Introduction
The pancake problem is a wellknown combinatorial optimization problem first studied by Bill Gates and Christos Papadimitriou in 1979 [2]. The problem is described as follows: a chef has a stack of pancakes, all of which have different sizes. Before serving the stack, the chef must sort the pancakes so that they are in ascending order – in size – from top to bottom. However, the chef can only reorder pancakes by using a spatula to flip some number of pancakes on top of the stack. Given an unsorted stack, the objective is to sort it using as few flips as possible.
This problem, which has recently been shown to be NPhard [1], has seen continued interest due to its relation to a number of applications. These include calculating genome similarity [3] and physical stacking problems, such as using a forklift to stack a set of differently sized boxes in a structurally sound way. The problem is also a standard benchmark in the heuristic search community, as search techniques have proven to be an effective way to optimally sort pancake stacks. In this work, we increase our understanding of the behaviour of such approaches on this problem by evaluating the accuracy of a stateoftheart heuristic function. We also propose three new heuristic functions that greatly improve problemsolving time.
Background
A stack of $n$ pancakes can be represented as a permutation of the numbers from $1$ to $n + 1$, where the permutation maps each location in the stack to the pancake at that location. In this representation, $n+1$ corresponds to the plate under the stack and can never be flipped. For example, $(2~1~4~3~5)$ represents a stack of $4$ pancakes in which the second smallest pancake is on top of the stack and is followed by the smallest pancake.
Iterative Deepening A\ (IDA*) [5] is a heuristic search algorithm that is commonly used to solve statespace search problems. This algorithm employs a heuristic function, which estimates the optimal cost of reaching a goal from any given state. In the pancake puzzle, this means that the heuristic function estimates the minimum number of flips needed to sort a given stack. IDA* is guaranteed to only find optimal solutions if the heuristic function never overestimates the cost of reaching the goal. Such heuristic functions are said to be admissible.
The stateoftheart heuristic function used for the pancake problem is the gap heuristic function [4]. A gap occurs whenever two adjacent pancakes in a stack are not adjacent in the sorted stack; this computation includes the plate as a pancake. The gap heuristic function then counts the number of gaps in the stack. For example, in stack $(2~1~4~3~5)$, there is a gap between every pair of consecutive pancakes except pancakes $4$ and $3$, and so the heuristic estimate for this stack is $3$. Since each flip can remove at most one gap, this heuristic function is admissible.
New Heuristic Functions
In this work, we introduce three new admissible heuristic functions for this problem. All three functions use the inverse of a permutation, which is a mapping from each pancake to its location in the stack. For example, the inverse of $(2~4~1~3~5)$ is $(3~1~4~2~5)$.
The first new function uses the inverse to quickly identify stacks in which no single flip will decrease the number of gaps. This top heuristic function increases the gap heuristic estimate by one in such cases. The top’ heuristic function is then given by the maximum of the values returned by the top heuristic function when applied to both a stack and its inverse. Top’ remains admissible since the number of flips needed to sort a stack is the same number needed to sort its inverse [6]. Finally, we have also developed a way to quickly look ahead to the top heuristic estimates of all successors of a stack, and thereby improve the estimate of the stack itself. The Ltop’ heuristic function is then given by the maximum found using this lookahead on both a stack and its inverse.
Experimental Analysis
To better understand the gap heuristic function, we analyzed its accuracy on all $10!$ possible $10$pancake stacks by calculating the heuristic error on each. This metric is defined as the difference between the heuristic estimate and the optimal cost of that problem. The first column of Table 1 shows a count of the number of states with each amount of heuristic error. The table shows that the gap heuristic function is off by no more than one on the majority of problems, though there are stacks where it is much more inaccurate.
These results suggest that it is highly likely that the gap heuristic function will be very accurate on any randomly generated stack, and we confirmed this holds on larger stacks by random sampling. This helps to explain the success of IDA*based approaches seen in the literature, in which random sampling is the standard way to generate test sets [4, 6].
Heuristic Error  Gap  Top  Top’  LTop’ 

0  1,717,763  1,839,701  1,924,425  1,992,492 
1  1,710,742  1,644,853  1,593,637  1,547,761 
2  194,280  141,609  109,304  87,645 
3  5,924  2,620  1,430  898 
4  91  17  4  4 
Table 1: Heuristic accuracy on all stacks of 10 pancakes. The entries are thenumber of problems with the corresponding heuristic error.
Table 1 also displays heuristic error information for our new heuristic functions. It shows that these functions are increasingly accurate, so we tested them as part of an IDA*based solver to see if the improved accuracy translated into better performance. We experimented on two different test sets. The first, “random $50$”, consists of $1000$ randomly generated stacks of $50$ pancakes. The second, “constructed $32$”, consists of $1000$ stacks of $32$ pancakes with a higher average heuristic error for the gap heuristic function.
Table 2 shows the performance of each heuristic function on both test sets. For each function, the table shows the improvement seen relative to the gap heuristic function, where a larger value indicates a greater speedup. Clearly, the new heuristics lead to substantial speedups on both test sets.
Problem  Gap  Top  Top’  LTop’ 

random 50  1.0  1.28  1.74  2.51 
constructed 32  1.0  1.95  4.31  6.94 
Table 2: The relative performance of IDA* when using different heuristic functions on two problem sets. The table shows the speedup seen relative to the gap heuristic function.
Conclusion
In this work, we identified that the gap heuristic function is very accurate on randomly generated pancake problems, though there are stacks for which it is inaccurate. We then introduced three new heuristic functions that are more accurate and greatly improved search performance. Our work both improves our ability to solve pancake problems, and increases our understanding of this wellstudied problem.
References

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