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Given a DNF formula on n variables, the two natural size measures are the number of terms or size s(f), and the maximum width of a term w(f). It is folklore that short DNF formulas can be made narrow. We prove a converse, showing that narrow formulas can be sparsified. More precisely, any width w DNF irrespective of its size can be ϵ-approximated by a width w DNF with at most (w(1/ϵ))^O(w) terms. We combine our sparsification result with the work of Luby and Velikovic to give a fasterarXiv:1205.3534v1 fatcat:e23ypu6yafgzni35gvz6eqzqgi