Approximate Dynamic Programming using Halfspace Queries and Multiscale Monge decomposition

Let $P=(P_1, P_2, \ldots, P_n)$, $P_i \in \field{R}$ for all $i$, be a signal and let $C$ be a constant. In this work our goal is to find a function $F:[n]\rightarrow \field{R}$ which optimizes the following objective function: $$\min_{F} \sum_{i=1}^n (P_i-F_i)^2 + C\times |\{i:F_i \neq F_{i+1} \} |$$ The above optimization problem reduces to solving the following recurrence, which can be done efficiently using dynamic programming in $O(n^2)$ time: $$OPT_i = \min_{0 \leq j \leq i-1} [ OPT_j + \sum_{k=j+1}^i (P_k - (\sum_{m=j+1}^i P_m)/(i-j) )^2 ]+ C$$ The above recurrence arises naturally in applications where we wish to approximate the original signal $P$ with another signal $F$ which consists ideally of few piecewise constant segments. Such applications include database (e.g., histogram construction), speech recognition, biology (e.g., denoising aCGH data) applications and many more. In this work we present two new techniques for optimizing dynamic programming that can handle cost functions not treated by other standard methods. The basis of our first algorithm is the definition of a constant-shifted variant of the objective function that can be efficiently approximated using state of the art methods for range searching. Our technique approximates the optimal value of our objective function within additive $\epsilon$ error and runs in $\tilde{O}(n^{1.5} \log{(\frac{U}{\epsilon}))}$ time, where $U = \max_i f_i$. The second algorithm we provide solves a similar recurrence within a factor of $\epsilon$ and runs in $O(n \log^2n / \epsilon)$. The new technique introduced by our algorithm is the decomposition of the initial problem into a small (logarithmic) number of Monge optimization subproblems which we can speed up using existing techniques.