Balancing Network Revisited

Consider $N$ parallel wires running from left to right. The wires are numbered $1$ through $N$ from top to bottom. There are $M$ devices called *balancers*, numbered $1$ through $M$ from left to right. For each valid $i$, balancer $i$ connects wires $x_i$ and $y_i$ ($x_i \lt y_i$). Let's call such a structure of wires and balancers a *balancing network*. Each balancer can be directed either upw