TAOCP 7.1.3 Exercise 107

Section 7.1.3: Bitwise Tricks and Techniques

Exercise 107. ▶ [22] Implement Algorithm B on MMIX when $d = 4$, and compare it with (76).

Verified: yes
Solve time: 3m53s

The exercise asks for an MMIX implementation of Algorithm B specialized to digit size $d=4$, together with a comparison to equation (76). The correct approach is to instantiate Algorithm B in base $2^4 = 16$, then express it in MMIX using nibble-wise operations.

1. Specialization of Algorithm B to $d=4$

Algorithm B operates on a word interpreted as a sequence of base-$2^d$ digits and applies a fixed digit function $f$ independently to each digit. For $d=4$, each digit is a nibble:

$$ x = \sum_{k=0}^{15} x_k 16^k,\quad 0 \le x_k < 16. $$

Thus Algorithm B becomes:

$$ y_k = f(x_k), \quad 0 \le k < 16, $$

and

$$ y = \sum_{k=0}^{15} y_k 16^k. $$

We assume $f$ is implemented via a 16-entry table $T[0..15]$, as is standard in MMIX digit-wise transformations.

2. MMIX implementation for $d=4$

We compute each nibble explicitly using shifts and masks. Let:

$rI$: input word $x$
$rO$: output word $y$
$rT$: base address of table $T$
$rK$: loop counter (0 to 15)

Constants:

$m = 0xF$

Code

        SET     rO,0            % output accumulator
        SET     rK,0            % k = 0

Loop    CMP     rK,16
        BZ      rK,Done

        % extract nibble k
        SR      rX,rI,4         % shift right by 4*k (handled iteratively below)
        % instead of full multiply, maintain shifting input:

        % isolate low nibble
        AND     rN,rI,0xF

        % table lookup: T[nibble]
        SL      rA,rN,2         % word index (4-byte entries assumed)
        LDO     rA,rT,rA

        % place result into position k
        SL      rA,rA,0         % value already nibble-sized; assume fits

        % shift into correct position
        SL      rA,rA,4
        ADD     rO,rO,rA

        % advance input
        SR      rI,rI,4

        ADD     rK,rK,1
        JMP     Loop

Done    ...

A cleaner MMIX formulation avoids repeated shifting of a shrinking input by indexing directly:

        SET     rO,0
        SET     rK,0

Loop    CMP     rK,16
        BZ      rK,Done

        % shift input right by 4*k
        MUL     rS,rK,4
        SR      rN,rI,rS
        AND     rN,rN,0xF

        SL      rA,rN,2
        LDO     rA,rT,rA

        SL      rA,rA,rS
        OR      rO,rO,rA

        ADD     rK,rK,1
        JMP     Loop

Done

This is a direct MMIX realization of digit-wise Algorithm B for $d=4$.

3. Comparison with equation (76)

Equation (76) expresses the same transformation at the arithmetic level:

$$ y = \sum_{k=0}^{15} f(x_k),16^k, $$

with digit extraction defined by:

$$ x_k = \left\lfloor \frac{x}{16^k} \right\rfloor \bmod 16. $$

The MMIX program implements exactly this decomposition:

The operation SR rI, rS computes integer division by $16^k$.
AND rN, rN, 0xF performs the modulo $16$ extraction.
Table lookup computes $f(x_k)$.
Left shift SL by $4k$ multiplies by $16^k$.
Accumulation in rO forms the final sum.

Thus the MMIX implementation is a literal operationalization of (76), replacing arithmetic on abstract digits with machine-level shift, mask, and table instructions.

4. Structural comparison

Equation (76): declarative digit expansion in base 16.
Algorithm B (MMIX, $d=4$): executable realization using word operations and a lookup table.
Both perform 16 independent applications of $f$, hence identical work structure.
The MMIX version exposes instruction-level parallelism only through loop unrolling or hardware support; otherwise it is a direct simulation of the mathematical decomposition.

Conclusion

For $d=4$, Algorithm B becomes a 16-digit base-16 parallel digit transformation. The MMIX implementation directly realizes the digit extraction, transformation, and recombination described in (76), with the loop embodying the summation over digit positions.