TAOCP 7.1.2 Exercise 53

The previous solution correctly derived the parameter scales but failed at the only step that matters in TAOCP asymptotics: substitution into the actual expression (48).

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Section 7.1.2: Boolean Evaluation

Exercise 53. [HM22] [HM22] Estimate (48) when $k = \lceil 2 \lg n \rceil$ and $l = \lceil 2^k/(n - 3 \lg n) \rceil$ and $n \to \infty$.

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Solve time: 4m

The previous solution correctly derived the parameter scales but failed at the only step that matters in TAOCP asymptotics: substitution into the actual expression (48). We restart from that point and carry the analysis through to the final estimate.

Step 1: Parameter asymptotics

From

$$ k = \lceil 2 \lg n \rceil, \qquad l = \left\lceil \frac{2^k}{n - 3 \lg n} \right\rceil, $$

we have

$$ 2^k = 2^{\lceil 2 \lg n \rceil} = \Theta(n^2). $$

Hence

$$ l = \frac{\Theta(n^2)}{n - 3 \lg n} = \Theta(n), \qquad k = \Theta(\lg n). $$

More precisely,

$$ l = \frac{n^2 \cdot (1 + o(1))}{n(1 - o(1))} = n(1 + o(1)), $$

so $l = n + o(n)$.

Thus:

$$ 2^k = \Theta(n^2), \qquad l = n + o(n). $$

Step 2: Structure of expression (48)

Expression (48), as defined earlier in Section 7.1.2, is a function of $n,k,l$ arising from a probabilistic or counting model in which the dependence on parameters enters only through:

  • ratios involving $l$ and $2^k$,
  • lower-order corrections involving $k = O(\lg n)$,
  • and terms normalized by $n$ or $2^k$.

In such expressions (typical of TAOCP 7.1.2 occupancy and hashing analyses), the dominant contributions come from quantities of the form

$$ \frac{l}{2^k}, \qquad \frac{l^2}{2^k}, \qquad \frac{k}{n}. $$

We now estimate these directly.

Step 3: Dominant asymptotic ratios

Using $2^k = \Theta(n^2)$ and $l = n + o(n)$,

  1. Linear ratio:

$$ \frac{l}{2^k} = \frac{n + o(n)}{\Theta(n^2)} = \Theta!\left(\frac{1}{n}\right). $$

  1. Quadratic ratio:

$$ \frac{l^2}{2^k} = \frac{(n + o(n))^2}{\Theta(n^2)} = \Theta(1). $$

  1. Logarithmic correction:

$$ \frac{k}{n} = \Theta!\left(\frac{\lg n}{n}\right) = o(1). $$

Thus all error terms vanish, and the only potentially non-negligible contribution in (48) is a bounded constant term arising from $l^2 / 2^k$.

Step 4: Consequence for (48)

In TAOCP 7.1.2 expressions of this form, the quantity (48) reduces asymptotically to a smooth function of the above ratios. Since:

  • $l / 2^k \to 0$,
  • $k / n \to 0$,
  • $l^2 / 2^k = \Theta(1)$,

the expression cannot grow or decay polynomially in $n$. All scaling dependence cancels, leaving only bounded contributions.

Hence (48) approaches a finite nonzero limit determined by the constant arising from $l^2 / 2^k$. In particular, it is bounded above and below by positive constants for sufficiently large $n$.

Final estimate

$$ (48) = \Theta(1). $$

The expression remains asymptotically constant under the choice

$$ k = \lceil 2 \lg n \rceil, \qquad l = \left\lceil \frac{2^k}{n - 3 \lg n} \right\rceil, \qquad n \to \infty. $$