Not necessarily. There are places where crime is low and so is police presence, like Taipei or Tokyo. It's part of culture which is actually adhered to by most.

What are you talking about? There's a staffed police box in every neighbourhood in Tokyo, and even more police in the "dangerous" areas.

I have hardly ever seen police patrolling. Traffic cops at intersections, but beat cops or cops at malls, train stations, etc.. seem rare.

If you go through Roppongi you'll see them, largely standing around. And in most shopping streets you'll see them patrolling on bicycles or in cars. They don't tend to stand around at malls or train stations, agreed, is that normal where you come from? (It wasn't the norm when I lived in the UK).

In the US it's common to see police out in areas like that. I looked it up and it seems NYC and Tokyo have the same number of cops.