It is always surprising that so many people are stopped by the police and arrested for DUI when the reason they were stopped in the first place had nothing to do with DUI. I think more DUI arrests result from vehicle defects then from weaving.
The Fourth Amendment says that the police cannot stop you anytime they feel like it. They have to have a reason to stop you. The amount of evidence police must have to stop you depends on the situation. There are two levels of evidence that justify a stop. These are Reasonable and Articulable Suspicion, and Probable Cause.
Reasonable and Articulable Suspicion is the amount of evidence necessary to justify a belief that a person might be violating the law. That belief must be reasonable in the circumstances and based on evidence to support that reason. The officer must be able to articulate that reason as the reason for the stop. Also the reason for the stop must be related to the subsequent investigation. This means that the officer cannot just invent a reason as a pretext. The police cannot stop you on a mere hunch.