Minimum moves to "stabilize" an array

Question

We have given an array of size $N$, consisting of positive integers, and an integer $H$. We also know that an array is stable iff $|A_i - A_{i-1}| \leq H$ for each $1<i<N-1$. We can modify the element by adding +1 to some element or by adding -1 to some element, we don't have a limit on how much we can use those operations, but we want to make the array stable in minimum moves.

I started solving the task by writing a dynamic programming relation, described by the state $f(i, j)$ meaning the minimum moves to stabilize the first $i$ elements of the array, such that the last element is equal to $j$.

However it seems that this method is slow for big values of $N$. How would you approach solving this problem?

Answer 1

Let $M=\max_i|A_i|$.

Besides the linear programming method proposed in this post (which is suitable for small $N$ and large $M,H$), you can optimize your dynamic programming method by building a larger table to reduce the time for querying the minimum value among $f(i-1,j-H),\ldots,f(i-1,j+H)$.

Let $f(i,j,k)$ be the minimum moves to stabilize the first $i$ elements of the array, such that the last element is in the range $\left[j,j+2^k\right)$. Then we have $f(0,j,0)=0$, $$f(i,j,0)=|A_i-j|+ \min\left\{f\left(i-1,j-H,\hat{k}\right),f\left(i-1,j+H+1-2^{\hat{k}},\hat{k}\right)\right\}$$ where $2^{\hat{k}}>H$, and $$f(i,j,k)= \min\left\{f(i,j,k-1),f\left(i,j+2^{k-1},k-1\right)\right\}$$ for $k>0$. Note each entry can be calculated in $O(1)$ time if we calculate them in some proper order.

Note in an optimal solution, the final $A_N$ must be in the range $[-(M+NH),M+NH]$ (otherwise the final elements are all either greater than $M$ or less than $-M$, which is not optimal), so we need to calculate $f(i,j,k)$ for all $i,j,k$ such that $0\le i\le N$, $j\ge -(M+(2N-i)H)$, $k\ge0$ and $j+2^k\le M+(2N-i)H+1$. The time complexity is $O\left(N(M+NH)\log (M+NH)\right)$, while the naive $f(i,j)$ solution takes $O(N(M+NH)H)$ time.

You can further use the range minimum query technique to reduce the time complexity to $O(N(M+NH))$.