Partial Solution :
This question is really two: how to connect and how to program.
(Of course it is relevant to know how many ports you have ...)
I'll just talk about the link.
The LEDs only come on if the chain goes in the right direction. Therefore it is possible to place 2 leds between each pair of doors (with opposite senses).
If between each pair of doors (n*(n-1))/2
you put 2 leds obtens
a " Charlieplexer "
If you have 11 ports (and lots of wire ...), you can directly use the Arduino in "CarliePlexing" controlling 11 * 10 leds.
(The Arduino ports may be in the "+" "-" "off" state).
Unfortunately we can not have the two leds (p1 = +, p2 = -) and
(p2 = -, p1 = +) connected at the same time. But with the controller we switch between the two with speed, we will see the two connected.
link
If you have "few" ports you may need additional hardware (multiplexers or similar)
Update: Using external hardware
(-> render arduino + multiplexer in google uncle ...)
Multiplexers allow you to "reduce" outputs / inputs.
For example a multiplexer / demultiplexer of 16 will need 4 pins to select the desired output and one more for the signal.
link
Multiplexers can be set up ...
For example:
link
you'll find an arduino with 3-channel 8-channel multiplexer.