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Let $P=(P_1, P_2, \ldots, P_n)$, $P_i \in \field{R}$ for all $i$, be a signaland 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 toapproximate the original signal $P$ with another signal $F$ which consistsideally of few piecewise constant segments. Such applications include database(e.g., histogram construction), speech recognition, biology (e.g., denoisingaCGH data) applications and many more.  In this work we present two new techniques for optimizing dynamic programmingthat can handle cost functions not treated by other standard methods. The basisof our first algorithm is the definition of a constant-shifted variant of theobjective function that can be efficiently approximated using state of the artmethods for range searching. Our technique approximates the optimal value ofour 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 introducedby our algorithm is the decomposition of the initial problem into a small(logarithmic) number of Monge optimization subproblems which we can speed upusing existing techniques.

Approximate Dynamic Programming using Halfspace Queries and Multiscale Monge decomposition