Hello Codeforces! The 6th Stage of the 1st Universal Cup is coming. It will be held from Mar 4th to 5th.
The contest is based on an official ICPC Asia Pacific regional contest. It has been used in the 2022 ICPC Asia Taoyuan Regional Programming Contest, which was held on November 19, 2022 — November 21, 2022. Thanks to the judges and authors for preparing this great contest and sharing with us as a Universal Cup Stage.
As usual, we have three time windows for participating. You can participate in the contest in the following three time windows:
- Mar 4th 13:00 — 18:00 (UTC +8)
- Mar 4th 19:00 — 24:00 (UTC +8)
- Mar 5th 02:00 — 07:00 (UTC +8)
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Can someone give some idea for problem H? How to find the optimal cut while calculating a particular DP[K][N] in O(1) as the editorial says?
This is how we did it.
Let's look at $$$O(N*N*K)$$$ dp solution.
$$$dp[k][i] = min(dp[k][i], dp[k-1][j-1]+cost(j,N))$$$ for all $$$i<j<=N$$$
Create a helper array where $$$changes[i]$$$ contains all $$$i<j$$$ such that $$$cost(i,j-1) \neq cost(i,j)$$$. The updated dp states are -
$$$dp[k][i] = min(dp[k][i], dp[k-1][j-1]+cost(j,N))$$$ for all $$$j$$$ present in $$$changes[i]$$$.
Editorial claims that sizes of these $$$changes[i]$$$ array is $$$O(1)$$$ because of random inputs. I do not have any rigorous proof for this claim.