![]() ![]() If I sort this permutation and reconstruct:, 3->3 4 and 4->5 6 I get, 1 2 3 4 5 6. Step 3: do above translation on array from step one. ![]() , so we know that, in the previous array, 1 maps to 1, 2 to 2, 3 to 3, 4 to 5. Step 1: replace runs with the head element of the run and inserting data into a hash table replacing the list in step1 with the map created in step2 and updating the hashtable for translation.sorting to create a map to the missing digits with it's sorted index.replacing the runs with the head element of the run and inserting that data into a hashtable (for recoverability). ![]() I have thought of the naive approach, curious if anyone has some clever trick that can do it better then mine, I'm at like O( 2n + n log n), Really the second part is the hard part to make a clean solution. (3 4) (1 2) (5 6) => 2 1 3 // Mappings: 2->3 4, and 1->1 2 and 3->5 6įor now, I am passing over the list and making a list of lists, grouping the runs. The resultant permutation shouldn't have any runs in it.įor example (i've highlighted the runs): (1 2 3 4 5 6) => 1 //Mappings: 1->1 2 3 4 5 6 If I sort the reduced permutation then expand the elements inside from the mapping, and sort the original permutation I will get the same result. the first run would be 1, 4 transforms to 2, and transforms to 3, to hold the second criteria. In the above, we have to reduce the two runs. I want to replace these with a single number, a minimum, so if, after removing runs, we have a permutation of 4 elements, the permutation uses the numbers 1. In the list, 1 2 3 5 6 4 there are two runs, 1 2 3 and 5 6. So a run is a sequential set of numbers in a permutation that is sorted and in-order. Since we confine our attention to permutations of \).I need an algorithm that can map the runs in a permutation to a single number, but also reduce the subsequent numbers. We remark that a Golomb ruler of order n is defined to be a sequence of n distinct positive integers such that all of the entries in its difference triangle are distinct. In terms of its difference triangle, the integers in each row are distinct. If we think of a permutation matrix as a configuration of points in the Euclidean plane at the integral positions ( i, π i) for i = 1,…, n, then for a Costas permutation, no two of the line segments determined by these points have both the same length and the same slope, and thus all the line segments they determine are distinct. Since π is a permutation, its zeroth order differences are distinct since π has only one ( n − 1)st difference, no restriction is placed on ( n − 1)st order differences. A Costas permutation (or Costas array or Costas permutation matrix) is a permutation π of order n such that for each k = 0,1,2,…, n − 1, its k th order differences in row k of its difference triangle are distinct. The difference triangle is used and analyzed extensively in the literature on Costas arrays. Permutation matrices and more general classes of (0,1)-matrices are treated in. We refer to the book on permutations and their descents. The notion of the derivative of a permutation captures the changes in consecutive entries of a permutation π, and therefore contains information about e.g. ![]()
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