Arduino Uno Rev3 and WS2811 Power Supply

I'm wondering exactly what power supply I should be supplying to an Arduino Uno unit along with 4 strings of 5V WS2811 lights that are connected.

There's 200 lights total (50 lights per string) rated at the following:

Max power: 0.6W/pcs; 0.3W/pcs
Size: 12mm/50pcs
Wire Color: Green/Black/Red White Blue
Working voltage: DC 12V/5V

I'm wiring it in a fashion similar to the below attachment. I'm wondering if I should have a separate power supply for the lights and if so what would be the appropriate DC adapter to use? And also what should I be supplying to the Arduino unit itself? Thank you!

Yes you need a separate power supply.

You also need to inject power at the start of each string (ie, in 4 places with your setup)

What I did was ziptie an extra pair of wires with connectors onto each string, with a 5v dc-dc step down (buck) converter at the start of each string, I Input connected to that extra pair of wires, and then powered that pair with a 19v laptop supply (the 3.5 A ones are perfect size), and let the 5v from that power the pro mini controller - though dont fo thsy if powering from the USB

Yes, you definitley need a seperate power supply.

The specs you posted said 0.3W at 5V. That would mean, that with 200pcs you would need 60W of power. 60W / 5V = 12Amps(!).

Alternatively you can power each strip individually then it would be "only" 3 Amps per string.

In terms of connecting them, just watch out that the connecter you are using is rated for the currentand that all grounds are connected together.