Dishwasher Drain - Not by sink
I would use the Air Admittance Valve. When you make up your new drain, install a tee with the side outlet for the DW and also a piece of pipe going up as high as possible with the AAV. Remember, you still want to install the DW drain hose with a high loop in it to prevent the water from being sucked out during the wash cycle. Air Gaps were required a while back but many areas found that they caused more problems than they solved. Food in the discharge caused clogging. Some area still require them though.
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