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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 conditionarXiv:1109.6749v1 fatcat:2i3bgtix3rctnmvaxz6zte62ai