If only the LCS LENGTH is needed (not the actual subsequence string), the DP table can be stored in

Each cell depends only on the row above and the cell to the left in the current row. So keep two rolling rows; the older one can be discarded once it's been used to compute the newer. O(min(m, n)) space if you orient rows along the shorter string. You can't get O(1) because you do need the current and previous row; reconstructing the actual LCS does need the full table.