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Multiplying a matrix M by either or on either the left or the right will permute either the rows or columns of M by either π or π −1.The details are a bit tricky. To begin with, when we permute the entries of a vector (, …,) by some permutation π, we move the entry of the input vector into the () slot of the output vector.
Then the n queens problem is equivalent to choosing a subset of the rows of this matrix such that every primary column has a 1 in precisely one of the chosen rows and every secondary column has a 1 in at most one of the chosen rows; this is an example of a generalized exact cover problem, of which sudoku is another example. n-queens completion
Any systematic rule for choosing column c in this procedure will find all solutions, but some rules work much better than others. To reduce the number of iterations, Knuth suggests that the column-choosing algorithm select a column with the smallest number of 1s in it.
Another way of finding superpermutations lies in creating a graph where each permutation is a vertex and every permutation is connected by an edge. Each edge has a weight associated with it; the weight is calculated by seeing how many characters can be added to the end of one permutation (dropping the same number of characters from the start ...
In theory P (or Q) are obtained by permutations of rows (or columns) of the identity matrix, in practice the corresponding permutations are applied directly to rows (or columns) of A. Matrix A of side n {\displaystyle n} has n 2 {\displaystyle n^{2}} coefficients while two triangle matrices combined contain n ( n + 1 ) {\displaystyle n(n+1 ...
Because permutations of an array can be made by altering some array A through the removal of an element x from A then tacking on x to each permutation of the altered array, it follows that Heap's Algorithm permutes an array of size +, for the "buffer" in essence holds the removed element, being tacked onto the permutations of the subarray of ...
A matrix representation of a permutation, a square matrix with exactly one 1 in each row and column, and all other elements 0. Persymmetric matrix A matrix that is symmetric about its northeast–southwest diagonal, i.e., a ij = a n − j +1, n − i +1 .
Both the width of the rows and the permutation of the columns are usually defined by a keyword. For example, the keyword ZEBRAS is of length 6 (so the rows are of length 6), and the permutation is defined by the alphabetical order of the letters in the keyword. In this case, the order would be "6 3 2 4 1 5".