## Sunday, January 7, 2024

### A 1:1 week

For the third consecutive week, the competitive programming scene consisted of a Universal Cup round, and a Codeforces round, both on Saturday. The Universal Cup round was called Stage 17: Jinan (problems, results, top 5 on the left, analysis). The usual suspects occupied the first two places (well done!), but quite unusually the scores of two teams that participated in the original ICPC regional, both from Peking University according to Google Translate, were enough for 3rd and 4th. I guess we need to pay attention to Peking University at the upcoming World Finals indeed :)

The Codeforces round was called Hello 2024 (problems, results, top 5 on the left, analysis). I started relatively quickly, and was in 2nd place after solving problem F using the excellent AtCoder library segment tree template that helps focus on defining the monoid in question (search for "Data op" function in my solution). This has left me with 3 problems to solve: problem E, where the contour of a quadratic dynamic programming solution was more or less clear after reading the problem, and problems G and H, which looked very interesting to solve but where I did not have any good idea quickly. I've decided to implement E before thinking more about G and H — after all, how long can figuring out the details and implementing a relatively straightforward quadratic dynamic programming take? It turns out, almost two hours :(

ecnerwala did not get stuck on problem E, and therefore had plenty of time for the two harder problems, and he needed that time as he solved G with just 7 minutes remaining in the round and claimed the first place. Congratulations!

Let me highlight problem D from this round: consider a rooted tree where each node either has a leaf, or has exactly two children. For each node with two children, one of the edges to the children has weight 0, while the other has weight 1. We run a depth-first traversal of this tree starting from the root, and for each leaf we print the sum of weights on the path from the root to this leaf. Now we are given just the printed sequence and not the tree, and need to answer if there exists a tree as described above that could lead to this sequence being printed? For example, 2 1 0 1 1 can be printed, but 1 0 2 1 3 can't. The length of the given sequence is up to 2*105.

In my previous summary, I have mentioned a Universal Cup problem: your program is executed twice. In the first run, you are given a graph with n=1000 (note that it is not n<=1000, but exactly n=1000) vertices, and 2000<=m<=5000 edges. It is guaranteed that this graph was generated randomly by adding a uniformly random non-existing edge m times. You can make at most 5 changes to this graph, where one change is adding an edge between two vertices not connected by an edge, or removing an edge. The first run of your program ends there. The graph with your changes applied is then shuffled (randomly permute vertices, edges, and ends of an edge), and is given as input to the second run of your program. The trick is that your program is not told whether it is the first run or the second run, and instead needs to detect it itself. In other words, you need to apply such changes in the first run that you'd be able to detect that the graph was changed from a random one in the second run.

As this comment suggests, there are so many things one can try in this problem. When I solved it earlier today, I've decided to go for the most general approach: let us choose some hash of a graph that is reasonably fast to compute, and that does not change when we shuffle the graph. Now we say we are in the second run if this hash modulo 10000 is equal to 0, and otherwise we just keep trying random edges to add to make this hash modulo 10000 equal to 0. This will require 10000 iterations on average to find, which should be fast enough as each iteration is O(m), and this will incorrectly work on the first run with probability one in 10000, which is tolerable as the problem likely has on the order of 100 testcases.

As for the concrete hash function, I've used the (generally incorrect) idea of the hash described in "Problem 2" in this post with 10 iterations, but using the polynomial combination method from "Problem 4" to quickly combine the values of adjacent nodes since it avoids sorting. Most likely any not extremely bad hash would work since the given graphs are random. This was accepted from the first attempt, I did not even have to do any hyperparameter tuning.

Finally, Mike Mirzayanov has decided to tackle a long-standing Codeforces issue: people using multiple accounts (and maybe also the same account being used by multiple people?). Of course, it escalated rather quickly :) As for me, the social aspect — competing against other people, seeing their progress over the years, learning what type of problems they solve best, talking to them on the forums, and so on — is very important in competitive programming, and it complements the problem solving part very nicely. I think people violating the 1:1 property of the person-to-account mapping do ruin the social aspect, especially if they compete at the highest level (in this case there were two accounts in the top 10 by rating), and therefore I support Mike's decision to act. It is also quite sad that it has come to this at all — competitive programming depends on participants' integrity in a big way (it is very hard to detect if one consults with others during the round, for example), and seeing top-level participants violate one rule that it hard to enforce does not lend confidence that other rules that are hard to enforce are still standing.

Thanks for reading, and check back next week!