Before revision:  This sampling algorithm thus offers improvement over the affine-invariant Dikin Walk for polytopes (which mixes in (mn) steps from a warm start) for applications in which m \gg n. Furthermore, we describe an (mn^{\omega+1 + n^{2\omega+2}) algorithm for finding a suitably approximate John's ellipsoid for a symmetric polytope based on Vaidya's algorithm, and show the mixing time is retained using these approximate ellipsoids (where \omega < 2.373 is the current value of the fast matrix multiplication constant)}  containing x, }\\{|q-x|}\ is bounded above by a polynomial in n} .
After revision:  This sampling algorithm thus offers improvement over the affine-invariant Dikin Walk for polytopes (which mixes in  (mn^{\omega+1 + n^{2\omega+2}) algorithm for finding a suitably approximate John's ellipsoid for a symmetric polytope based on Vaidya's algorithm, and show the mixing time is retained using these approximate ellipsoids (where \omega < 2.373 is the current value of the fast matrix multiplication constant)} containing x, }\\{|q-x|}\ is bounded above by a polynomial in n} .