Sequential Commitment Against a Mutable Load Surface

In live air routing, the airspace is divided into cells $c \in \mathcal{C}$. Each cell carries a current load $L(c)$ — the number of flights currently routed through it — and has a fixed capacity $U(c)$. The collection of values $\{L(c)\}_{c\in\mathcal{C}}$ is the load surface.

A route $r$ is the set of cells a flight traverses. Committing a flight to route $r$ updates the load surface:

$$L(c) \;\leftarrow\; L(c) + 1 \quad \text{for every } c \in r.$$

Cells off the route are unchanged. After this update, the next flight to be planned sees a different surface than the one before — the load surface is mutable.

For example, suppose cells $c_1,\dots,c_5$ currently carry loads

$$L = (2,\, 0,\, 3,\, 1,\, 0).$$

Committing a flight to the route $r = \{c_2,c_3,c_4\}$ produces

$$L' = (2,\;0+1,\;3+1,\;1+1,\;0) = (2,\,1,\,4,\,2,\,0).$$

Only the three cells on $r$ were incremented.