Today was an upsolving day. There were some tasks that had been implemented before, but hadn’t got AC.

One of these tasks is B from Umnik’s Day 3. The idea of solution came from his analysis. The problem is: you have an array h of length n and m queries of type (l, r, w) — maximal minimum among all segments of length w inside range [l, r]. 1 ≤ w ≤ r — l + 1.

Solution: let’s do parallel binary search, that is binary search by answer for each query simultaneously. Binary search query is element X and l, r. We should answer maximal length of segments inside range [l, r] that consist of elements ≥ X. We put 1 if element X ≥ 1, 0 — else. So we want maximal segment of ones inside [l, r]. So we will do scanline by X in descending order. For some new X we put 1 in all positions of X in array h. Then if we have query X, l, r, we should do calculations like in the picture below:

To be able to join segments and answer what segment consists of this point we need disjoint set union (DSU) and to find maximal length of segments strictly inside some range we need segment tree (ST) on maximum. So do scanline log(ans) times and maintain binary search queries for each problem’s query. Everything is with help of DSU and ST.

My solutions had been getting WAs because of the bug. My realization of joining point in DSU was so, that I didn’t any changes if this point is not connected to some existing segment. That’s absolutely wrong, because I need to put this point in DSU and update ST with 1 in this point.

**Conclusion: ***EVERY TIME THINK ABOUT CORNER CASES!!!*

Task C from Umnik’s Day 3. String of digits and sequence of operations are given. Every operation is digit->string. You should say the number, represented by resulting string by modulo. I saw no way how to do it moving from operation i to operation i + 1. But what if we will iterate from the last operation to the first? Then you can maintain for each digit in what string it turns into and the size of this string. That’s all we needed. Great.

**Conclusion: ***SOMETIMES YOU SHOULD PROCESS SOMETHING FROM THE END INSTEAD OF BEGINNING*

In last story I put some words about task E:

Task E can be reduced to checking the uniqueness of the perfect matching.

After huge number of submissions and unsuccessful attempts to optimize my solution to get AC instead of TL I decided to look at analysis. So the solution is: let’s run algorithm that at every step finds vertex with degree one (so it has only one edge) and add its edge to answer. When queue is empty it stops. If all empty vertices are added to answer, so it is unique solution. Otherwise it is not. Why? Vertices of degree one cannot be left. If vertices of degree 0 are left, there is no solution at all. If vertices of degree 2 and more are left, so in this bipartite graph number edges ≥ number of vertices, so we have alternating cycle (edge in matching, not in matching, in matching, …). We can change the “orientation” of edges in this cycle and get new answer. So it is not unique. i.e. this solution uses the idea of matching and cycle too, but it does not need to build it. Cool.

**Conclusion: ***YOU CAN USE IDEAS BASED ON BIPARTITE GRAPHS AND MATCHINGS BUT MAY BE YOU DO NOT NEED TO BUILD THEM*