# big-o notation Published articles for big-o notation. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Software Performance Begins with Time Complexity and Memory DevFeed: [Software Performance Begins with Time Complexity and Memory]() Original publisher: [Read original article]() Author: Furkan Kolcu Published: 2025-09-10T17:39:37Z Content type: tutorial Language: en Sources: [Furkan Kolcu - Software Engineer Blog]() Topics: [Software](), [Algorithms, Complexity](), [Scalability](), [Programming](), [Data structures](), [Go](), [C]() Tags: [big-o-notation](), [binary-search](), [c](), [complexity](), [go](), [memory](), [performance](), [scalability](), [software-engineering](), [sorting](), [space-complexity](), [technology](), [time-complexity]() ### AI overview This tutorial explains how time complexity and memory usage affect application performance as input size, system scale, and traffic increase. It introduces Big O notation, compares common complexity classes, and discusses trade-offs involving algorithms, data structures, and memory, with examples in Go and C. ### Source excerpt Memory usage and time complexity shape the performance of every application. This post explains why they matter, how they affect scalability, and how to think about the trade-offs behind every decision. ## Zero-One Laws for Random Graphs DevFeed: [Zero-One Laws for Random Graphs]() Original publisher: [Read original article]() Published: 2015-02-09T09:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Graphs](), [Math and Logic]() Tags: [big-o-notation](), [connectivity](), [countability](), [distribution](), [erdos-renyi](), [graphs](), [logic](), [logical](), [mathematics](), [model-theory](), [network-science](), [parameter](), [random-graph](), [random-graphs](), [statement](), [vertex]() ### AI overview This article introduces zero-one laws for Erdős-Rényi random graphs. It explains that many graph properties, including properties expressible in first-order logic, have probabilities that tend toward zero or one as the graph grows, with behavior determined by relevant thresholds or constant edge probabilities. ### Source excerpt Last time we saw a number of properties of graphs, such as connectivity, where the probability that an Erdős-Rényi random graph $ G(n,p)$ satisfies the property is asymptotically either zero or one. And this zero or one depends on whether the parameter $ p$ is above or below a universal threshold (that depends only on $ n$ and the property in question). To remind the reader, the Erdős-Rényi random "graph" $ G(n,p)$ is a distribution over graphs that you draw from by including each edge independently with probability $ p$. ## The Giant Component and Explosive Percolation DevFeed: [The Giant Component and Explosive Percolation]() Original publisher: [Read original article]() Published: 2015-02-02T09:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Graphs](), [Data analysis]() Tags: [big-o-notation](), [erdos-renyi](), [first-moment-method](), [giant-component](), [graph](), [graphs](), [inequality](), [markov](), [method-of-moments](), [network-science](), [percolation](), [probability-theory](), [random-graph](), [random-graphs](), [random-variables]() ### AI overview This article develops rigorous results about Erdős-Rényi random graphs, following a conjecture about connectivity at edge probability p = 5/n. It introduces threshold theorems and begins explaining the first moment method, Markov's inequality, and isolated vertices. ### Source excerpt Last time we left off with a tantalizing conjecture: a random graph with edge probability $ p = 5/n$ is almost surely a connected graph. We arrived at that conjecture from some ad-hoc data analysis, so let's go back and treat it with some more rigorous mathematical techniques. As we do, we'll discover some very interesting "threshold theorems" that essentially say a random graph will either certainly have a property, or it will certainly not have it. ## The Two-Dimensional Fourier Transform and Digital Watermarking DevFeed: [The Two-Dimensional Fourier Transform and Digital Watermarking]() Original publisher: [Read original article]() Published: 2013-12-30T19:24:05Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [fourier transform](), [watermarking](), [Algorithm](), [Code]() Tags: [algorithm](), [animations](), [big-o-notation](), [calculus](), [dimension](), [fft](), [fourier-analysis](), [fourier-transform](), [github](), [graphics](), [image-manipulation](), [images](), [james-hance](), [mathematics](), [matrices](), [programming](), [python](), [star-wars](), [up](), [watermarking]() ### AI overview This tutorial introduces the multidimensional Fourier transform, explains its relationship to the one-dimensional transform, describes an FFT-style algorithm for computing it, and applies it to digitally watermarking images. ### Source excerpt We've studied the Fourier transform quite a bit on this blog: with four primers and the Fast Fourier Transform algorithm under our belt, it's about time we opened up our eyes to higher dimensions. Indeed, in the decades since Cooley & Tukey's landmark paper, the most interesting applications of the discrete Fourier transform have occurred in dimensions greater than 1. But for all our work we haven't yet discussed what it means to take an "n-dimensional" Fourier transform. ## Optimism in the Face of Uncertainty: the UCB1 Algorithm DevFeed: [Optimism in the Face of Uncertainty: the UCB1 Algorithm]() Original publisher: [Read original article]() Published: 2013-10-28T11:42:55Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Algorithm](), [Mathematics](), [Data Science]() Tags: [algorithm](), [bandit-learning](), [bandits](), [big-o-notation](), [calculus](), [confidence-bounds](), [exploitation](), [exploration](), [machine-learning](), [mathematics](), [programming](), [python](), [random-variables](), [randomized-algorithm](), [science]() ### AI overview This tutorial introduces the multi-armed bandit problem as a model of the exploration-exploitation tradeoff and presents the UCB1 algorithm, which the article describes as performing close to optimally. ### Source excerpt startups The software world is always atwitter with predictions on the next big piece of technology. And a lot of chatter focuses on what venture capitalists express interest in. As an investor, how do you pick a good company to invest in? Do you notice quirky names like "Kaggle" and "Meebo," require deep technical abilities, or value a charismatic sales pitch? When it comes to innovation in software engineering and computer science, and that as a society we should value big pushes forward much more than we do. ## P vs. NP, A Primer (And a Proof Written in Racket) DevFeed: [P vs. NP, A Primer (And a Proof Written in Racket)]() Original publisher: [Read original article]() Published: 2012-02-23T19:57:35Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Computing](), [Algorithms, Complexity](), [Racket](), [Programming]() Tags: [algorithm](), [algorithms](), [analysis](), [big-o-notation](), [class](), [computational-complexity](), [computing](), [decidability](), [efficiency](), [fast](), [halting-problem](), [input](), [language](), [machine](), [np-completeness](), [p-versus-np](), [primer](), [racket](), [solve](), [theory](), [turing-machine](), [turing-machines]() ### AI overview A primer on the distinction between decidability and computational efficiency, introducing the complexity class P and its definition in terms of polynomial-time decision by a Turing machine. It also discusses efficient algorithms and gives a card-sorting example. ### Source excerpt Decidability Versus Efficiency In the early days of computing theory, the important questions were primarily about decidability. What sorts of problems are beyond the power of a Turing machine to solve? As we saw in our last primer on Turing machines, the halting problem is such an example: it can never be solved a finite amount of time by a Turing machine. However, more recently (in the past half-century) the focus of computing theory has shifted away from possibility in favor of determining feasibility. ## Inner Product Spaces--A Primer DevFeed: [Inner Product Spaces--A Primer]() Original publisher: [Read original article]() Published: 2011-07-25T00:29:00Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic](), [Computing]() Tags: [big-o-notation](), [eigenvectors](), [inner-product](), [mathematics](), [matrices](), [orthogonality](), [symmetry](), [vector-spaces](), [vectors]() ### AI overview This primer explains why vector spaces can be extended with a dot-product-like operation and defines inner products through conjugate symmetry, linearity, additivity, and positive definiteness. It introduces inner product spaces and discusses their connection to linear functionals. ### Source excerpt Vector spaces alone are not enough to do a lot of the interesting things we'd like them to do. Since a vector space is a generalization of Euclidean space, it is natural for us to investigate more specific types of vector spaces which are more akin to Euclidean space. In particular, we want to include the notion of a dot product. By admitting additional structure to a vector space, we may perform more computations, and hopefully get more interesting results. ## Big-O Notation--A Primer DevFeed: [Big-O Notation--A Primer]() Original publisher: [Read original article]() Published: 2011-06-14T16:26:27Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Algorithms, Complexity](), [Algorithms](), [Mathematics]() Tags: [algorithm](), [algorithms](), [analysis](), [big-o-notation](), [big-theta-notation](), [complexity](), [computational-complexity](), [example](), [limits](), [mathematics](), [measurement](), [primer](), [real-analysis](), [time]() ### AI overview This primer introduces Big-O notation as a way to describe an algorithm's runtime growth. It defines the notation mathematically and illustrates it with a list-summing algorithm characterized as O(n), while also distinguishing Big-Theta notation. ### Source excerpt The Quest to Capture Speed Companies and researchers spend hundreds of millions of dollars for the fruits of their algorithms. Whether one is indexing websites on the internet for search, folding proteins, or figuring out which warehouse is the most cost-effective to ship a product from, improvements in algorithm speed save immense amounts of money. It's no surprise then, that a similarly immense amount of money has gone into the mathematics behind algorithm analysis.