In solving large sparse linear least squares problems $Ax \cong b$, several different numeric methods involve computing the same upper triangular factor $R$ of $A$. It is of interest to be able to compute the nonzero structure of $R$, given only the structure of $A$. The solution to this problem comes from the theory of matchings in bipartite graphs. The structure of $A$ is modeled with a bipartite graph and it is shown how the rows and columns of $A$ can be rearranged into a structure from which the structure of its upper triangular factor can be correctly computed. Also, a new method for solving sparse least squares problems, called block back-substitution, is presented. This method assures that no unnecessary space is allocated for fill, and that no space is needed for intermediate fill.