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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...

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Google’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...

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Turing 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...

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Low 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...

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P 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...

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Other 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...

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Classic 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...

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Kolmogorov 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),...

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Ramsey 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...

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Miller-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...

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On 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...

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An 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...

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Parameterizing 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...

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On 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....

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The 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...

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Zero 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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