in polynomial time, and polynomial-time randomized algorithms for uniform generation. More specifically, we show that each problem in this second class admits a fully polynomial-time randomized approximation scheme (FPRAS) and a polynomial-time Las Vegas algorithm (with preprocessing) for uniform generation. Remarkably, the key idea to prove these results is to show that the fundamental problem # NFA admits an FPRAS, where # NFA is the problem of counting the number of strings of length n (given in unary) accepted by a non-deterministic finite automaton (NFA). While this problem is known to be P-complete and, more precisely, SpanL -complete, it was open whether this problem admits an FPRAS. In this work, we solve this open problem and obtain as a welcome corollary that every function in SpanL admits an FPRAS.

Fast Sampling and Counting k -SAT Solutions in the Local Lemma Regime

We give new algorithms based on Markov chains to sample and approximately count satisfying assignments to k -uniform CNF formulas where each variable appears at most d times. For any k and d satisfying kd &lt; n o(1) and k ≥ 20 log k + 20 log d + 60, the new sampling algorithm runs in close to linear time, and the counting algorithm runs in close to quadratic time. Our approach is inspired by Moitra (JACM, 2019), which remarkably utilizes the Lovász local lemma in approximate counting. Our main technical contribution is to use the local lemma to bypass the connectivity barrier in traditional Markov chain approaches, which makes the well-developed MCMC method applicable on disconnected state spaces such as SAT solutions. The benefit of our approach is to avoid the enumeration of local structures and obtain fixed polynomial running times, even if k = ω (1) or d = ω (1).

Faster exponential-time algorithms for approximately counting independent sets

Counting the independent sets of a graph is a classical #P-complete problem, even in the bipartite case. We give an exponential-time approximation scheme for this problem which is faster than the best known algorithm for the exact problem. The running time of our algorithm on general graphs with error tolerance ε is at most O(20.2680n) times a polynomial in 1/ε. On bipartite graphs, the exponential term in the running time is improved to O(20.2372n). Our methods combine techniques from exact exponential algorithms with techniques from approximate counting. Along the way we generalise (to the multivariate case) the FPTAS of Sinclair, Srivastava, Štefankovič and Yin for approximating the hard-core partition function on graphs with bounded connective constant. Also, we obtain an FPTAS for counting independent sets on graphs with no vertices with degree at least 6 whose neighbours' degrees sum to 27 or more. By a result of Sly, there is no FPTAS that applies to all graphs with maximum degree 6 unless P=NP.

Changes in Tornado Climatology Accompanying the Enhanced Fujita Scale

AbstractUnited States tornado records form the basis for a variety of meteorological, climatological and disaster-risk analyses, but how reliable are they in light of changing standards for rating, as with the 2007 transition of Fujita (F) to Enhanced Fujita (EF) damage scales? To what extent are recorded tornado metrics subject to such influences that may be nonmeteorological in nature? While addressing these questions with utmost thoroughness is too large of a task for any one study, and may not be possible given the many variables and uncertainties involved, some variables that are recorded in large samples are ripe for new examination. We assess basic tornado-path characteristics—damage rating, length, width, and occurrence time, as well as some combined and derived measures—for a 24-yr period of constant path-width recording standard that also coincides with National Weather Service modernization and the WSR-88D deployment era. The middle of that period (in both time and approximate tornado counts) crosses the official switch from F to EF. At least minor shifts in all assessed path variables are associated directly with that change, contrary to the intent of EF implementation. Major and essentially stepwise expansion of tornadic path widths occurred immediately upon EF usage, and widths have expanded still further within the EF era. We also document lesser increases in path lengths, and in tornadoes rated at least EF1 compared to EF0. These apparently secular changes in the tornado data can impact research dependent on bulk tornado-path characteristics and damage-assessment results.

Fast Algorithms for General Spin Systems on Bipartite Expanders

A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs. In this work, we consider arbitrary spin systems on bipartite expander Δ-regular graphs, including the canonical class of bipartite random Δ-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Roughly, this guarantees that the spin system is in the so-called low-temperature regime. Our approach generalises the techniques of Jenssen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to “bicliques” of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in Õ( n 2 ) time, where n is the size of the graph.

50 Years Ago in TheJournalofPediatrics: To Every Season Turner, Turner, Turner

Hsu LY, Hirschhorn K. Unusual Turner mosaicism (45,X/47,XXX; 45,X/46,XXqi; 45,X/46,XXr): detection through linear deceleration from normal linear growth or secondary amenorrhea. J Pediatr 1971;79;2:276-81. It is said that the “post-human genome-sequencing project era” of genomic medicine is like no other. New genes, genomic mechanisms, and associated human phenotypes are being deciphered at a staggering pace. In fact, we are walking a worn path, laid by pioneers of the medical cytogenetics era like Drs Hsu and Hirschhorn. The current number of human chromosomes were not accurately identified until 1956.1Tjio J.H. Levan A. The chromosome number of man.Hereditas. 1956; 42: 1-6Crossref Scopus (566) Google Scholar A short 3 years later, the association between 45,X and Turner syndrome was recognized and, along with it, the first report of sex chromosome mosaicism.2Ford C.E. Jones K.W. Polani P.E. De Almeida J.C. Briggs J.H. A sex-chromosome anomaly in a case of gonadal dysgenesis (Turner’s syndrome).Lancet. 1959; 276: 711-712Abstract Scopus (531) Google Scholar,3Ford C.E. Polani P.E. Briggs J.H. Bishop P.M. A presumptive human XXY/XX mosaic.Nature. 1959; 183: 1030-1032Crossref PubMed Scopus (36) Google Scholar Only 15 years after the number of human chromosomes was announced, Hsu and Hirschhorn described 3 unusual 45,X mosaic variants and their phenotypes. They identify that growth deceleration, not just short stature from birth, and secondary amenorrhea, not just primary, can be associated with variant Turner syndrome. This march of refining and expanding the phenotype and prognosis associated with a new genetic syndrome is familiar. In the same way today, new gene/disease associations are made based on the most “severely” affected individuals. Then, as less obviously affected individuals are confirmed or identified genetically, the list of features and our understanding of the highly variable expression of syndromic presentations expands. This process of phenotype identification, its refinement, and analysis of potential mechanism continues perhaps indefinitely. Even now, we are identifying new phenotypes associated with classical Turner syndrome (eg, hyperinsulinism).4Gibson C.E. Boodhansingh K.E. Li C. Conlin L. Chen P. Becker S.A. et al.Congenital hyperinsulinism in infants with turner syndrome: possible association with monosomy X and KDM6A haploinsufficiency.Horm Res Paediatr. 2018; 89: 413-422Crossref PubMed Scopus (16) Google Scholar In the most recent decade, with the advent of clinically noninvasive prenatal screening for approximate sex chromosome count, the phenotype of 45,X-mosaic individuals is widening yet further. Today, these patients would have been screened for further mosaicism and Y-chromosome by genome-wide array. Our nomenclature too has shifted, in ways small and large. Terms like “sexual infantilism” are considered value-laden and no longer acceptable, nor is revealing photography like the fully unclothed woman with a small black oval over the eyes. Yet many things are relatively unchanged. In conjunction with new diagnostic tools like noninvasive prenatal screening and genome-wide arrays, the karyotype is still key for diagnosis of structures like the ring-X described in patient three. By carefully, painstakingly phenotyping, and genetically characterizing their patients, Hsu and Hirschhorn were able to expand Turner variant phenotypes and the relationship to diagnostic karyotypes. The tools and knowledge base of genomic medicine are vastly expanded, but 50 years later, our approach is fundamentally the same.