Pancake sorting

Demonstration of the primary operation. The spatula is flipping over the top three pancakes, with the result seen below. In the burnt pancake problem, their top sides would now be burnt instead of their bottom sides.

Pancake sorting is a variation of the sorting problem in which the only allowed operation is to reverse the elements of some prefix of the sequence. Unlike a traditional sorting algorithm, which attempts to sort with the fewest comparisons possible, the goal is to sort the sequence in as few reversals as possible. This operation can be visualized by thinking of a stack of pancakes in which one is allowed to take the top k pancakes and flip them. A variant of the problem is concerned with burnt pancakes, where each pancake has a burnt side and all pancakes must, in addition, end up with the burnt side on bottom.

The pancake problems

The original pancake problem

The maximum number of flips required to sort any stack of n pancakes has been shown to lie between (15/14)n and (18/11)n, but the exact value is not known.

The simplest pancake sorting algorithm requires at most 2n−3 flips. In this algorithm, a variation of selection sort, we bring the largest pancake not yet sorted to the top with one flip, and then take it down to its final position with one more, then repeat this for the remaining pancakes. Note that we do not count the time needed to find the largest pancake, only the number of flips; if we wished to create a real machine to execute this algorithm in linear time, it would have to both perform prefix reversal (flips) and be able to find the maximum of a range of consecutive numbers in constant time.

On September 17, 2008, a team of researchers at the University of Texas at Dallas led by Founders Professor Hal Sudborough announced the acceptance by the journal Theoretical Computer Science of a more efficient algorithm for pancake sorting than the one proposed by Gates and Papadimitriou. This establishes a new upper bound of (18/11)n, improving upon the existing bound of (5/3)n from 1979.

On November 2, 2011, a paper was submitted to the arXiv announcing a proof that the problem is NP-hard, thereby answering a question open for over three decades. It is worth noting, however, that the NP-hard problem consists of computing the minimum number of flips required to sort n pancakes, and not the actual sorting of the pancakes. As discussed above, the sorting can be trivially computed in O(n) (see Big O notation) time, which places it in the polynomial class of problems.

The burnt pancake problem

In a more difficult variation called the burnt pancake problem, the bottom of each pancake in the pile is burnt, and the sort must be completed with the burnt side of every pancake down. In 2008, a group of undergraduates built a bacterial computer that can solve a simple example of the burnt pancake problem by programming E. coli to flip segments of DNA which are analogous to burnt pancakes. DNA has an orientation (5' and 3') and an order (promoter before coding). Even though the processing power expressed by DNA flips is low, the high number of bacteria in a culture provides a large parallel computing platform. The bacteria report when they have solved the problem by becoming antibiotic resistant. An animation depicting this process has been produced.

The pancake problem on strings

The above discussion presumes that each pancake is unique, i.e. no two of them are identical. Hence the sequence on which the prefix reversals are performed is a permutation, i.e. the sequence in which every symbol occurs exactly once. However, "strings" are sequences in which a symbol can repeat. Chitturi and Sudborough (2010) and Hurkens et al. (2007) independently showed that the complexity of transforming a compatible string into another with prefix reversals is NP-complete. They also gave bounds for the same. Hurkens et al. gave an exact algorithm to sort binary and ternary strings. Chitturi (2011) proved that the complexity of transforming a compatible signed string into another with prefix reversals, i.e. the burnt pancake problem on strings, is NP-complete.

History

Although seen more often as an educational device, pancake sorting also appears in applications in parallel processor networks, in which it can provide an effective routing algorithm between processors.^{[citation needed]}

The problem is notable as the only well-known mathematics paper ever written by Microsoft founder Bill Gates (as William Gates), entitled "Bounds for Sorting by Prefix Reversal". Published in 1979, it describes an efficient algorithm for pancake sorting. In addition, the most notable paper published by Futurama co-creator David X. Cohen (as David S. Cohen) concerned the burnt pancake problem. Their collaborators were Christos Papadimitriou (then at Harvard, now at Berkeley) and Manuel Blum (then at Berkeley, now at Carnegie Mellon University), respectively.

The connected problems of signed sorting by reversals and sorting by reversals were also studied more recently. Whereas efficient exact algorithms have been found for the signed sorting by reversals see [Kaplan, Shamir, Tarjan, 1997], the problem of sorting by reversals has been proven to be hard even to approximate to within certain constant factor [Berman, Karpinski, 1999], and also proven to be approximable in polynomial time to within the approximation factor 1.375 [Berman, Karpinski, Hannenhalli, 2002].

Related integer sequences

Number of stacks of given height that require N flips to get sorted

Height	N
	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
1	1
2	1	1
3	1	2	2	1
4	1	3	6	11	3
5	1	4	12	35	48	20
6	1	5	20	79	199	281	133	2
7	1	6	30	149	543	1357	1903	1016	35
8	1	7	42	251	1191	4281	10561	15011	8520	455
9	1	8	56	391	2278	10666	38015	93585	132697	79379	5804
10	1	9	72	575	3963	22825	106461	377863	919365	1309756	814678	73232
11	1	10	90	809	6429	43891	252737	1174766	4126515	9981073	14250471	9123648	956354	6
12	1	11	110	1099	9883	77937	533397	3064788	14141929	49337252	118420043	169332213	111050066	13032704	167

Sequences from The Online Encyclopedia of Integer Sequences of Neil Sloane:

•  A058986 – maximum number of flips
•  A067607 – number of stacks requiring the maximum number of flips (above)
•  A078941 – maximum number of flips for a "burnt" stack
•  A078942 – the number of flips for a sorted "burnt-side-on-top" stack
•  A092113 – the above triangle, read by rows

Further reading

• Mohammad, H.H. and Hal Sudborough, I. "On the Diameter of the Pancake Network," Journal of Algorithms 25 (1), 67-94, 1997.
• Roney-Dougal, C. and Vatter, V. "Of pancakes, mice and men," Plus Magazine 54, March 2010.
• Chitturi, B. and Sudborough, H. "Prefix reversals on strings". Proc. Int. Conf. Bioinformatics and Computational Biology, Vol. 2 (2010)591–598.
• Hurkens, C., Iersel, L. V., Keijsper, J., Kelk, S., Stougie, L. and Tromp J. "Prefix reversals on binary and ternary strings". SIAM J. Discrete Math. 21(3)(2007) 592–611.
• Chitturi, B. "A note on complexity of genetic mutations". DMAA 3(3) (2011)269-286 doi:10.1142/S1793830911001206.

External links

• Cut-the-Knot: Flipping pancakes puzzle, including a Java applet for the pancake problem and some discussion.
• Douglas B. West's "The Pancake Problems"
• Weisstein, Eric W., "Pancake Sorting", MathWorld.
• Animation explaining the bacterial computer that can solve the burnt pancake problem.