Generate all permutations of {1, 2, 3, 4} by a. the bottom-up minimal-change algorithm...

b.

Johnson-Trotter algorithm:

Johnson-Trotter algorithm generates n! without explicitly generating permutations for smaller values of n.

Algorithm process:

In this process, a direction is associative with each component k in a permutation and direction is represented by an arrow above the component.

The component k is said to be mobile, if the arrow points to a smaller value adjacent to it.

Given set is {1, 2, 3, 4}: size of the set as n = 4 and largest mobile component is shown in bold.

Largest element in the set k = 4, and swap k with adjacent component.

Reverse the direction component k and largest element in the remaining set k = 3, and swap k with adjacent component.

Largest element in the set k = 4, and swap k with adjacent component.

Reverse the direction component k and largest element in the remaining set k = 3, and swap k with adjacent component.

Largest element in the set k = 4, and swap k with adjacent component.

Largest element in the set k = 4, and swap k with adjacent component.

Reverse the direction component k and largest element in the remaining set k = 3, and swap k with adjacent component.

Largest element in the set k = 4, and swap k with adjacent component.

Reverse the direction component k and largest element in the remaining set k = 3, and swap k with adjacent component.

Largest element in the set k = 4, and swap k with adjacent component.

Therefore, the final list would be listed as a dictionary if the numbers were interpreted as letters of an alphabet, for n = 4,

c.

Lexicographic-order algorithm:

• In lexicographic ordering, a permutation A = a1a2 a3……an is less than (or precedes) a permutation B = b₁ b₂b₃…….b_n, if

• For example, take two sets, the first two positions and. But they differ in the third position. Since, the then permutation 2134 precedes the permutation 2143 in lexicographic ordering.

• For instance, least significant item in the list at the head and the next greater element than (1 2 3 4) is (2 1 3 4). Since, the most significant tail of the list doesn’t change, and the next greater permutation differs in the least significant elements.

• In the given set, there are 4! = 24 permutations of the set {1, 2, 3, 4}. To make a greater permutation, some element of the list must be replaced by a larger element to its left; to make the next permutation, the replacement must happen as far to the left as possible, in the least significant position, and the replacing element must be as small as possible.

The permutations of the list (1 2 3 4) in lexicographic order are as follows: