In this paper we consider the wavelet synopsis construction problem without the restriction that we only choose a subset of coefficients of the original data. We provide the first near optimal algorithm. We arrive at the above algorithm by considering space efficient algorithms for the restricted version of the problem. In this context we improve previous algorithms by almost a linear factor and reduce the required space to almost linear. Our techniques also extend to histogram construction, and improve the space-running time tradeoffs for V-Opt and range query histograms. We believe the idea applies to a broad range of dynamic programs and demonstrate it by showing improvements in a knapsack-like setting seen in construction of Extended Wavelets.