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Abstract: Given a surface P, the curve complex of P is a graph in which the vertices correspond to isotopy classes of essential curves on P. Two vertices are connected by an edge if the corresponding isotopy classes of curves have disjoint representatives. If the surface P is a Heegaard splitting of a 3-manifold M, then d(P) is the minimum distance in this graph between vertices corresponding to curves that bound compressing disks on opposite sides of P. By a theorem of Hartshorn, it is known that if M is an irreducible compact orientable 3-manifold and P is a Heegaard surface for M, then d(P) is bounded above by the genus of any properly embedded essential surface. We prove that d(P) is similarly bounded by the genus of alternate Heegaard surface Q as long as Q is not isotopic to a stabilization of P.