Introselect is a more complex version of the same idea.
But basically both quickselect and median of medians are divide and conquer. Median of medians is an expensive way to guarantee a good division. Quickselect hopes for one.
If quickselect goes well, then median of medians will be relatively cheap when run because it is run on a much shorter list. If quickselect does not go well, the median of medians passes guarantee progress. So average behavior is only slightly worse, and worst case behavior is still nicely bounded (albeit with a much worse constant).
But basically both quickselect and median of medians are divide and conquer. Median of medians is an expensive way to guarantee a good division. Quickselect hopes for one.
If quickselect goes well, then median of medians will be relatively cheap when run because it is run on a much shorter list. If quickselect does not go well, the median of medians passes guarantee progress. So average behavior is only slightly worse, and worst case behavior is still nicely bounded (albeit with a much worse constant).