The mapping of optimal paths in the strong disorder limit to the strands of invasion percolation clusters is shown to lead to a new universal property of these clusters. We suggest that the corresponding strands arising in the annealed Eden growth process are in the same universality class as directed polymers in weak quenched disorder with an upper critical dimension #6. (S0031-9007(96)00188-3) PACS numbers: 64.60.Ak We address several issues in this Letter. What are the geometries of the optimal polymer in a strongly disordered medium in higher dimensions? What is the upper critical dimensionality for this problem? Is the geometry of the polymer in a strongly disordered medium universal? Our study is carried out in the context of a growing invasion percolation cluster. In this procedure, the bonds of the lattice are assigned strengths in a quenched random manner, and a cluster grows by invading the weakest interfacial bond. We will show that both bond and site variants of percolation lead to the same universality class. We then go on to study an analogous model with annealed instead of quenched disorder. In this case all interfacial bonds have an equal probability of being invaded. We present arguments and numerical evidence that even though the randomness is annealed, the effects of quenched disorder are self-generated within the model leading to geometries that are self-affine and characterized by the roughness exponent aDP . Thus, within the same process, the interface of the Eden cluster is characterized by a dynamical exponent zKPZ, whereas the static wandering exponent of the strands of the cluster is given by 1yzKPZ (a strand is defined as the unique path that excludes dead ends from an arbitrary site to a central seed site). Our results have a wide range of applicability— the strong disorder limit is relevant up to a correlation length in a variety of situations (7) including transport in amorphous semiconductors at low temperatures, electrical conduction and fluid flow in porous rocks, and the magnetic properties of doped semiconductors. Further, there are novel forms of percolation that are equivalent to the problem of the optimal polymer in a strongly disordered environment (6). Our prediction of the self- generated quenched randomness ought to be observable in Eden growth and other random invasion processes. We begin with an alternative way to view the geometry of the polymer in a strongly disordered environment. We