# Breaking a number in two different ways

I am interested in knowing if there is a name for this process:

Suppose I have positive reals $a_1,a_2,\ldots, a_k, b_1,b_2,\ldots, b_m$ such that $\sum_{i=1}^k a_i = \sum_{j=1}^m b_j.$

Then, I can come up with a sequence $c_1,c_2,\ldots, c_l$ where $k,m\leq l\leq k+m$ recursively as follows starting with $c_0 = 0.$

$$c_r = \min\left\{\min_{t: \sum_{i=1}^t a_i > c_{r-1}} \sum_{i=1}^t a_i, \min_{s: \sum_{j=1}^s b_j > c_{r-1}} \sum_{j=1}^s b_j \right\}$$

The $c_r$'s are an ordered list of partial sums.

Let us also define $d_r = c_r - c_{r-1},$ the difference sequence.

Is there a name for this construction of $c_r$'s or $d_r$'s?

I am working on a paper on network flows where I have to split flows in this particular way. It is easy to see what I mean from a picture but cumbersome to understand from the equation. If there is a well-known name for this, I could use it. Thanks.

• Hey. I edited the question to make it clear. Is it clear now? – Hedonist Jan 9 '14 at 0:44
• Thanks, the edit is helpful. Perhaps you might add some motivation and background as well. Why are you interested in this sequence? – j.c. Jan 9 '14 at 1:04
• Hi j.c. I am working on a paper on network flows where I have to split flows in this particular way. It is easy to see what I mean from a picture but cumbersome to understand from the equation. If there is a well-known name for this, I could use it. Thanks. – Hedonist Jan 9 '14 at 1:09
• That would be useful information to include when asking the question! See e.g. meta.mathoverflow.net/a/883/353 – j.c. Jan 9 '14 at 1:14
• Are you just taking the union of the sets of partial sums? If not, please give an example. – Douglas Zare Jan 9 '14 at 2:09