TAOCP 7.2.1.2 Exercise 17
Introduce an additional array $a'_{1}\ldots a'_{n}$ alongside Algorithm P, where at all times $a'_{k}=j$ if and only if $a_{j}=k$.
Section 7.2.1.2: Generating All Permutations
Exercise 17. ▶ [20] [20] Extend Algorithm P so that the inverse permutation a′ 1 . . . a′ n is available for processing when a1 . . . an is visited in step P2. (The inverse satisfies a′ k = j if and only if aj = k.)
Verified: no
Solve time: 2m42s
Solution
Introduce an additional array $a'{1}\ldots a'{n}$ alongside Algorithm P, where at all times $a'{k}=j$ if and only if $a{j}=k$. This is enforced as an invariant throughout the execution.
The initial permutation $a_{1}\ldots a_{n}$ is assumed to be the input sequence of distinct elements $x_{1},\ldots,x_{n}$ in step P1. At initialization, set $a_{j}\leftarrow x_{j}$ for $1\le j\le n$ exactly as in Algorithm P. Simultaneously set $a'{x{j}}\leftarrow j$ for $1\le j\le n$, so the defining relation $a'{k}=j \Leftrightarrow a{j}=k$ holds initially.
No modification is required in steps P2, P3, P4, P6, or P7. The only change occurs in step P5, where Algorithm P performs an interchange $a_{j-c_{j}+s}\leftrightarrow a_{j-q+s}$ of two positions in the array $a$.
Let
$u=j-c_{j}+s,\qquad v=j-q+s.$
Step P5 exchanges the values stored at positions $u$ and $v$. Denote
$x=a_{u},\qquad y=a_{v}$
immediately before the interchange. After executing the interchange, the updated array satisfies $a_{u}=y$ and $a_{v}=x$.
To preserve the invariant for the inverse permutation, the locations of the values $x$ and $y$ must be updated consistently. Before the interchange, $a'{x}=u$ and $a'{y}=v$. After the interchange, these relationships become $a'{x}=v$ and $a'{y}=u$, while all other entries of $a'$ remain unchanged.
Thus step P5 is extended by replacing the single interchange with the following coupled update:
$a_{u}\leftrightarrow a_{v},\qquad a'{a{u}}\leftrightarrow a'{a{v}}.$
Equivalently, using the pre-swap values $x$ and $y$, perform
$a'{x}\leftarrow v,\qquad a'{y}\leftarrow u.$
After this update, every value $k$ satisfies $a'{k}=j$ if and only if $a{j}=k$, since the only affected values are $x$ and $y$, whose positions have been exchanged consistently in both arrays. All other values preserve their locations.
Since Algorithm P changes $a$ only in step P5 and each such change is mirrored by the corresponding update in $a'$, the invariant is preserved at every visit in step P2. Therefore $a'{1}\ldots a'{n}$ is available at each visited permutation as the inverse of $a_{1}\ldots a_{n}$.
This completes the proof. ∎