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On the problem of sorting burnt pancakes. (English) Zbl 0831.68029
Summary: The “pancake problem” is a well-known open combinatorial problem that recently has been shown to have applications to parallel processing. Given a stack of $$n$$ pancakes in arbitrary order, all of different sizes, the goal is to sort them into the size-ordered configuration having the largest pancake on the bottom and the smallest on top. The allowed sorting operation is a “spatula flip”, in which a spatula is inserted beneath any pancake, and all pancakes above the spatula are lifted and replaced in reverse order. The problem is to bound $$f(n)$$, the minimum number of flips required in the worst case to sort a stack of $$n$$ pancakes. Equivalently, we seek bounds on the number of prefix reversals necessary to sort a list of $$n$$ elements. Bounds of $$17n/16$$ and $$(5n+5)/3$$ were shown by Gates and Papadimitriou in 1979. In this paper, we consider a traditional variation of the problem in which the pancakes are two sided (one side is “burnt”), and must be sorted to the size- ordered configuration in which every pancake has its burnt side down. Let $$g(n)$$ be the number of flips required to sort $$n$$ “burnt pancakes”. We find that $$3n/2 \leq g(n) \leq 2n-2$$, where the upper bound holds for $$n \geq 10$$. We consider the conjecture that the most difficult case for sorting $$n$$ burnt pancakes is $$-I_n$$, the configuration having the pancakes in proper size order, but in which each individual pancake is upside down. We present an algorithm for sorting $$- I_n$$ in $$23n/14 + c$$ flips, where $$c$$ is a small constant, thereby establishing a bound of $$g(n) \leq 23n/14 + c$$ under the conjecture. Furthermore, the longstanding upper bound of $$f(n)$$ is also improved to $$23n/14 + c$$ under the conjecture.

##### MSC:
 68P10 Searching and sorting
pancake problem
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##### References:
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