The concept of a permutation has numerous applications in computer science, for example, in the analysis of sorting algorithms and in implementations of distributed systems. Formally, a permutation is a bijection from a set onto itself. For simplicity, let us restrict ourselves to the sets [n] := {1, 2, . . . , n} of the first n positive integers, and denote by Sn the set of all permutations on [n]. It is often convenient to represent a permutation σ by the n-tuple (σ(1), σ(2), . . . , σ(n)). Swapping two elements of a permutation yields another permutation. For i, j ∈ [n] let Tij : Sn → Sn be the mapping that swaps the images of i and j, that is, if σ ∈ Sn, then τ := Tij (σ) is given by τ (i) = σ(j), τ (j) = σ(i), and τ (k) = σ(k) if i 6= k 6= j. Example. Let σ = (2, 4, 3, 1). Then T13(σ) = (3, 4, 2, 1). Now, consider allowing only swaps with some pairs (i, j). For any integer d ≥ 0, let
Pd := {(i, j) ∈ [n] × [n] : i = j or d ≤ j − i ≤ n − d}
Say that a permutation τ is (d, )-reachable from σ if there are (i1, j1),(i2, j2), . . . ,(i, j) ∈ Pd such that the corresponding swaps transform σ to τ , that is, τ = Tij` ◦ · · · ◦ Ti2j2 ◦ Ti1j1(σ).
PROBLEM :
Prove that for any positive integer n, the permutation (n, n − 1, . . . , 1) is (1, bn/2c) - reachable from (1, 2, . . . , n).
To find out whether τ is (d,
)-reachable from σ, one may apply bidirected search as follows: First generate all permutations that are (d, b/2c)-reachable from σ. Next generate all permutations that are (d, d`/2e)-reachable from τ . Finally, report “YES” if the two generated sets intersect, and otherwise report “NO”. Describe the algorithm using pseudocode; in particular, describe how the sets of permutations are generated. (You may assume the given arguments are valid: you need not implement error handling.)Implement the algorithm of the previous task using Java Programming Language. Show your code. Use your implementation to solve, for all d = 1, 2, . . . , 4 and
= 1, 2, . . . , 9, whether (9, 8, . . . , 1) is (d,) - reachable from (1, 2, . . . , 9). Show the results as a 4 × 9 matrix.
