Honestly I wouldnt consider max flow as an advanced algorithm since it appears in almost every ICPC contest and all you have to do is to convert the problem to a max flow problem and copy paste your max flow templates. and centroid related problems are basically dfs problems it is really not something that you yourself cant figure out ! however I have never used the decomposition but finding centroid of a tree is used in many interactive problems. the others I have no clue but i read about FFT yesterday and converting problems to fft is much harder than flow problems

Whenever you feel interested in it.

I can't speak to other topics. either I haven't learned them / I haven't practiced them enough / I haven't found any problems on them in the contests I have attempted, however, Flows, FFT and Convex Hull Trick are all fairly easy to understand once you give it some time, and I have seen quite a few problems on them as well.
The hardest part for all three is realising that the problem might be solvable by it and then converting it to a eg. max-flow problem so I suggest learn them and practice a lot.

Well...today

If you are purple in CF those are appropriate.

IMHO my CF level could rarely require FFT and Convex Hull Trick. Others are absolutely useless here, on CF.

On the other side, some different (say, ICPC or IOI-styled) contest could require some more provisional algos too. I think I have seen non-learning problems, involving each of the listed topics on some other platforms. So it generally depends on what do you want prepare for.

if you are only doing it for codeforces, then learn the topic only after the first time you fail to solve a problem that you reached in a contest because of the topic!

For what it's worth, the only topics here I've seen more than a handful of times on CF are FFT and flow. (Markov chains might implicitly come up on a semi-regular basis, but I've never encountered a problem I solved using formal Markov chain methods I've studied as opposed to general intuition.)

I'd argue that everyone Gold+ should have at least a basic familiarity with both FFT and flow. In recent history, these topics have rarely appeared below the ~2500 level, but they're not actually difficult to learn at a basic level: FFT can just be black-boxed (i.e., all you really need to do is find some template code, know that we can multiply polynomials efficiently, and realize that convolutions are often useful in combinatorics problems), and max flow is not especially complicated (and can also be black-boxed if you aren't trying to solve particularly difficult problems). Some basic knowledge of these ideas may be helpful when you end up attempting problems above your level.

In case of codeforces its after you become a Master I think, but for other platforms, when you are a stable expert on codeforces because on atcoder E problems are expert level or some times candidate master level problems but I know 600 points problem that use all of those techniques You've written except HLD. Similarly I have seen codechef giving problems on HLD and FFT for 6 star rated contests and DMOJ also start giving you advance concept near 20 points problems or even less. Thing is codeforces keeps the "knowing of advanced concept to solve a problem" to the minimum. So minimum that concepts needed to become expert on codeforces are less than concept needed to crack an average interview (although problems are way harder) but when you go to other sites concept barrier is way higher, so it kinda depends what platforms you give contests on.

I know all of them, and haven't used any in a contest. I still don't think its useless to learn them, because they give you more intuition, and its much easier to train understanding difficult things on something concrete and well known like hld, rather than working on a specific hard problem. I think some of these are just genius anyways, and you should learn them regardless of if they really help improve your rating, rating is not the end all be all.

On that note, it doesn't mean I support having more knowledge heavy tasks on codeforces, I think while they're really cool, its not fun to do contests when the bottleneck is how much cp specific tricks do you know.

Thanks, everyone for sharing ur experience. I think I've found what I was looking for.