My first reaction to the "fastest traceroute" question would be, is this a trick question? This is because many routers rate-limit the number of ICMP packets they forward. Causing a lot of ICMP replies could easily exceed this limit and result in missing hops. This requires being deliberately slow. A lot of the early Internet measurement studies had to deal with these problems, though ideally a single traceroute is less likely to trigger this behavior.

However, your hint indicates that there's something other than inducing a number of TTL exceeded packets. Can't verify as I'm on mobile, but are you thinking of the TCP Record Route option?