"Fast? No actually I'm slow and lumbering. I am however very perceptive. I have worked a lot on my sensitivity. They may start first but I arrive first. My advice is for you to cultivate sensitivity and deep understanding and you won't need speed. They will need to catch up to you instead of the other way round."
I personally believe this is a great strategy for jiujitsu as well, with one slight modification. I believe that a if you're faster than me, but I force you to go a longer, more treacherous way, I will win the race every time.
To put in striking terms: if I position myself in such a way and at such an angle that I have a clear shot but you don't and you have to take an indirect shot, I will always win.
To put in grappling terms: if I position myself in such a way and at such an angle that I can deliver heavy attacks directly while you have to re-align your body before you can properly respond then, again, I will always win.
How can we develop the sensitivity the Peter is talking about? By completely shifting the focus of our training from winning to chasing perfection.
Aim to make everything you do on the mat perfect, or not at all.
Study what perfect gripping means then aim to only grip perfectly. If you get a "this will do" grip, don't fool yourself and proceed to the next part of the technique. Sure, you might get the sweep or submission against a surprised, tired or inferior opponent but at the cost of perfecting the move so it works against top level competition. Here's a beautiful and recent example of Roger Gracie sweeping and submitting the fantastic Rodrigo Comprido at the 2015 IBJJF black belt league finals: