Would you like to give some example terms?

just from skimming over the paper and without knowing what background you already have, you might check out some of the videos from this playlist (specifically lecture 2 covers the relational algebra operators I think you might be referring to, and lectures 4 and 11-13 cover other relevant topics): https://www.youtube.com/playlist?list=PLSE8ODhjZXjYutVzTeAds8xUt1rcmyT7x

this, for instance, is mostly incomprehensible to me

I'll probably mess this up a little, but a couple pages before it defines Q as the join of three relations where the relational algebra translates roughly to:

SELECT R.*, S.*, T.*
FROM R NATURAL JOIN S NATURAL JOIN T;

SELECT B, C
FROM Q
WHERE A = ?;

SELECT * FROM 
R NATURAL JOIN T WHERE A = ?;

thanks! I'll have to read what you just wrote a few times... definitely need to brush up on the basics, but it's slowly making more and more sense

so Q is like all the "loops" that can be traced from A to B to C in this diagram?

and then SELECT B, C FROM Q WHERE A = ? could be broken into WHERE A = ?, which are all the "loops" for that include the given A, and then SELECT B, C grabs the values of B and C for all of those loops?

and isn't R JOIN S a mistake in "In other words, the value a_i is heavy if its contribution to the size of intermediate relation R JOIN S is greater than its contribution to the size of the output"? shouldn't it be R JOIN T?

I think the inequality should've been for R JOIN S instead of R JOIN T but I would need to read the rest more carefully. Either way pretty sure you're right that it's a mistake and that text and the left side of the inequality should refer to the same two relations.

thanks!