Class DeviationDistanceNormalized2005

  • All Implemented Interfaces:
    NormalizedPermutationDistanceMeasurerDouble, PermutationDistanceMeasurerDouble

    public final class DeviationDistanceNormalized2005
    extends Object
    implements PermutationDistanceMeasurerDouble, NormalizedPermutationDistanceMeasurerDouble
    Normalized Deviation Distance:

    The original version of Normalized Deviation distance (Ronald, 1998) is the sum of the positional deviation of the permutation elements divided by N-1 (where N is the length of the permutation). The positional deviation of an element is the difference in its location in the two permutations. Normalizing by dividing by N-1 causes each element's contribution to distance to be in the interval [0,1].

    Sevaux and Sorensen (2005) suggested a different normalizing factor that provides a distance in the interval [0,1]. Maximal distance occurs for an inverted permutation. The normalizing factor is (N2/2) when N is even and (N2-1)/2 when N is odd.

    For example, consider p1 = [0, 1, 2, 3, 4, 5] and p2 = [1, 0, 5, 2, 4, 3]. Element 0 is displaced by 1 position. Likewise for elements 1 and 2. Element 3 is displaced by 2 positions. Element 4 is in the same position in both. Element 5 is displaced by 3 positions.

    Sum the deviations: 1 + 1 + 1 + 2 + 0 + 3 = 8.

    The length is 6, which is even, so we'll divide by 18. So, normalized deviation distance is 8 / 18 = 0.444...

    If instead, p2 = [5, 4, 3, 2, 1, 0], then 0 and 5 are both displaced by 5 positions, 1 and 4 are displaced by 3 positions, and 2 and 3 are displaced by 1 position. Sum of deviations is then: 2 * 5 + 2 * 3 + 2 * 1 = 18. The length is still 6, so we again divide by 18, and distance is 1.

    Runtime: O(n), where n is the permutation length.

    Original normalized deviation distance was introduced in:
    S. Ronald, "More distance functions for order-based encodings," in Proc. IEEE CEC. IEEE Press, 1998, pp. 558–563.

    This version of normalized deviation distance was introduced in:
    M. Sevaux and K Sorensen, "Permutation distance measures for memetic algorithms with population management," in Proc. of MIC2005, 2005.