We present a new reduced scaling multireference singles and doubles configuration interaction (MRSDCI) algorithm based upon the combination of local correlation and pseudospectral methods. Taking advantage of the locality of the Coulomb potential, the weak-pairs approximation of Saebo and Pulay is employed to eliminate configurations having simultaneous excitations out of pairs of distant, weakly interacting orbitals. In conjunction with this, the pseudospectral approximation is used to break down the most time-consuming two-electron integrals into a product of intermediate quantities depending on no more than two orbital indices. The resulting intermediate quantities are then used directly in the CI equations to substantially reduce the number of floating point operations required for diagonalization of the Hamiltonian. Additionally, our CI algorithm is based upon the symmetric group graphical approach CI (SGGA-CI) of Duch and Karwowski. For the purpose of developing reduced scaling CI algorithms, this approach has some important advantages. The most important of these advantages are the on-the-fly calculation of integral coupling coefficients and the separation of the spin and spatial parts of the wave function, which simplifies implementation of local correlation approximations. We apply the method to determine a series of binding energies in hydrocarbons and show that the approximate method predicts binding energies that are within a few kcal/mol of those predicted by the analytic nonlocal method. For large molecules, the local pseudospectral method was shown to be over 7 times as fast as the analytic nonlocal method. We also carry out a systematic study on the performance of different basis sets in the weak-pairs method. It was determined that triple-zeta basis sets were capable of recovering only 99.0% of the correlation energy, whereas double-zeta basis sets recovered 99.9% of the correlation energy.