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Once the longer number starts repeating digits, then it's not n^2 anymore. Multiplies get replaced with lookups. And we're only counting the multiplies. That's all they counted in the article. Not the adds, not the shifts.

No, that's taken into account.

Ok, maybe I don't understand the problem, but it seems obvious that it should never be greater than O(min(n1, n2) * 10), where n1 and n2 are the lengths in digits of each argument, and assuming we are multiplying decimal numbers.

Take the first digit of the longer number. Multiply it by the shorter number and store the result. Take the second digit of the longer number. If it matches the first digit, do a lookup of the last result and use that, else multiply and store. Repeat.

There will be a maximum of 10 * (length of the shorter number) multiplies, because there are only 10 unique digits. After that every operation is a lookup.

You could even do a tiny optimization by skipping the multiplication for the zero digit.

Worst case, the two numbers are the same length, in which case it's O(n/2 * 10), which is a heck of a lot better than O(n log n).

What am I missing here?

EDIT to respond to the comments: in the article, they are only counting the number of multiplies in the O() value. They are not including the adds.

I'm wondering how many millions in developer productivity are being lost every minute.

It's not clear to me whether Anthropic's limitations are technical or merely contractual. Is Anthropic actually putting the limitations in their prompts, so that the model would refuse to answer a question on how to do certain things?

If so, that's a major problem. If the military is using it in some mission critical way, they can't be fighting the model to get something done. No such limitations would ever be acceptable.

If the limitations are contractual, then there is some room for negotiation.

This looks great, but it's not clear to me how to use it for a practical task. I need to transcribe about 10 years worth of monthly meetings. These are government hearings with a variety of speakers. All the videos are on YouTube. What's the most practical and cost-effective way to get reasonably accurate transcripts?

Thanks. That's exactly the right thing to say.

This area has a lot of hills.

yes, quite a bit

Irked me too. I'm running for state representative in House District 9, which covers much of that area. The district is drop dead gorgeous and it's growing rapidly. There's a lot here.

178 billion? That's nothing. I did trillions just this morning. I went to the grocery store and picked an item off the shelf, effectively filtering out the trillions of other products that I could have picked but didn't.

They did not process 178 billion rows per second. They did a search that found something in a large data set by eliminating the parts of the data set that could not have contained the item. Same way I did by picking one grocery store and going straight to the shelf.