Proving the correctness of an algorithm and calculating its complexity - how to figure it out?

Question

Mathematics

Hello everyone,
I have to solve / write a paper on the topic "Algorithms for reorganizing and optimizing websites".
And if, for example, we understand the essence of the algorithm’s work, it’s more or less normal for me, then the mathematical part is not very good.
For example, there is an algorithm described verbally and mathematically in a scientific publication:
There is a large site - thousands of pages, they are somehow non-optimally linked to each other, users need to make a large number of transitions to find the desired page, the search for the desired content is slow.
A site linking model is created in the form, for example, of a cladogram (either a tree or a graph), and then it is proposed to reorganize this linking according to a certain algorithm in order to optimize finding the necessary information on the site in accordance with a certain parameter, for example, the frequency of visiting a page by a link by users.
The complexity (O) is calculated, the mathematical/logical proof of the correctness of the algorithm is displayed (completes, gives the correct output). Then implement all this in pseudocode/flowchart.
My difficulty arises precisely with the mathematical part and the proof of correctness, so I will be grateful for recommendations on what to read / watch on this particular topic
Also, if anyone has any ideas about what other algorithms can be used when optimizing sites / Internet resources, please share.
If someone is well versed in the mathematical side of the work of algorithms and can explain / advise on this profile, I will be glad to talk.
Thank you in advance

Answer 1

you solve an optimization problem, you need to prove the "optimality" (well, or close to optimal) solution, and not some kind of correctness (any relinking can be correct, but not optimal).
You enter the optimality criterion yourself, and evaluate your algorithm according to it.
For example, you can introduce the concepts of "transfer cost" and the value of individual "pages" and then calculate this cost for different linking options, then optimize it, show that on average, for example, your option will be more profitable, and so on.