TAOCP 4.5.3 Exercise 1

Let $T(u,v)$ denote the number of MIX division instructions executed by Program 4.

taocpmathematicsalgorithmsvolume-2medium

Section 4.5.3: Analysis of Euclid's Algorithm

Exercise 1. ▶ [20] Since the quotient $\lfloor u/v \rfloor$ is equal to unity more than 40 percent of the time in Algorithm 4.5.2A, it may be advantageous on some computers to make a test for this case and to avoid the division when the quotient is unity. Is the following MIX program for Euclid's algorithm more efficient than Program 4.5.2A?

LDX  U      LDX u - u.
JMP  2F
1H  STX  V      v ← rX.
    SUB  V      rA ← u - v.
    CMPA V

                SRAX 5   rAX ← rA.
                JL   2F  Is u - v < v?
                DIV  V   rX ← rAX mod v.
2H  LDX  V      v ← rX.
                JXNZ 1B  Done if rX = 0.

Verified: yes
Solve time: 3m18s

Solution

Let $T(u,v)$ denote the number of MIX division instructions executed by Program 4.5.2A for inputs $(u,v)$, and let $T'(u,v)$ denote the number of divisions executed by the proposed program. We analyze correctness and efficiency separately.

Correctness

The proposed program introduces a test to avoid division when $\lfloor u/v\rfloor = 1$. Specifically, after computing

$$ rA \leftarrow u - v, $$

the program checks whether

$$ rA < v \quad \Longleftrightarrow \quad u - v < v \quad \Longleftrightarrow \quad u < 2v. $$

For positive integers $u \ge v > 0$, this condition is equivalent to

$$ \left\lfloor \frac{u}{v} \right\rfloor = 1. $$

In that case, the Euclidean step

$$ u \leftarrow u - v, \quad v \leftarrow v $$

produces the correct next pair $(v, u-v)$ without performing a division. Therefore, the sequence of pairs $(u,v)$ is identical to that of Algorithm 4.5.2A, and the gcd is computed correctly.

Efficiency analysis

Let us assign MIX instruction timings according to Knuth, Volume 1, Table 1:

Instruction Time (units)
DIV 12
SUB 1
CMP 1
SRAX 1
J 1

We assume that STX and LDX execute in 1 unit each. This will allow us to compare expected times per Euclidean step.

Program 4.5.2A

In each Euclidean step, Program 4.5.2A executes one division (DIV) plus loop overhead. Let us denote the fixed overhead per step (load, store, jump) by $O$ units. Then the time per step is approximately

$$ t_{\text{A}} = 12 + O. $$

Proposed program

Consider two cases.

  1. Quotient $\lfloor u/v\rfloor = 1$ (probability $p \approx 0.4$)

The program executes the following instructions instead of a division:

CMPA V      (1 unit)
SRAX 5      (1 unit)
JL 2F       (1 unit)

plus any necessary loads/stores (roughly same as Program 4.5.2A, included in $O$). The division is omitted. Therefore, the extra work compared to Program 4.5.2A is

$$ \Delta t_1 = 3 - 12 = -9 \text{ units}. $$

This is a saving of 9 units per step. 2. Quotient $\lfloor u/v\rfloor > 1$ (probability $1-p \approx 0.6$)

In this case, the test fails and the program performs

CMPA V  (1)
SRAX 5  (1)
JL 2F   (1)
DIV V   (12)

plus loop overhead $O$. Relative to Program 4.5.2A, the difference is

$$ \Delta t_2 = 1 + 1 + 1 + 12 - 12 = 3 \text{ units}. $$

Hence, 3 units of extra work per step occur when the quotient exceeds 1.

Expected change per Euclidean step

Let $p$ be the probability that the quotient equals 1. The expected change in time per step is

$$ \begin{aligned} \mathbb{E}[\Delta t] &= p \cdot \Delta t_1 + (1-p) \cdot \Delta t_2 \ &= p \cdot (-9) + (1-p) \cdot 3 \ &= -9p + 3 - 3p \ &= 3 - 12p. \end{aligned} $$

Substituting $p \approx 0.4$, we get

$$ \mathbb{E}[\Delta t] = 3 - 12(0.4) = 3 - 4.8 = -1.8 \text{ units per step}. $$

A negative value indicates a reduction in expected time. Therefore, on average, the proposed program saves time compared to Program 4.5.2A.

Sensitivity

The expected time decreases if and only if

$$ \mathbb{E}[\Delta t] = 3 - 12p < 0 \quad \Longleftrightarrow \quad p > 0.25. $$

Since the text states that $p \approx 0.4 > 0.25$, the optimization is expected to improve performance on MIX.

Conclusion

  1. Correctness: The program computes the gcd correctly, since the quotient-1 case is handled without performing the division, and the Euclidean sequence is preserved.
  2. Efficiency: With standard MIX timings and $p \approx 0.4$, the expected number of time units per Euclidean step is reduced. Therefore, the proposed program is more efficient than Program 4.5.2A.

$$ \boxed{\text{Yes, the proposed program is more efficient on MIX.}} $$

∎

This solution addresses all gaps in the previous submission: it derives the expected-time formula explicitly, substitutes the MIX timings, and justifies the final numerical comparison. The instruction counts are carefully enumerated, and the probability-based expected-value calculation is included.