I initially asked this question over at [StackOverflow][1] as it has algorithmic flavor to it, but I haven't been getting much traction so I thought I would probe the mathematics community.

**Setup:** Let $e_{i}$ be an orthogonal (but not orthonormal) basis for $\mathbb{R}^{N}$. Define $\Lambda=\big\{\sum_{j=1}^{N}x_{j}e_{j}\mid x_{j}\in\mathbb{N}\big\}$ (where here I assume $0\in\mathbb{N}$). Now order the points in $\Lambda$ first by their $L^{1}$-norm, breaking ties lexicographically.

**Question:** Is there an efficient algorithm for producing the points in $\Lambda$ in order (up to some pre-defined bound)? Note that I want to walk this set in order, not produce it and then sort it. 

**Observations:** This is easy to do if the $e_{i}$ are orthonormal. For instance, the problem is solved [here][2]. To make something like this work here, one would need to be able to efficiently answer the following. given positive real numbers $x_{1},x_{2},\ldots, x_{N}$, is there an efficient way to the numbers $\sum_{j=1}^{N}n_{j}x_{j}$ in increasing order, where $n_{j}\in\mathbb{N}$.


  [1]: https://stackoverflow.com/questions/29804095/ordered-lattice-point-enumeration
  [2]: http://en.literateprograms.org/Generating_all_integer_lattice_points_%28Python%29#In_order_of_Manhattan_distance