Why.. the 5v tied to the 5v on the power supply?
You don't have to do that, but it is a convenient way to power the Arduino. You could power the Arduino from some other PSU or usb if you had a reason to do that. But if you do, don't connect the 5V on the Arduino to the 5V on the strip. You should never connect + of two separate power supplies together, even if they say they are the same voltage. They are never exactly the same, one will always be slightly higher than the other, and connecting them together could damage them.