## A Review of Metrics on Permutations for Search Landscape Analysis

Venue: | COMPUTERS AND OPERATION RESEARCH |

Citations: | 4 - 0 self |

### BibTeX

@ARTICLE{Schiavinotto_areview,

author = {Tommaso Schiavinotto and Thomas Stützle},

title = { A Review of Metrics on Permutations for Search Landscape Analysis},

journal = {COMPUTERS AND OPERATION RESEARCH},

year = {},

volume = {34},

pages = {3143--3153}

}

### OpenURL

### Abstract

Search landscape analysis has become a central tool for analysing the dependency of the performance of stochastic local search algorithms on structural aspects of the spaces being searched. Central to search landscape analysis is the notion of distance between candidate solutions. This distance depends on some underlying basic operator and it is defined as the minimum number of operations that need to be applied to one candidate solution for transforming it into another one. For operations on candidate solutions that are represented by permutations, in almost all researches on search landscape analysis surrogate distance measures are applied, although efficient algorithms exist in many cases for computing the exact distances. This discrepancy is probably due to the fact that these efficient algorithms are not very widely known. In this article, we review algorithms for computing distances on permutations for the most widely applied operators and present simulation results that compare the exact distances to commonly used approximations.

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Citation Context ...cribed in [29]; this algorithm has a higher complexity of O(n log(n)) when compared to the best version of the previous one, but it allows for an easier implementation. This algorithm is also used in =-=[32]-=-. The diameter of GNI is n − 1. 3.5 Other operators There are several other operators that are widely applied to permutations. An example is the well-known 2-edge-exchange operator used in the Travell... |

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Citation Context ... sort and dNS(π ′ , π) = dNS (π −1 · π ′ , ı) = s(π −1 · π ′ ). Interestingly, the sortedness and the number of swaps to be done in bubble sort correspond also to the precedence based distance metric =-=[30, 40]-=-, which is defined as � � � � # 〈i, j〉 �1 ≤ i < j ≤ n, posπ ′(πj) < posπ ′(πi) . The diameter of the search space for the swap operator is n(n − 1)/2. 4s3.3 Interchange Operator The interchange operat... |

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Citation Context ...= i , δ ij � (π1 . . . πi−1 πi+1 . . . πj πi πj+1 . . . πn) i < j I (π) = (π1 . . . πj πi πj+1 . . . πi−1 πi+1 . . . πn) i > j With this operator, the distance dNI (s, s ′ ) is known as Ulam’s metric =-=[6, 13]-=-: dNI = n − |lis(π −1 · π ′ )|, where lis(π) is the longest increasing subsequence (LIS) of the permutation π and |·| denotes the length of the given sequence. An increasing subsequence is formed by t... |

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Citation Context ...moved is bounded [16] and for sorting by short block-moves, where the block moved is of maximum length three [17]. The latter is of particular interest because it corresponds to the Or-exchange moves =-=[28]-=-; the authors of [17] conjecture that a polynomial-time algorithm exists for solving this problem; however, presently such an algorithm is not known. 4 Metrics used as Approximation Fitness-distance c... |

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Citation Context ...umerated. All the permutations of all LISs of π −1 · π ′ correspond to all possible paths on GNI to reach π ′ from π and vice-versa. Another algorithm for finding an LIS in a sequence is described in =-=[29]-=-; this algorithm has a higher complexity of O(n log(n)) when compared to the best version of the previous one, but it allows for an easier implementation. This algorithm is also used in [32]. The diam... |

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Citation Context ...= i , δ ij � (π1 . . . πi−1 πi+1 . . . πj πi πj+1 . . . πn) i < j I (π) = (π1 . . . πj πi πj+1 . . . πi−1 πi+1 . . . πn) i > j With this operator, the distance dNI (s, s ′ ) is known as Ulam’s metric =-=[6, 13]-=-: dNI = n − |lis(π −1 · π ′ )|, where lis(π) is the longest increasing subsequence (LIS) of the permutation π and |·| denotes the length of the given sequence. An increasing subsequence is formed by t... |