Big-O Notation – A Primer
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...
View ArticleGoogle’s Page Rank – The Final Product
Dangling Nodes and Non-Uniqueness Recall where we left off last time. Given a web with no dangling nodes, the link matrix for has 1 as an eigenvalue, and if the corresponding eigenspace has dimension...
View ArticleTuring Machines and Conway’s Dreams
Additional Patterns Last time we left the reader with the assertion that Conway’s game of life does not always stabilize. Specifically, there exist patterns which result in unbounded cell population...
View ArticleLow Complexity Art
The Art of Omission Whether in painting, fiction, film, landscape architecture, or paper folding, art is often said to be the art of omission. Simplicity breeds elegance, and engages the reader at a...
View ArticleP vs. NP, A Primer (And a Proof Written in Racket)
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...
View ArticleOther Complexity Classes
Not Just Time, But Space Too! So far on this blog we’ve introduced models for computation, focused on Turing machines and given a short overview of the two most fundamental classes of problems: P and...
View ArticleClassic Nintendo Games are NP-Hard
Problem: Prove that generalized versions of Mario Brothers, Metroid, Donkey Kong, Pokemon, and Legend of Zelda are NP-hard. Solution: http://arxiv.org/pdf/1203.1895v1.pdf Discussion: Three researchers...
View ArticleKolmogorov Complexity – A Primer
The Complexity of Things Previously on this blog (quite a while ago), we’ve investigated some simple ideas of using randomness in artistic design (psychedelic art, and earlier randomized css designs),...
View ArticleRamsey Number Lower Bound
Define the Ramsey number to be the minimum number of vertices required of the complete graph so that for any two-coloring (red, blue) of the edges of one of two things will happen: There is a red...
View ArticleMiller-Rabin Primality Test
Problem: Determine if a number is prime, with an acceptably small error rate. Solution: (in Python) Discussion: This algorithm is known as the Miller-Rabin primality test, and it was a very important...
View ArticleOn Coloring Resilient Graphs
I’m pleased to announce that another paper of mine is finished. This one is submitted to ICALP, which is being held in Copenhagen this year (this whole research thing is exciting!). This is joint work...
View ArticleAn Un-PAC-learnable Problem
In a previous post we introduced a learning model called Probably Approximately Correct (PAC). We saw an example of a concept class that was easy to learn: intervals on the real line (and more...
View ArticleParameterizing the Vertex Cover Problem
I’m presenting a paper later this week at the Matheamtical Foundations of Computer Science 2014 in Budapest, Hungary. This conference is an interesting mix of logic and algorithms that aims to bring...
View ArticleOn the Computational Complexity of MapReduce
I recently wrapped up a fun paper with my coauthors Ben Fish, Adam Lelkes, Lev Reyzin, and Gyorgy Turan in which we analyzed the computational complexity of a model of the popular MapReduce framework....
View ArticleThe Complexity of Communication
One of the most interesting questions posed in the last thirty years of computer science is to ask how much “information” must be communicated between two parties in order for them to jointly compute...
View ArticleZero Knowledge Proofs for NP
Last time, we saw a specific zero-knowledge proof for graph isomorphism. This introduced us to the concept of an interactive proof, where you have a prover and a verifier sending messages back and...
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