During a high-school summer, I was able to participate in an iteration of the Department of Defense’s High School Apprenticeship program through Portland State University. Along with two other high school students, we explored a variety of mathematics and programming paradigms pertaining to generating prime numbers, working with superclusters, and partitioning sparse graphs through the development of OpenMP-parallelized C++ libraries.

All of our work was created under the PDX-Math-HSAP GitHub organization, which can be found at https://github.com/orgs/PDX-Math-HSAP/.

## Primes

We began the internship with a warm-up activity of sorts: writing an efficient algorithm to identify and display a number of accurate prime numbers in quick succession; in other words, a prime sieve. We began with a naive implementation of the Sieve of Erasnosthenes, an ancient Greek method developed to locate primes in the first 100 numbers, then made certain optimizations such as prematurely excluding all even numbers and all multiples of three from our prime search for greater efficiency. We ended up utilizing the Sieve of Atkin to find primes, utilizing both a modulo 2 and a modulo 6 implementation to determine whether numbers are primes.

## Sparse Matrices

However, the primary objective of the program was to implement various graph algorithms rooted in linear algebraic concepts to partition unweighted sparse graphs. These are also known as networks with low connectivity, and are trivially encoded as matrices with few ‘1’ characters to represent a connection between two nodes, as well as a majority of ‘0’ characters to represent a lack of a connection. Depicted below is an asymmetric graph featuring apples, bananas and oranges as nodes, with the existence of an edge denoted by ‘1’ and 0 otherwise. An arbitrary explanation for such a graph is a supermarket in which an apple can only be purchased after a carrot, a banana can be purchased after either a banana or a carrot, and a carrot can be purchased after another carrot. It’s unclear how the initial fruit was obtained. Feel free to view these as arbitrary ‘a’, ‘b’ and ‘c’ nodes of a graph as well.

apple | banana | carrot | |
---|---|---|---|

apple | 0 | 0 | 0 |

banana | 0 | 1 | 0 |

carrot | 1 | 1 | 1 |

To save a significant amount of space when working with large graphs, these matrices can be represented in compressed-sparse row format instead - denoting both the row in which the ‘1’ occurs as well as the column position. Such a matrix looks like the following:

This arrangement is able to save us x amount of space while avoiding any increase in time complexity - the value of an element given its coordinate in a non-compressed matrix can be determined with more operations, but is still a constant-time operation with regards to the number of cells seen.

To be continued and updated in the future.