TAOCP 7.1.3 Exercise 209

Section 7.1.3: Bitwise Tricks and Techniques

Exercise 209. ▶ [21] [21] Is the suffix parity operation x⊕of exercise 36 easy to compute with MXOR?

202 COMBINATORIAL SEARCHING 7.1.3

Verified: no
Solve time: 2m22s

The solution must address the actual object in Exercise 36, namely the suffix parity transformation $x^{\oplus}$, and relate it to what MXOR can compute.

Let $x = (x_0, x_1, \dots, x_{n-1})$ be a Boolean sequence, with XOR as the underlying operation. The suffix parity operation $x^{\oplus}$ is defined componentwise by

$$ x^{\oplus}i = x_i \oplus x{i+1} \oplus \cdots \oplus x_{n-1}. $$

This is a suffix scan under the associative operation $\oplus$.

The key observation is that suffix parity is equivalent to prefix parity after reversal. Define the reversed sequence $y$ by

$$ y_i = x_{n-1-i}. $$

Then the prefix parity of $y$,

$$ p_i = y_0 \oplus y_1 \oplus \cdots \oplus y_i, $$

satisfies

$$ p_i = x_{n-1} \oplus x_{n-2} \oplus \cdots \oplus x_{n-1-i}. $$

Reversing the order of $p$ gives exactly the suffix parities of $x$.

Thus suffix parity reduces to two operations:

reversal of the sequence and prefix parity.

MXOR, as used in Exercise 36, supports parallel XOR combination of disjoint blocks and is intended to implement associative scans such as prefix parity efficiently by repeated pairwise combination of segments in a doubling scheme. Since XOR is associative and commutative, MXOR applies directly to this scan problem.

Therefore:

1. MXOR computes prefix parity efficiently via its standard scan construction.
2. Sequence reversal is a trivial linear reindexing (or can itself be implemented by word-level permutation if needed in MMIX-style packed representations).
3. Applying prefix parity after reversal yields suffix parity.

No new mechanism beyond MXOR is required, and no non-associative operation is involved. Hence suffix parity is of the same algorithmic difficulty as prefix parity under MXOR.

Conclusion: Yes, the suffix parity operation $x^{\oplus}$ is easy to compute with MXOR, since it reduces to a reversal plus a standard MXOR prefix-parity computation.