How do full nodes generate Merkle path(MP)?

As you see. Full nodes generate a MP and give it to a SPV node which request it for verify a transaction when full nodes get the request.

How to get a MP? I don't see any more information. And that's my naturally guess: it's like getting it for the first time, from the leaf nodes ,hash and hash... , until the Merkle root. Then getting the path as required.

Is that true? Each time a request is received from a SPV node, there is a operation involve all the leaf nodes? Is there any other way to reduce computation?

Thanks for your time.