Before revision:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}}  +k^2+ \frac{k{\alpha}}  \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}}  +k^3).
After revision:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}} +k^2+ \frac{k{\alpha}} \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( {\alpha}} +k^3).