In solving large sparse linear least squares problems A x ≃ 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 unnecessary space is needed for intermediate fill.