If the two paths didn't share a vertex then you could construct a new path using the two longest paths (since the graph is connected) that is longer than the longest paths, resulting in a contradiction.
I was very interested in this subject a while back, enough to do a summer research project on (https://prism.ucalgary.ca/handle/1880/108831). The benefit of having an extended alphabet would differ based on the encoding scheme. For instance, my research was focused on the DNA Fountain algorithm (https://science.sciencemag.org/content/355/6328/950), which mapped every nucleotide to 2 bits. With 8 nucleotides, we could increase this to 3 bits. I remember reading about some encoding schemes that didn’t have a strict 1 nucleotide to sequence of bits conversion, and the benefits for these schemes would be different.
If I recall correctly, P and Z nucleotides (which I used in my research, and is also part of the hachimoji DNA) had lower PCR fidelity rates than natural nucleotides. This could mean that to store the same amount of data, more copies of DNA would be necessary to counterbalance more errors being introduced by biological processes, which diminishes the benefit outlined above.
I’m Steven Munsu Hwang and a co-founder of AsOne. Thomas sold me on his passion and vision for AsOne, and I’ve been working on it ever since.
My research area of interest is theoretical computer science, and I want to figure out which particulars of TCS I want to pursue for graduate school (hopefully by interacting with posters of https://asone.ai/topic/compsci!).
Speaking of brain machine interfaces, how excited are you for Neuralink’s upcoming update and what were your general thoughts about their last presentation?
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