  We show that if a set $A$ is computable from every superlow 1-random set,
then $A$ is strongly jump-traceable. This theorem shows that the computably
enumerable (c.e.) strongly jump-traceable sets are exactly the c.e.\ sets
computable from every superlow 1-random set.
  We also prove the analogous result for superhighness: a c.e.\ set is strongly
jump-traceable if and only if it is computable from every superhigh 1-random
set.
  Finally, we show that for each cost function $c$ with the limit condition
there is a 1-random $\Delta^0_2$ set $Y$ such that every c.e.\ set $A \le_T Y$
obeys $c$. To do so, we connect cost function strength and the strength of
randomness notions. This result gives a full correspondence between obedience
of cost functions and being computable from $\Delta^0_2$ 1-random sets.
