## 3 Methods
### Fitness Landscape Analysis
#### 3.3.3 Search Landscape Calculus  
Let \(\eta(p)\) be the neighbors of \(p\) (i.e., solutions reachable with one mutation) and \(f(p)\) is fitness. Search landscape calculus [32] defines the average local rate of fitness change:  
\[\Delta[f](p)=\frac{1}{|\eta(p)|}\sum_{p^{\prime}\in\eta(p)}\bigl{|}f(p)-f(p^{ \prime})\bigr{|}. \tag{41}\]  
Search landscape calculus then defines \(\Delta[f]\) as the average of \(\Delta[f](p)\) over all \(p\). It is infeasible to directly compute \(\Delta[f]\) for the true fitness function \(f\). Therefore, the search landscape calculus focuses on topological properties that influence fitness, such as absolute positions, relative positions, and element precedences for permutation problems, replacing \(f\) by a distance function \(\delta\) relevant to the context of a problem feature. Thus, define \(\Delta[\delta](p)\):  
\[\Delta[\delta](p)=\frac{1}{|\eta(p)|}\sum_{p^{\prime}\in\eta(p)}\delta(p,p^{ \prime}), \tag{42}\]  
and from this derive \(\Delta[\delta]\) as the average of \(\Delta[\delta](p)\) over all points \(p\). The distance \(\delta\) must correspond to the topological property under analysis, and its maximum must be proportional to the permutation length \(n\) (i.e., \(\max\{\delta(p_{1},p_{2})\}\in\Theta(n)\)). We focus on two topological features, absolute element positions and edges, and thus require corresponding distance functions. We use exact match distance [67] (denoted as \(\delta_{em}\)), which is the count of the number of positions containing different elements, as a measure of how different two permutations are within the context of absolute positioning; and we use cyclic edge distance [68] (denoted as \(\delta_{cc}\)) as a measure of how different two permutations are within the context of relative positions. Both meet the requirement that the maximum distance is proportional to \(n\). It is exactly \(n\) in both cases. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mutation Operator** & **TSP** & **QAP** & **LCS** \\ \hline Cycle(\(\alpha\)) & \(-0.0569\) & \(0.0213\) & \(-0.0278\) \\ Cycle(\(5\)) & \(0.1801\) & \(0.1339\) & \(-0.5342\) \\ Cycle(\(4\)) & \(0.1667\) & \(0.1737\) & \(-0.3984\) \\ Cycle(\(3\)) & \(0.2482\) & \(0.2210\) & \(-0.6180\) \\ Swap & \(0.3318\) & \(0.2245\) & \(-0.6355\) \\ Insertion & \(0.5277\) & \(0.0305\) & \(-0.3547\) \\ Reversal & \(0.8459\) & \(0.0189\) & \(-0.0350\) \\ Scramble & \(0.0117\) & \(0.0048\) & \(-0.0340\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: FDC for combinations of mutation operator and problem. Table 4 summarizes \(\Delta[\delta_{cm}]\) and \(\Delta[\delta_{ce}]\) for both forms of cycle mutation, and the other mutation operators considered. The rates of change of the topological properties of swap, insertion, reversal, and scramble were determined elsewhere [32]. The \(\Delta[\delta_{cm}]\) is the average number of elements whose absolute positions are changed by a single application of the operator. For cycle mutation, this is the average length of the induced cycle, determined earlier in the article--Cycle(\(\alpha\)) in Section 3.1.3 and Cycle(\(kmax\)) in Section 3.1.2. To compute \(\Delta[\delta_{ce}]\) we need the average number of edges changed by the operator. For cycle mutation, this depends upon the cycle length, and it also depends upon whether any cycle elements are adjacent. As the permutation length \(n\) increases, the probability of adjacent cycle elements decreases. For sufficiently large \(n\), the probability of adjacent cycle elements approaches 0, in which case the number of edges replaced by cycle mutation is, on average, twice the length of the cycle. Thus, \(\Delta[\delta_{ce}]=2\Delta[\delta_{cm}]\) for both forms of cycle mutation. In the search landscape calculus, when \(\lim_{n\to\infty}\Delta[\delta]=\infty\), the fitness landscape exhibits very large local changes in fitness, often due to many deep local optima that are difficult to escape.