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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.\n  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.\n  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 $c$. 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