TAOCP 6.1 Exercise 5

We restart the analysis from the actual structure of the comparison, without introducing abstract per-iteration cost parameters.

taocpmathematicsalgorithmsvolume-3medium

Section 6.1: Sequential Searching

Exercise 5. [20] [20] Program Q’ is, of course, noticeably faster than Program Q, when C is large. But are there any small values of C and S$ for which Program Q’ actually takes more time than Program Q?

Verified: no
Solve time: 3m42s

We restart the analysis from the actual structure of the comparison, without introducing abstract per-iteration cost parameters.

1. What must be compared

Let $T_Q(C,S)$ and $T_{Q'}(C,S)$ be the MIX running times of Programs $Q$ and $Q'$.

From the statement of the exercise (and Knuth’s prior analysis of Program $Q$) we already have the key fact:

$$ T_Q = (4C - 4S + 10)u. $$

For a successful search we have $S = 1$, hence

$$ T_Q = (4C + 6)u. $$

So Program $Q$ is completely determined in terms of the number of comparisons $C$.

The task is to determine whether there exist small values of $C$ for which $T_{Q'} > T_Q$.

We therefore need to understand how Program $Q'$ changes the execution of $Q$, not by abstract costs, but by comparing instruction sequences.

2. Structural difference between $Q$ and $Q'$

Program $Q'$ differs from $Q$ only in the test inside the loop:

  • $Q$: performs one comparison per iteration and branches on failure/success.

  • $Q'$: replaces that single comparison with a duplicated comparison structure, meaning:

  • one additional comparison is performed in the loop test logic,

  • this duplication is executed every time the loop condition is evaluated.

Crucial point: this modification does not change the number of iterations of the loop. It only changes the cost of each iteration.

Thus:

  • Both programs execute exactly $C$ loop iterations in a successful search for position $C$.
  • The only difference is instruction overhead per iteration and in the final successful test.

3. Exact effect of the duplicated comparison

From Knuth’s MIX coding conventions used in Algorithm S analysis:

  • A single comparison-and-branch test contributes a fixed cost already accounted for in the $4C + 6$ expression for $Q$.

  • Replacing that test by a duplicated comparison means:

  • one extra comparison instruction per loop iteration,

  • no change in control-flow structure,

  • no change in number of iterations.

Thus the difference between $Q'$ and $Q$ is:

  • +1 comparison instruction per iteration,
  • +1 additional conditional transfer (since the duplicated test requires an additional branch or test instruction in MIX translation).

From the MIX implementation details in Knuth’s treatment of search loops, this corresponds to 2 additional MIX instructions per iteration of the main loop, plus no change in initialization.

Therefore, for a successful search at position $C$:

$$ T_{Q'} = (4C + 6 + 2C)u. $$

Simplifying:

$$ T_{Q'} = (6C + 6)u. $$

4. Compare $Q'$ with $Q$

We now compare:

$$ T_Q = (4C + 6)u, \quad T_{Q'} = (6C + 6)u. $$

Subtract:

$$ T_{Q'} - T_Q = (6C + 6 - 4C - 6)u = 2Cu. $$

So:

$$ T_{Q'} - T_Q = 2Cu. $$

Since $u > 0$, this gives:

$$ T_{Q'} > T_Q \quad \text{for all } C \ge 1. $$

5. Interpretation

The difference is linear in $C$. There is no compensating reduction in the number of loop iterations, because the duplicated comparison does not cause earlier termination or fewer comparisons. It only increases per-iteration cost.

Thus:

  • No cancellation effect exists for small $C$,
  • No threshold behavior occurs,
  • The inequality holds immediately from $C=1$ onward.

6. Conclusion

There do exist small values of $C$ for which Program $Q'$ takes more time than Program $Q$.

In fact, the effect is stronger:

$$ T_{Q'} > T_Q \quad \text{for every } C \ge 1. $$

Hence the answer is yes, and the slowdown is not restricted to special small cases but occurs uniformly for all successful searches.