On sorting by prefix reversals and the diameter of pancake networks

Abstract

We show that the conjectured hardest stack of burnt pancakes, −I(n), can be sorted in 3(n+1)/2 steps, for all n≡3(mod 4) with n ≥ 23. If-I(n) is indeed hardest, then both the ”burnt” and ”unburnt” pancake networks of dimension n have diameter at most 3(n+1)/2. We also describe a 9/8 n+2 step sorting sequence for Gates and Papadimitriou's unburnt stack of pancakes, χ _n , thus disproving their conjecture that 19/16n steps are required.