Ulam_Distance — Permutations

Computes the Ulam metric distance of two permutations.

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Interface

#include <codecogs/maths/combinatorics/permutations/ulam_distance.h>

using namespace Maths::Combinatorics::Permutations;

inline int ulam_distance(int n, int *a, int *b)

Consider two permutations of the same order $n$

$$\sigma = \left( \begin{array}{cccc} 1 & 2 & \ldots & n \cr \sigma(1) & \sigma(2) & \ldots & \sigma(n) \end{array} \right)$$

(1)

and

$$\tau = \left( \begin{array}{cccc} 1 & 2 & \ldots & n \cr \tau(1) & \tau(2) & \ldots & \tau(n) \end{array} \right)$$

(2)

If we let $L(\alpha)$ be the length of the longest ascending subsequence of a permutation $\alpha$, then the Ulam metric distance between $\sigma$ and $\tau$ is given by:

$$U = n - L(\overline{\tau} \circ \sigma)$$

(3)

where $\overline{\tau}$ is the inverse of the $\tau$ permutation.

This function calculates the value of $U$ based on the formula given above.

References

SUBSET, a C++ library of combinatorial routines, http://www.csit.fsu.edu/~burkardt/cpp_src/subset/subset.html

Example 1

#include <codecogs/maths/combinatorics/permutations/ulam_distance.h>
#include <iostream>
int main()
{
  int sigma[5] = {1, 2, 3, 4, 5}, tau[5] = {5, 2, 1, 4, 3},
  distance = Maths::Combinatorics::Permutations::ulam_distance(5, sigma, tau);
  std::cout << "The Ulam metric distance of Sigma and Tau is: ";
  std::cout << std::endl << distance << std::endl;
  return 0;
}

Output:

The Ulam metric distance of Sigma and Tau is:
3

Parameters

the size of the permutations

the first permutation

the second permutation

Returns

the value of the Ulam metric distance between a and b