DP on Trees - Combining Subtrees

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

Solution

For two vectors $a$ and $b$, define the vector $c=a\oplus b$ to have entries $c_i=\min_{k=0}^i\left(a_k+b_{i-k}\right)$ for each $0\le i < \text{size}(a)+\text{size}(b)-1$.

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

The editorial naively computes a bound of $\mathcal{O}(N^3)$ on the running time of this solution. However, this actually runs in $\mathcal{O}(N^2)$!

Time Complexity of Merging Subtrees

The complexity can be demonstrated with the following problem:

You have an list of $N$ ones and a counter initially set to $0$. While the list has greater than one element, remove any two elements $a$ and $b$ from the list, add $a\cdot b$ to the counter, and add $a+b$ to the list. In terms of $N$, what is the maximum possible value of the counter at the end of this process?

Problems