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DP on Trees - Combining Subtrees

Author: Benjamin Qi

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StatusSourceProblem NameDifficultyTagsSolution
CFNormalCheck CF

This was the first problem I saw that involved this trick.

Solution

For two vectors aa and bb, define the vector c=abc=a\oplus b to have entries ci=mink=0i(ak+bik)c_i=\min_{k=0}^i\left(a_k+b_{i-k}\right) for each 0i<size(a)+size(b)0\le i < \text{size}(a)+\text{size}(b).

Similar to the editorial, define dp[x][0][g]\texttt{dp[x][0][g]} to be the minimum cost to buy exactly gg goods out of the subtree of xx if we don't use the coupon for xx, and define dp[x][1][g]\texttt{dp[x][1][g]} to be the minimum cost to buy exactly gg goods out of the subtree of xx if we are allowed to use the coupon for xx. We update dp[x][0]\texttt{dp[x][0]} with one of the child subtrees tt of xx by setting dp[x][0]=dp[x][0]dp[t][0]\texttt{dp[x][0]}=\texttt{dp[x][0]}\oplus \texttt{dp[t][0]}, and similarly for dp[x][1]\texttt{dp[x][1]}.

Code

The editorial naively computes a bound of O(N3)O(N^3) on the running time of this solution. However, this actually runs in O(N2)O(N^2)!

Time Complexity of Merging Subtrees

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Problems

StatusSourceProblem NameDifficultyTagsSolution
COCINormal
CFNormalCheck CF
DMOJNormal
CFNormal
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DP

DMOJHard
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NT

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