 This provides an alternative proof of the previously known result, as well as new structural insights into these diagonal flip paths\left({2}}}\right)  .
<meaning-changed>  This provides an alternative proof of the previously known result, as well as new structural insights into these diagonal flip paths\left({2}}}\right)  .
 This provides an alternative proof of the previously known result, as well as new structural insights into these diagonal flip paths\left({2}}}\right) time. As a further improvement over previous results, in many natural cases, our algorithm runs in time linear in the length of the minimum flip path .