General Information

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Project Description

I will be investigating the relationship between edit distance and q-gram distance, two measures of comparison between strings, as they apply to read mapping to reference genomes. Read mapping consists of matching shorter sequences of DNA sequences to a longer, reference sequence. Edit distance and q-gram distance are related by the following equation:

QD ≤ 2qED

My goal is to further define the relationship between edit distance and q-gram distance under various cases and possibly determine the probability distribution of the q-gram distance with respect to edit distance.

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Weekly Log

Week 1:

I met with my mentor and co-mentor to discuss my goals for the project, which is to investigate the relationship between q-gram distance and edit distance. After discussing with my mentor/co-mentor and doing some initial reading, I created my first presentation that was delivered on Friday.

Week 2:

I did further reading and discussed with my mentor/co-mentor about the theorems and proofs that I will need to understand and possibly expand upon. After getting a grasp of these concepts, I worked out a few examples for my own benefit and then attempted to apply the ideas to slightly different cases.

Week 3:

I discussed with my mentor and co-mentor to further define some short term goals. I was also introduced to Latex. I worked on developing a proof that describes the upper bound of the probability of the L1 distance when considering 1-grams and one indel. I typed up my proof in Latex.

Week 4:

I made revisions to my 1-gram proof as per suggestions from my mentor and co-mentor. I also started working towards creating a similar proof for 2-grams. I had two meetings with my mentor and co-mentor in which we discussed ideas for the 2-gram proof.

Week 5:

I made further revisions to my 1-gram proof and continued working on the 2-gram proof. Unfortunately, we have not been able to formulate a good argument yet, but my co-mentor came up with an idea that involves Eulerian paths that I will try to further define next week.

Week 6:

I finished up my 1-gram proof for now and focused on developing the 2-gram proof. I did some reading on Eulerian paths and worked out some complications with the 2-gram proof.

Week 7:

We formulated a promising conjecture for the 2-grams, although we must now develop a solid proof for it. I created my final presentation that was delivered on Thursday.

Week 8:

In Prague.

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Presentations

• First Presentation

• Second Presentation

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References

• Approximate string-matching with q-grams and maximal matches, Esko Ukkonen
• Indel-tolerant Read Mapping with Trinucleotide Frequencies using Cache-Oblivious kd-Trees, Md Pavel Mahmud, John Wiedenhoeft, and Alexander Schliep.

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Additional Information

• My Mentor: Dr. Alexander Schliep
• My Co-Mentor: Md Pavel Mahmud