Denoting n {\displaystyle n} as the number of variables and L {\displaystyle L} as the number of bits of input to the algorithm, Karmarkar's algorithm requires O ( n 3.5 L ) {\displaystyle O(n^{3.5}L)} operations on O ( L ) {\displaystyle O(L)} digit numbers, as compared to O ( n 6 L ) {\displaystyle O(n^{6}L)} such operations for the ellipsoid algorithm. The runtime of Karmarkar's algorithm is thus